varundus 2026-09-11 08:00

Zotero/001_artiklid/Road Safety Indicators/gundogduapplyinglinearanalysismethods2010.md

@@ -0,0 +1,254 @@
+---
+category: literaturenote
+citekey: gundogduapplyinglinearanalysismethods2010
+title: "Applying linear analysis methods to GIS-supported procedures for preventing traffic accidents: Case study of Konya"
+authors: "Gundogdu, Ismail Bulent"
+year: 2010
+date: 2010-00-00 2010
+doi: 10.1016/j.ssci.2010.02.016
+publication: Safety Science
+url: "https://doi.org/10.1016/j.ssci.2010.02.016"
+zotero_key: NN23P7ZA
+zotero_storage: 5D9PX62X
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Gundogdu, Ismail Bulent"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+## Contents lists available at [ScienceDirect](http://www.sciencedirect.com/science/journal/09257535)
+
+# Safety Science
+
+journal homepage: [www.elsevier.com/locate/ssci](http://www.elsevier.com/locate/ssci)
+
+![](_page_0_Picture_5.jpeg)
+
+# Applying linear analysis methods to GIS-supported procedures for preventing traffic accidents: Case study of Konya
+
+Ismail Bulent Gundogdu \*
+
+Department of Geomatic Engineering, Selcuk University, Konya, Turkey
+
+#### article info
+
+Article history: Received 26 August 2009 Received in revised form 10 February 2010 Accepted 20 February 2010
+
+Keywords: Linear analysis GIS Traffic accident Highest Probable Hot Spot Hot Pieces
+
+### abstract
+
+This study develops methods to obtain maps to determine traffic Hot Spots in Konya, Turkey, by applying linear analysis supported by Geographical Information Systems (GIS).
+
+Hot Spot analysis is known method but the study differs from former researches at the point of determining of risky zones, classification and illustration of them on the maps with the different accident parameters. The aim is not to contribute another Hot Spot analysis using a number of statistical methods, but to determine the Highest Potential Hot Spots (HPHS), which are inter-sectional clusters, and to use different parameters, such as number of accidents, number of fatalities and injured, and number of accidents with only financial loss.
+
+In addition, apart from classical illustrational techniques, Hot Pieces (HPCS) on roads divided into 1 km segments are shown by their grading according to their numerical values. Hence, thematic illustration distinguishes them from others.
+
+Another aspect of this study is that, besides investigation of Hot Spots by means of data of accidents of previous years', Probable Hot Spots (PRHS) were illustrated and highly potential Hot Spots were determined. These latter are candidates for Hot Spots in the near future. Thus, premature accidents can be anticipated easily.
+
+The main intention of this study is to emphasize the importance of using criteria, other than total accident number, to illustrate intersection Hot Spots and to constitute a model of accident severity and variety. It is anticipated that the results obtained from highway accidents data will guide improvement of the route segments.
+
+-2010 Elsevier Ltd. All rights reserved.
+
+### 1. Introduction
+
+Traffic accidents have been recognized as one of the major causes of human and economic losses, both in developed and developing countries. This problem is of great concern in developing countries, because of its seriousness and the limited resources available to develop feasible countermeasures for reducing this ever-growing challenge [\(Berhanu, 2004](#page-6-0)).
+
+A report from the [World Health Organization and the World](#page-6-0) [Bank \(2004\)](#page-6-0) on road traffic accidents and injuries estimated that 1.2 million people are killed in road crashes each year and as many as 50 million are injured worldwide [\(Rathinam et al., 2007; Rutter](#page-6-0) [and Quine, 1996; Hilakivi et al., 1989\)](#page-6-0). According to estimations of the World Bank, traffic accidents will be the third most frequent cause of deaths in 2020.
+
+Statistics from the Turkish Statistical Institute (TUIK) reveal that, in 2006, there were 665, 618 traffic accidents in Turkey, 2629 of which resulted in mortality, 77,644 resulted in injuries and 585,345 in economical damage. The damage was estimated as \$870 million.
+
+According to a questionnaire from [Arslan \(2007\)](#page-6-0) among 800 people whose relatives lost their lives in traffic accidents in European countries revealed that 37% of those had a tendency to commit suicide and 64% of those had depression in the subsequent 3 years. Another questionnaire applied in Turkey revealed that 50% of 240 people whose relatives lost their lives in traffic accidents suffered from insomnia and 39.2% suffered from hysteria, in addition to depression.
+
+Several precautionary procedures have been conducted to prevent or minimise the destructive problem of traffic accidents. Plotting, which needs to be conducted by analysing causes of accidents through a large amount of data, is one of the major precautionary measures that might prevent accidents.
+
+According to a report from the [Turkish Ministry of Public Works](#page-6-0) [and Settlement \(2007\)](#page-6-0), there was, between 2003 and 2005, a 64% decrease in the number of accidents, a 94% decrease in road fatalities and a 65% decrease in the number of people injured on rehabilitated segments of roads.
+
+<sup>\*</sup> Tel.: +90 555 7214769; fax: +90 332 2410635. E-mail address: [bgundogdu@selcuk.edu.tr](mailto:bgundogdu@selcuk.edu.tr)
+
+<span id="page-1-0"></span>The most frequent method in defining a Hot Spot is how they are determined. A Hot Spot can be defined as a point where there is a particularly frequent recurrence of an event. These spots are determined statistically and have various visual methods and usage fields that can be described as follows ([Eck and Weisburd,](#page-6-0) [1995\)](#page-6-0). Areas of concentrated crime are often referred to as Hot Spots. Researchers and police use the term in many different ways. Some refer to Hot Spot addresses [\(Eck and Weisburd, 1995; Sher](#page-6-0)[man et al., 1989\)](#page-6-0), others refer to Hot Spot blocks ([Weisburd and](#page-6-0) [Green, 1994; Taylor et al., 1984\)](#page-6-0), and others examine clusters of blocks [\(Block and Block, 1995](#page-6-0)).
+
+The number of accidents on a given road section during a certain period of time is probabilistic in nature and is a non-negative integer. Despite the fact that accidents are random and unpredictable at micro-level, statistical models can predict reliable estimates of expected accidents by relating aggregates of accidents to various explanatory measures of flow, site characteristics, and road geometry at macro-level. Numerous empirical relationships between vehicle accidents and these explanatory variables have been established in several previous studies (e.g. [Maher and Summersgill,](#page-6-0) [1996; Miaou, 1994; Miaou and Lum, 1993](#page-6-0)). Such accident predicting models are useful in identifying the most critical variables of safety, assessing design and management alternatives, and improving safety standards for new roads.
+
+The significantly different areas detected by the applied statistics (usually termed LISA: Local Indicators of Spatial Autocorrelation) are, traditionally, termed Hot Spots, hotpoints or, occasionally, hotbeds [\(Ratcliffe and McCullagh, 1999](#page-6-0)). The statistics used to define the hotbed coverage may result in circular, elliptical or amoeboid shapes depending on the method or application used. There is a wide choice of approaches available ([Mccullagh, 2006\)](#page-6-0). The numerical analysis of spatial point distributions has interested academics for many years. A very early work that resulted is the famous Moran's I statistic by [Moran](#page-6-0) [\(1948\).](#page-6-0) An early important collection of work on general spatial analysis is that edited by [Berry and Marble \(1968\).](#page-6-0) The development of spatial point analysis theory is illustrated by later works by [Cliff and Ord \(1973, 1981\), Ord and Getis \(1995\), Anselin](#page-6-0) [\(1995\) and Gatrell et al. \(1996\).](#page-6-0) The overview by [Getis and Ord](#page-6-0) [\(1996\)](#page-6-0) has led to large number of recent studies, such as [Cho](#page-6-0) [\(2003\),](#page-6-0) using the techniques that have been developed over the last 50 years ([Mccullagh, 2006](#page-6-0)).
+
+The Linear Referencing Model (LRM) was used to determine Hot Spots in this study.
+
+An LRM can be defined as a mechanism for finding and stating the location of an unknown point along a network by referring it to a known point [\(Vonderohe and Chou, 1997\)](#page-6-0). More specifically, an LRM is a process for determining a previously unknown location based on a defined path along the underlying transportation network; a distance along that path measured from a known datum location; and, optionally, an offset from the path. There are several types of LRMs, which differ based on the parts of the network used for referencing and the ways in which measures and offsets are calculated [\(Nyerges, 1990](#page-6-0)).
+
+The following tasks must be undertaken in a study which has the same aim:
+
+- - Determining the locations where an increased number of accidents occur.
+- - Detailed functional analysis of a location where an accident has occurred for the purpose of determining the factors causing accidents.
+- - Developing methods used for determining elements, such as traffic notices, and lights controlling traffic flow that cause danger.
+- - Inferring the effect of physical features of a road such as curve, pavement material, and turning directions on accident occurrence.
+
+![](_page_1_Figure_13.jpeg)
+
+Fig. 1. General map of roads in Konya Province and location of the related highways.
+
+- Handling traffic problems of the province in a comprehensive context by assessing all highways.
+
+It is not adequate processing just the total number of accidents in carrying out these tasks. All criteria must be evaluated separately. For this purpose, there must be an acceptable and detailed accident report for the long term. It may be possible to explain the effects of different accident information details by comparing four base criteria. These main criteria, detailed below, are the total number of accidents, fatalities, injured and with only financial loss. In particular, result base maps related to base criteria have been produced for this purpose. In addition, the question of how many and which segments are problematic? Will be explained. That the problems are either changing according to period of year or structure of road will be determined.
+
+## 2. Study area and accident data
+
+# 2.1. Study area
+
+Highway transport is very important in Turkey. Konya is the biggest city of the country, and has 2957 km of highway, out of a total 61,939 km of roads.
+
+According to statistics in 2006, the city is eighth in the rank of total accident rates, third in the rank of mortality rates, fifth in the rank of injuries. The region has been one of the most important transport connection points for some time. The city is in the centre of the country and connects crowded cities and tourist regions. A large amount of agricultural crops are carried from east to west by trucks with the help of seven routes of the city.
+
+Having a smooth topography and straight routes enables high– speed traffic. Thus, in addition accidents are inevitable, and the severity of accidents and losses may be at a very disturbing level. [Fig. 1](#page-1-0) shows digitized routes on the map.
+
+#### 2.2. Traffic accident data
+
+The most consuming phase of the study is detailed registration of accident data. In this study, 3856 accidents data for the last 10 years, obtained from the Konya Police Department. All detailed data have been prepared by using MS Access to evaluate by Arc GIS 9.2 program. Seven highways totally 724 km have been digitized with the scale of 1/25,000 maps and connected with coordinates of accident points and other details by the program.
+
+Because of the changing transport density in certain seasons, it will be possible to evaluate old data with new data or a different period of year and different purposes for the future.
+
+## 3. Methodology
+
+It can be easily understood, through statistical tests, whether or not the inputs are meaningful in determining Hot Spots. The Poisson distribution technique is commonly used for this purpose. Recent studies (e.g. [Meel and Safr, 2008](#page-6-0)) have looked at on application of the Poisson distribution technique to traffic accidents, analysing high-crash locations ([Kristen and Mark, 2002\)](#page-6-0) black spots and Poisson distribution [\(Ding and Pei, 2007\)](#page-6-0) explaining dangerous locations with four categories [\(Hu et al., 2007\)](#page-6-0), crash counts and prediction modelling [\(Ma et al., 2008; Caliendo et al.,](#page-6-0) [2007; Mitra and Washington, 2007; Ng et al., 1995](#page-6-0)). Poisson assumptions for the crash densities ([Li et al., 2008\)](#page-6-0) and time series ([Quddus, 2008](#page-6-0)) can be arranged.
+
+The main features of Poisson distribution are that it was developed for specific circumstances of randomly occurring events over a time and for repetitive experiments of events with low potential for occurrence. When formation of Hot Spots is analysed, it is seen that independently occurring individual accidents form Hot Spots by assembling in clusters. This is why Poisson distribution is the most fitting statistical distribution for determining Hot Spots by handling randomly occurring traffic accidents.
+
+In order to use Poisson distribution for the purpose of determining Hot Spots along a road, the road must be divided into certain segments. The area where an accident happened is recorded officially to the nearest kilometre. It is, therefore, sensible to divide the road into 1 km segments. After accident probability for each individual segment is calculated, 1 km segments exceeding the threshold value are characterized as Hot Spots. In this case, the number of accidents is calculated for each part using the formula below (n: number of accidents; k: number of segments).
+
+$$\lambda = \frac{n}{k}$$
+
+Accident potential for each segment is estimated using the formula
+
+$$f(k;\lambda) = \frac{\lambda^k e^{-\lambda}}{k!}$$
+
+With a probability rate of 0.05, where e is the base of the natural logarithm (e = 2.71828...), k is the number of occurrences of an event – the probability of which is given by the function, k! is the factorial of k, and k is a positive real number, equal to the expected number of occurrences that occur during the given interval.
+
+In order to determine which pieces, whose accident potential has previously been estimated with Poisson distribution, have the character of Hot Spots, the Getis–Ord Gi analysis must be applied. The statistics Gi and Gi, introduced by [Getis and Ord](#page-6-0) [\(1996\)](#page-6-0) for the study of local patterns in spatial data, were extended and re-written (below) in 1995 to redefine Gi as a standard variate and to allow for non-binary classifications of the distance d ([Ord](#page-6-0) [and Getis, 1995](#page-6-0)). In this analysis, the Gi(d) statistic is used as follows ([Khan et al., 2005](#page-6-0)):
+
+$$G_{i}(d) = \frac{\sum_{j=1}^{N} w_{ij}(d)x_{j} - \bar{x}_{i}\sum_{j=1}^{N} w_{ij}(d)}{S(i)\sqrt{\frac{\left[\binom{N-1}{j-1}w_{ij}^{N}(d) - \left(\sum_{j=1}^{N} w_{ij}(d)\right)^{2}\right]}{\frac{N-2}{N-2}}}, j \neq i$$
+
+where
+
+$$\bar{x}_i = \frac{\sum_{j=1, i \neq j}^{N} X_j}{N-1} \quad \text{and} \quad S(i) = \sqrt{\frac{\sum_{j=1, i \neq j}^{N} X_j^2}{N-1} - \left(\bar{x}_i\right)^2}$$
+
+$$G_{i}^{*}(d) = \frac{\sum_{j=1}^{N} w_{ij}(d)x_{j} - \bar{x}\sum_{j=1}^{N} w_{ij}(d)}{S\sqrt{\frac{\left[\sum_{j=1}^{N} w_{ij}^{2}(d) - \left(\sum_{j=1}^{N} w_{ij}(d)\right)^{2}\right]}{N-1}}}$$
+
+where
+
+$$\bar{x} = \frac{\sum_{j=1}^{N} x_j}{N}$$
+ and  $S = \sqrt{\frac{\sum_{j=1}^{N} x_j^2}{N} - (\bar{x})^2}$ 
+
+where Gi is the Getis–Ord Gi z-score value excluding the value at segment i, G <sup>i</sup> is Getis–Ord G <sup>i</sup> z-score value including the value at segment i, xi is relative crash rate of segment i, xj is relative crash rate of neighboring locations to segment i, d is fixed band radius around segment i, wij is spatial weight matrix for all segment j within distance d, and N is the number of weighted points.
+
+Getis and Ord's Gi analysis calculates the 'z' value for each segment. If the number of accidents at a certain point significantly differs from the z values calculated from the median, z values must be
+
+<span id="page-3-0"></span>![](_page_3_Figure_2.jpeg)
+
+Fig. 2. Hot Spots are shown according to: (a) number of accidents, (b) rate of mortality, (c) number of people injured, (d) quantity of damage, and (e) the inter-sectional cluster of all showing the highest risk points.
+
+over the threshold value of 1.645. Thus, Hot Spots can be easily determined by an enquiry based on z values.
+
+## 3.1. Analysis of Hot Spots with unusual criteria and the Highest Potential Hot Spots
+
+By means of this method, at which point and according to which criterion rehabilitation of roads must be performed can be predetermined, [Fig. 2](#page-3-0) shows Hot Spots according to different criteria. Hot Spots have been determined by the number of accidents so far. The upper limit of analysis (1.645) has been processed in the accident number column. It must be mentioned that other criteria are as important as accident numbers. These are the number of fatalities, number of injured and amount of damage. In this study, Hot Spots were determined on the basis of not only the number of accidents, but also the number of people killed/injured and the amount of damage. This is so that a more realistic analysis can be suggested. [Fig. 2](#page-3-0)a shows the classical method used with accident numbers, but the numbers and sizes of Hot Spots vary with other criteria, as seen in [Fig. 2b](#page-3-0)–d. It must not be forgotten that different maps enable different analyses to take into consideration different criteria. Another important point relates to the intersection segments of the four criteria, as seen in [Fig. 2](#page-3-0)e. The results of this study indicate that, between 10–14 km and 55–65 km of route 3, 25–29 km of route 4 and 37–45 km of route 6 are intersection segments and they must be considered as priority areas.
+
+#### 3.2. Illustration of hazardous segments as Hot Pieces
+
+Maps are indispensable products of numerical data to illustrate or analyse various subjects. According to the type of map reader, or subject, different maps are produced for different purposes. Use of correct illustration techniques allows correct analysis. Occasionally, spot representation may suit any subject, but linear or area may not be sufficient. So, in this study, various maps have been produced by different illustrating types for future investigations.
+
+Hot Spots have been shown in speckles so far. When involved in traffic accidents, it is seen that there is not just one event at any one point. For this reason, it would be wrong to be concerned with only one point in determining Hot Spots. A lot of factors that have caused an accident may have been effective before the accident occurred. If any roads are improved with the help of the maps, segments (±500 m) of accident points must be shown on the maps. Graduated segments, in particular, can show the way at the beginning and end of points of improving of risky segments by degrees. So, in this study, linear clustering, as a new technique, was used to determine the hazardous segments to prevent of traffic accidents and take precautions on the highways. Specifically, the results of risk assessment of 1 km segments are illustrated in linear clusters.
+
+Assessment of Hot Pieces (HPCS) would be a more precise guideline for the study of risk factors and accident regions together in linear clusters, as seen in Fig. 3. By the same method, different maps, which show deaths, injuries and financial damage distributions, can be produced.
+
+# 3.3. Illustration of potential Hot Spots (Probable Hot Spots)
+
+Up to now, dangerous points have been shown as Hot Spots by points which exceed the upper limit. But when accident data are investigated, there are some points which will be on the Hot Spot limit after 1–2 accidents. Investigating these points is as important as for normal Hot Spot points. There are some points which are not Hot Spots but they will be Hot Spots after just one accident.
+
+In this study, another important subject is taking preventative measures before a point becomes a Hot Spot.
+
+Points that are not Hot Spots yet but have high-potential for being Hot Spots have not been mentioned in any applications so far. This study, it must be emphasized, includes segments with high-potential (95%) for being Hot Spots. In [Fig. 4,](#page-5-0) Probable Hot Spots (PRHS) were determined and illustrated for the region approaching 1.645 value, at a rate of 95%. These probable Hot Spots can be compared with [Fig. 2a](#page-3-0)–d. On routes 5 and 6 particularly,
+
+![](_page_4_Figure_13.jpeg)
+
+Fig. 3. HPCS illustrations in linear clusters.
+
+<span id="page-5-0"></span>![](_page_5_Figure_2.jpeg)
+
+Fig. 4. Probable Hot Spots (PRHS).
+
+some points are not Hot Spots but they will be soon. Urgent precautions must be taken for these points, therefore.
+
+# 4. Conclusions and future work
+
+Interpretation of results on a map is one of the fundamental guides. In this study, the map user is supplied with different visualizations of linear analysis illustrated with different techniques.
+
+As a consequence of the application, however the hotspots have been evaluated by the number of accidents so far, studying with only accident numbers and mapping of them alone are not sufficient. Thematic maps show graduated accident parameters are the most important products for the study. It is important to determine which types of traffic accidents occur at which segments. The maps show the user not only kilometre values of accident points but also hazardous road segments. The number of people killed or injured, and the number of accidents with only financial damage, must be taken as information clusters individually and then HPHSs must be assessed through inter-sectional clusters of individual information clusters.
+
+When route maps producing different criteria are investigated, as seen in [Fig. 2](#page-3-0)a–d, Hot Spots appear at different points on the same routes. For instance, Hot Spots determined by mortality criteria are seen on straight routes because of the high speed on those sections. If there were an accident on this route, someone would normally be killed. Although the number of accidents with injuries is fewer than mortalities, the factors are more or less the same. Maps of damage Hot Spots show them appearing on international roads, which attract expensive vehicles. There is less mortality but more injured people and more accidental damage on international roads. These facts may give an idea about using goals and type of the routes and kinds of vehicles. However, the idea is not acceptable for the entire route. Nevertheless, the method should be applicable in other similar road networks especially using different accident parameters and considering all segments of roads.
+
+In pieces maps, some segments do not have any Hot Spots, but some segments meet all the criteria. In this case, Highest Potential Hot Spots, which are determined by the intersection of all criteria must be considered first. Hot Spot maps producing different criteria are the most important tools for road improvement.
+
+When we consider that the place where the accident happened is recorded with 1 km proximity, and the factors affecting the accident do not start just at the accident point, it is obvious that analysis of HPCS is critical. Table 1 shows the number of Hot Spots and priority status for all points, as another useful result for improving route planning.
+
+When the results of the study compared with existing accident data, some useful suggestions can be mentioned. HPHS and PRHS numbers and locations which are determined by this method are different from normal Hot Spot numbers or locations. Naturally priorities of precaution types will be various at different points or road segments.
+
+It can be seen that 22 high-potential Hot Spots could be real Hot Spots in the near future. These segments ought to be dealt with as a priority, in terms of preventative measures. All dangerous zones and the severity of accidents at the point of different criteria can be shown numerically from Table 1. So, both values and kilometres from maps must be used to improve current routes and planning new routes. The priority of the plan is reduction of mortality. It is also structurally problematic segments or points on routes that can be determined easily with the help of result values and terrestrial research. As an example, although route 5 does not have any
+
+Table 1 Critical pieces and their values (for seven routes).
+
+| Reference           | Hot Spot<br>numbers | Priority status                                                      |
+|---------------------|---------------------|----------------------------------------------------------------------|
+| Number of accidents | 38                  | Hot Spots estimated from data                                        |
+| Number of deaths    | 37                  | 1st priority spots                                                   |
+| Number of injuries  | 27                  | 2nd priority spots                                                   |
+| Financial damage    | 27                  | 3rd priority spots                                                   |
+| HPHS                | 04                  | Most dangerous spots (to be analysed<br>initially)                   |
+| HPCS                | 38                  | Accident zones                                                       |
+| PRHS                | 22                  | Highly potential hotspots<br>necessitating precautionary<br>measures |
+
+<span id="page-6-0"></span>Hot Spots determined for damage, it has the most number of deaths, since it is so straight and crowded.
+
+Investigating the dangerous spots and the road segments illustrated in the analysis maps prepared here, studying the contributory factors and taking precautions are the responsibility of the related administration.
+
+# References
+
+- Anselin, L., 1995. The local indicators of spatial association-LISA. Geographical Analysis 27 (2), 93–115.
+- Arslan, F., 2007. Emniyet Genel Müdür Yardımcısı. <[http://www.trafik.gov.tr/icerik/](http://www.trafik.gov.tr/icerik/bildiriler/A6-87.doc) [bildiriler/A6-87.doc>](http://www.trafik.gov.tr/icerik/bildiriler/A6-87.doc) (accessed May 2008).
+- Berhanu, G., 2004. Models relating traffic safety with road environment and traffic flows on arterial roads in Addis Ababa. Accident Analysis and Prevention 36 (5), 697–704.
+- Berry, B., Marble, D., 1968. Spatial Analysis. Prentice Hall, Englewood Cliffs.
+- Block, R., Block, C., 1995. Space, place and crime: hot spot areas and hot place of liquor-related crime. Crime and Place 4 (2), 145–184.
+- Caliendo, C., Guida, M., Parisi, A., 2007. A crash-prediction model for multilane roads. Accident Analysis and Prevention 39 (4), 657–670.
+- Cho, W.K.T., 2003. Contagion effects and ethnic contribution networks. American Journal of Political Science 47 (2), 368–387.
+- Cliff, A.D., Ord, J.K., 1973. Spatial Autocorrelation. Pion, London.
+- Cliff, A.D., Ord, J.K., 1981. Spatial Processes. Models and Application. Pion, London. Ding, J.M., Pei, Y.L. 2007. Identification of road black spots and safety evaluation. In: International Conference on Construction and Real Estate Management, Bristol, pp. 21–22.
+- Eck, J.E., Weisburd, D., 1995. Crime places in crime theory. Crime and Place 4, 1–33. Faaliyetlerimiz Ve Hedeflerimiz. 2007. Turkish Ministry of Public Works and Settlement Turkey.
+- Gatrell, A.C., Bailley, T.C., Diggle, P., Rowlingson, B.S., 1996. Spatial point pattern analysis and its application in Geographical epidemiology. Transactions of the Institute of British Geographers 21 (2), 256–274.
+- Getis, A., Ord, J.K., 1996. Local spatial statistics. An overview. In: Longley, P., Batty, M. (Eds.), Spatial Analysis: Modeling in a GIS Environment. Wiley, New York.
+- Hilakivi, I., Veilahti, J., Asplund, P., Sinivuo, J., Laitinen, L., Koskenvuo, K., 1989. A 16 factor personality test for predicting automobile driving accidents of young drivers. Accident Analysis and Prevention 21 (5), 413–418.
+- Hu, J.B., Fei, X.L., Liu, X.M., 2007. Statistical method for identifying dangerous location and countermeasures. Journal of Beijing University of Technology 33 (5), 465–469.
+- Khan, G., Qin, X., Noyce, A.D., 2005. Wisconsin Traffic Operations and Safety Laboratory. [<http://www.topslab.wisc.edu>](http://www.topslab.wisc.edu) (accessed June 2008).
+- Kristen, L.S.B., Mark, R.V., 2002. Improving the identification, analysis and correction of high-crash locations. ITE Journal 72 (1), 38–43.
+- Li, W., Carriquiry, A., Pawlovich, M., Welch, T., 2008. The choice of statistical models in road safety countermeasure effectiveness studies in Iowa. Accident Analysis and Prevention 40 (4), 1531–1542.
+
+- Ma, J., Kockelman, K.M., Damıen, P.A., 2008. Multivariate Poisson-lognormal regression model for prediction of crash counts by severity, using Bayesian methods. Accident Analysis and Prevention 40 (3), 964–975.
+- Maher, M.J., Summersgill, J., 1996. A comprehensive methodology for the fitting of predictive accident models. Accident Analysis and Prevention 28 (3), 281– 296.
+- Mccullagh, J.M., 2006. Detecting Hotspots in Time and Space. ISG06, University of Nottingham.
+- Meel, B.L., Safr, M.J., 2008. Fatal road traffic accidents in the Mthatha area of South Africa. South African Medical Journal 98 (9), 716–719.
+- Miaou, S.P., 1994. The relationship between truck accidents and geometric design of road sections: Poisson versus negative binomial regressions. Accident Analysis and Prevention 26 (4), 471–482.
+- Miaou, S.P., Lum, H., 1993. Modelling vehicle accidents and highway geometric design relationships. Accident Analysis and Prevention 25 (6), 689–709.
+- Mitra, S., Washington, S., 2007. On the nature of over-dispersion in motor vehicle crash prediction models. Accident Analysis and Prevention 39 (3), 459–468.
+- Moran, P., 1948. The interpretation of statistical maps. Journal of Royal Statistical Society 1 (2), 243–251.
+- Ng, C.H., Wong, Y.D., Lum, K.M., 1995. The applicability of the Poisson distribution in traffic accident count analysis – the Singapore experience. IES Journal 32 (2).
+- Nyerges, T.L., 1990. Locational referencing and highway segmentation in a geographic information system. ITE Journal 60 (3), 27–31.
+- Ord, J.K., Getis, A., 1995. Local spatial autocorrelation statistics: distribution issues and an application. Geographical Analysis 27 (4), 286–306.
+- Quddus, M.A., 2008. Time series count data models: an empirical application to traffic accidents. Accident Analysis and Prevention 40 (5), 1732–1741.
+- Ratcliffe, J.H., Mccullagh, M.J., 1999. Hotbeds of crime and the search for spatial accuracy. Journal of Geographical Systems 1 (4), 385–398.
+- Rathinam, C., Nair, N., Gupta, A., Joshi, S., Bansal, S., 2007. Self-reported motorcycle riding behaviour among school children in India. Accident Analysis and Prevention 39 (2), 334–339.
+- Rutter, D.R., Quine, L., 1996. Age and experience in motorcycling safety. Accident Analysis and Prevention 28 (1), 15–21.
+- Sherman, L.W., Gartin, P.R., Buerger, M.E., 1989. Hot Spots of predatory crime: routine activities and the criminology of place. Criminology 27 (1), 27–55.
+- Taylor, R.B., Gottfredson, S.D., Brower, S., 1984. Block crime and fear: defensible space, local social ties, and territorial functioning. Journal of Research in Crime and Delinquency 21 (4), 303–310.
+- Turkish Ministry of PublicWorks and settlement, 2007. [<http://www.bayindirlik.](http://www.bayindirlik.gov.tr/turkce/dosya/2003_2006faaliyetvehedefler.pdf) [gov.tr/turkce/dosya/2003\\_2006faaliyetvehedefler.pdf](http://www.bayindirlik.gov.tr/turkce/dosya/2003_2006faaliyetvehedefler.pdf)> (accessed October 2009).
+- Vonderohe, A., Chou, C., 1997. A Generic Data Model for Linear Referencing Systems. Transportation Research Board, Washington (DC).
+- Weisburd, D., Green, L., 1994. Defining the street level drug market. In: Mackenzie, D.L., UchidA, C. (Eds.), Drugs and Crime: Evaluating Public Policy Initiatives. Sage, Thousand Oaks, CA.
+- World Health Organisation World report on road traffic injury prevention: summary, 2004. <[http://www.who.int/world-health-day/2004/infomaterials/](http://www.who.int/world-health-day/2004/infomaterials/worldreport/en/summary_en_rev.pdf) [worldreport/en/summary\\_en\\_rev.pdf>](http://www.who.int/world-health-day/2004/infomaterials/worldreport/en/summary_en_rev.pdf) (accessed May 2008).

Zotero/001_artiklid/Road Safety Indicators/hadayeghitemporaltransferabilityupdatingzonal2006.md

@@ -0,0 +1,397 @@
+---
+category: literaturenote
+citekey: hadayeghitemporaltransferabilityupdatingzonal2006
+title: Temporal transferability and updating of zonal level accident prediction models
+authors: "Hadayeghi, Alireza; Shalaby, Amer S.; Persaud, Bhagwant N.; Cheung, Carl"
+year: 2006
+date: 2006-00-00 2006
+doi: 10.1016/j.aap.2005.12.003
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2005.12.003"
+zotero_key: ZUN7M7TU
+zotero_storage: B8H9E7F4
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Hadayeghi, Alireza"
+status: converted
+---
+![](_page_0_Picture_0.jpeg)
+
+Accident Analysis and Prevention 38 (2006) 579–589
+
+![](_page_0_Picture_2.jpeg)
+
+# Temporal transferability and updating of zonal level accident prediction models
+
+Alireza Hadayeghi <sup>a</sup>,∗, Amer S. Shalaby <sup>b</sup>,1, Bhagwant N. Persaud <sup>c</sup>,2, Carl Cheung <sup>c</sup>,<sup>3</sup>
+
+<sup>a</sup> *Department of Civil Engineering, University of Toronto and Synectics Transportation Consultants Inc., 36 Hiscott Street, St. Catharines, Ont., Canada L2R 1C8* <sup>b</sup> *University of Toronto, 35 St. George Street, Toronto, Ont., Canada M5S 1A4* <sup>c</sup> *Ryerson University, 350 Victoria Street, Toronto, Ont., Canada M5B 2K3*
+
+Received 3 March 2005; received in revised form 22 September 2005; accepted 5 December 2005
+
+#### **Abstract**
+
+This paper examines the temporal transferability of the zonal accident prediction models by using appropriate evaluation measures of predictive performance to assess whether the relationship between the dependent and independent variables holds reasonably well across time. The two temporal contexts are the years 1996 and 2001, with updated 1996 models being used to predict 2001 accidents in each traffic zone of the City of Toronto. The paper examines alternative updating methods for temporal transfer by imagining that only a sample of 2001 data is available. The sensitivity of the performance of the updated models to the 2001 sample size is explored. The updating procedures examined include the Bayesian updating approach and the application of calibration factors to the 1996 models. Models calibrated for the 2001 samples were also explored, but were found to be inadequate. The results show that the models are not transferable in a strict statistical sense. However, relative measures of transferability indicate that the transferred models yield useful information in the application context. Also, it is concluded that the updated accident models using the calibration factors produce better results for predicting the number of accidents in the year 2001 than using the Bayesian approach.
+
+© 2005 Elsevier Ltd. All rights reserved.
+
+*Keywords:* Temporal transferability; Model calibration; Bayesian approach; Safety planning models
+
+#### **1. Introduction**
+
+Accident prediction models are used to predict the number of accidents at intersections, on arterial road sections, on highways or on transportation networks. Of late, there is considerable interest in transportation planning applications, for which these models (usually developed at the planning zone level) are aimed at establishing a relationship between accident frequency and a number of explanatory variables such as traffic volume, road network characteristics and socioeconomic and demographic features. It is important for safety planning models to enable transportation planners to assess the safety implications of alternative transport planning options that are considered for medium to long-range implementation, in a way similar to how road designers make use of tools such as the Interactive Highway Safety Design Model (IHSDM)<sup>4</sup> to assess the safety implications of design decisions. For this reason and others, it is desirable to obtain reasonably accurate safety planning models that are transferable across time and have good predictive performance.
+
+In general, the importance of accident prediction models makes it crucial that they be properly calibrated ([Persaud et al.,](#page-10-0) [2002\).](#page-10-0) The quantity and quality of the data used to calibrate accident prediction models is very important to the success or failure of these models. Models calibrated with limited data will not perform adequately in terms of predictive accuracy. Therefore, having a reasonable dataset is very important in the development of accident prediction models. Moreover, the specification of an
+
+<sup>∗</sup> Corresponding author. Tel.: +1 905 704 0763; fax: +1 905 682 4495. *E-mail addresses:* ahadayeghi@synectics-inc.net (A. Hadayeghi), amer@ecf.utoronto.ca (A.S. Shalaby), bpersaud@ryerson.ca (B.N. Persaud), c22cheun@ryerson.ca (C. Cheung).
+
+<sup>1</sup> Tel.: +1 416 978 5907; fax: +1 416 978 5054.
+
+<sup>2</sup> Tel.: +1 416 979 5000x6464; fax: +1 416 979 5122.
+
+<sup>3</sup> Tel.: +1 416 979 5000; fax: +1 416 979 5122.
+
+<sup>4</sup> Interactive Highway Safety Design Model (IHSDM), Federal Highway Administration, US Department of Transportation.
+
+appropriate functional model form for calibration of accident prediction models is a complex task. It gets even more complicated for safety planning models because of the paucity of these.
+
+Data for transportation planning variables are usually obtained from surveys held periodically (for example, the Transportation Tomorrow Survey is conducted once every 5 years in the Greater Toronto Area). These surveys usually cover a comprehensive area and are therefore expensive in terms of the time and cost associated with data collection and analysis. As such, once a model is well specified to capture the decision process in one context, it is cost effective to transfer such a model to other contexts that have limited calibration data as long as the basic nature of the decision-making process remains the same [\(Badoe](#page-9-0) [and Miller, 1995a\).](#page-9-0)
+
+There are two aspects to model transferability—temporal and spatial. The first, results from application of a model estimated at one point of time for prediction at another point of time for the same spatial environment. This occurs whenever estimated models are applied in a forecasting framework. The second aspect – spatial transferability – involves the application of a model estimated in a specific geographic region for prediction in a different one for the same time period. Of course, there can be a combination of temporal and spatial transfers.
+
+Several research efforts have been conducted to address the temporal and spatial transferability of transportation planning models. Most of this research was on trip generation models and disaggregate and aggregate mode choice models [\(Ben-Akiva,](#page-9-0) [1981; Koppelman and Wilmot, 1982; Atherton and Ben-Akiva,](#page-9-0) [1976; Gunn et al., 1985; Gunn, 2001; Badoe and Miller, 1995b;](#page-9-0) [Ben-Akiva and Bolduc, 1987; Parody, 1977\).](#page-9-0)
+
+However, no similar efforts have been directed at safety planning models. This is likely in reflection of the reality that few safety planning models have been calibrated to date. A literature search indicated that only few studies have been undertaken to evaluate and quantify traffic safety impacts in the planning process [\(Hadayeghi et al., 2003; Lovegrove and Sayed, 2005;](#page-9-0) [Ladron de Guevara et al., 2004; NCHRP 8-44, in press\).](#page-9-0)
+
+Of most relevance to the current effort was some work by [Oh et al. \(2003\)](#page-9-0) that validated intersection accident prediction models intended for use in the IHSDM from two perspectives—internal validity and external validity. Internal model validity examines the strength of the original models in explaining the underlying phenomenon. External validity examines the ability of the models to predict crashes over space and time. The authors recommended that the validation over time could suffer from omitted variables that influence safety, poorly measured and surrogate variables, and model functional forms. On the other hand, the authors argue that internal validity and validation over space could suffer from omitted variables, site selection and countermeasure selection bias, poorly measured and surrogate variables, and misspecification of model functional form.
+
+This paper has two basic objectives. First, it examines the temporal transferability of the accident prediction models by using appropriate evaluation measures of predictive performance to assess whether the relationship between the dependent and independent variables holds reasonably well across time.
+
+The two temporal contexts are the years 1996 and 2001, with models estimated from 1996 data being used to predict 2001 accidents. A series of accident prediction models based on 1996 database were calibrated in the previous study ([Hadayeghi et al.,](#page-9-0) [2003\)](#page-9-0) to estimate the number of accidents in planning zones in the City of Toronto as a function of planning variables. It should be stressed that the spatial transferability of accident prediction models is outside the scope of this paper. However, some insights on this issue have been provided in some other papers [\(Persaud](#page-10-0) [et al., 2002\).](#page-10-0)
+
+The second objective of the paper is to compare alternative updating methods of temporal transfer and to estimate effects of the size of the application context data on model transferability. The updating procedures examined are the Bayesian updating and updating based on recalibration factors. Models calibrated for the 2001 samples were also explored, but were found to be inadequate, so were not tested in the 2001 application context. In addition, this study investigates the transfer bias caused by sample size. This issue was not addressed in previous studies which focused mainly on using only one method and one sample size. This is important since it is believed a large portion of differences in coefficients and predictions is caused by random variation due to selection of data. Also, the goodness of fit measures resulting from updated models are compared to the models estimated using the entire dataset in the application context. A key contribution of this paper is the variety in the validation methods and measures used, tested and compared.
+
+#### **2. Data**
+
+The data used in this study are comprised of accident, socioeconomic, demographic and road network characteristics, and traffic flow for the City of Toronto's 463 traffic zones for 1996 and 481 traffic zones for 2001. The zone definitions for 1996 and 2001 are not identical. For the purpose of this study, the 2001 zone definition is adopted for both 1996 and 2001.
+
+The data for this study were obtained from three sources. The accident data for 1996 and 2001, the first dataset, was provided electronically by the Traffic Data Centre (TDC) of the City of Toronto. The accident data are geocoded, which allows the grouping of accidents by traffic zone using GIS tools. The relevant information for each accident record includes severity (fatal or non-fatal injury and property damage only (PDO)) and period of day. The latter allows for separate models to be developed for the morning peak period (6:00–9:00 am) to which traffic intensity data pertained.
+
+The second dataset, consisting of socioeconomic and demographic data, was obtained from the 1996 and 2001 Transportation Tomorrow Survey (TTS), a 1-day travel survey conducted in the Greater Toronto and surrounding areas (GTA) on approximately 5% of its population. The survey is conducted to provide detailed demographic and travel information of GTA household members to local, regional, provincial road authorities and to transit operating agencies. The traffic demand data were obtained from the results of a traffic assignment of the morning <span id="page-2-0"></span>peak-period vehicle trips to the road network using the [EMME/2](#page-9-0) [\(1998\)s](#page-9-0)oftware package to estimate vehicle kilometers travelled (VKT) in each zone. This approximation is necessary because cordon counts, the only available source of traffic demand data collected consistently across the city, are made at a limited number of locations and therefore cannot be used directly and sufficiently to provide traffic flow estimates for each zone in the city. It is worth mentioning that it is a common practice in building travel demand and planning models to use traffic assignment model outputs.
+
+In addition to the above data, 1996 GIS road files were obtained from the Integrated Land Use, Transport and Environment (ILUTE) group at the University of Toronto, and the corresponding 2001 database, CanMap Street Files, was attained from University of Toronto Map Library.
+
+#### **3. Comparison of 1996 and 2001 conditions**
+
+Table 1 presents the magnitude of changes in various characteristics in the City of Toronto between 1996 and 2001. The number of households grew by 3.68%, from 908,499 in 1996 to 943,252 in 2001. The population of the City of Toronto grew by 2.7% in the 5-year period, while the total population of the Greater Toronto Area (GTA) increased by 9.3%. The difference in percent growth between the GTA and City of Toronto is indicative of the reality that more people lived in the suburbs in 2001 as a result of urban sprawl. During this period, the total employment in the City of Toronto increased by more than 7%.
+
+Total vehicle kilometers travelled (VKT) per person increased by 15%, from 0.75 in 1996 to 0.88 in 2001. However, there was little change in total major road kilometers in the same period, suggesting that traffic conditions should have worsened from 1996 to 2001. This observation is confirmed by the average volume to capacity (*V*/*C*) ratios in Table 1, which increased by almost 30% from 1996 to 2001. The percent average changes in zonal traffic volume from 1996 to 2001 in the City of Toronto, based on the 1996 zonal definitions, are presented in [Fig. 1](#page-3-0) . As seen, most of the zones experienced an increase in traffic volume in the 5-year period.
+
+Table 1 also shows that the total number of accidents increased by 27% from 1996 to 2001, an average annual increase of 5.41%. On the other hand, the number of severe (i.e., fatal plus injury) accidents increased only by a total of 5.64% in the 5-year period.
+
+[Fig. 2](#page-3-0) presents the zonal percentage change in total number of accidents for the City of Toronto. As can be seen, most of the zones outside of downtown have experienced a higher percentage change in total number of accidents than that for the downtown zones.
+
+## **4. Model specification**
+
+A series of macro-level, zonal accident prediction models that estimate the number of zonal accidents in the City of Toronto for 1996 was developed in the work described in [Hadayeghi et al.](#page-9-0) [\(2003\). A](#page-9-0) generalized linear modelling approach was employed, in which negative binomial regression models were developed separately for total accidents and for severe (fatal and non fatal injury) accidents as a function of socioeconomic/demographic, traffic demand and network data variables. A total of 22 possible explanatory variables describing zonal characteristics were explored. The definition of variables was based on (a) previous research for similar studies and (b) an iterative process in which various definitions were tried and the ones that gave the best results were ultimately selected.
+
+The following model form was used
+
+$$E(A) = a_0 \text{ VKT}^{b_0} \exp \sum_{i=1}^{n} b_i X_i$$
+ (1)
+
+Table 1 Change between 1996 and 2001 data in the City of Toronto
+
+| Attribute for independent variables                                               | 1996<br>2001 |         | Difference | Percent change (%) |  |
+|-----------------------------------------------------------------------------------|--------------|---------|------------|--------------------|--|
+| (a) Changes in urban structure, travel behavior and socioeconomic and demographic |              |         |            |                    |  |
+| Vehicle kilometer travelled (VKT)                                                 | 1728775      | 2089257 | 360482     | 17.25              |  |
+| Average volume capacity (V/C)                                                     | 0.22         | 0.31    | 0.09       | 30.18              |  |
+| Average posted speed                                                              | 42.08        | 42.48   | 0.40       | 0.95               |  |
+| Total population                                                                  | 2305600      | 2368700 | 63100      | 2.66               |  |
+| Number of households                                                              | 908499       | 943252  | 34753      | 3.68               |  |
+| Number of part-time employees                                                     | 181224       | 187890  | 6666       | 3.55               |  |
+| Number of full-time employees                                                     | 875225       | 943373  | 68148      | 7.22               |  |
+| Total number of employees                                                         | 1056449      | 1131263 | 74814      | 6.61               |  |
+| Total minor road kilometer                                                        | 4172.66      | 4406.24 | 233.58     | 5.30               |  |
+| Total major road kilometer                                                        | 859.83       | 900.73  | 4.90       | 0.54               |  |
+| Total number of intersections                                                     | 15498        | 16649   | 1151       | 6.91               |  |
+| Attribute for dependent variables                                                 | 1996         | 2001    | Difference | Percent change (%) |  |
+| (b) Change in Number of Accidents                                                 |              |         |            |                    |  |
+| Total accidents                                                                   | 53286        | 73043   | 19757      | 27.1               |  |
+| Severe (fatal and injury) accidents                                               | 15818        | 16764   | 946        | 5.64               |  |
+| Total accidents, morning peak period                                              | 5317         | 8691    | 3374       | 38.82              |  |
+| Severe accidents, morning peak period                                             | 1664         | 1971    | 307        | 15.58              |  |
+
+<span id="page-3-0"></span>![](_page_3_Figure_2.jpeg)
+
+Fig. 1. Percent change in average traffic volume from 1996 to 2001 in the City of Toronto based on 1996 zonal definitions.
+
+![](_page_3_Figure_4.jpeg)
+
+Fig. 2. Percent change in total number of accidents from 1996 to 2001 in the City of Toronto based on 1996 zonal definitions.
+
+<span id="page-4-0"></span>Table 2 Results of negative binomial regression for models I and II
+
+| Parameter                     | Yearly total accident model (Model I) |                        | Yearly severe accident model (Model II) |                                                                                                                                                         |  |
+|-------------------------------|---------------------------------------|------------------------|-----------------------------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------|--|
+|                               | 1996 Model                            | 2001 Model             | 1996 Model                              | 2001 Model                                                                                                                                              |  |
+| Intercept (log of α0)         | 4.1767 (0.2957, 199.47)               | 2.8065 (0.3572, 61.75) | 2.6618 (0.3239, 67.55)                  | 1.9593 (0.3736, 27.50)                                                                                                                                  |  |
+| Log of VKT                    | 0.2656 (0.0347, 58.76)                | 0.2845 (0.0540, 27.81) | 0.2415 (0.039, 38.41)                   | 0.1739 (0.0561, 9.60)<br>0.2423 (0.0208, 136.31)<br>0.0650 (0.0162, 16.03)<br>−0.0096 (0.0067, 2.07)<br>−0.3963 (0.3193, 1.54)<br>0.0013 (0.0012, 1.03) |  |
+| Total major road in kilometer | 0.1175 (0.0186, 40.05)                | 0.1960 (0.0197, 93.98) | 0.1225 (0.0202, 36.77)                  |                                                                                                                                                         |  |
+| Number of households (× 10−3) | 0.0864 (0.015, 33.27)                 | 0.0582 (0.0156, 13.89) | 0.0858 (0.0164, 27.38)                  |                                                                                                                                                         |  |
+| Speed                         | −0.0431 (0.0071, 36.61)               | −0.0124 (0.0065, 3.64) | −0.0312 (0.0078, 16.05)                 |                                                                                                                                                         |  |
+| Volume/capacity (V/C)         | −1.3678 (0.2766, 24.45)               | −0.6472 (0.3112, 4.33) | −1.2232 (0.3042, 16.18)                 |                                                                                                                                                         |  |
+| Intersection density          | 0.0052 (0.0012, 19.87)                | 0.0024 (0.0012, 3.98)  | 0.0038 (0.0013, 9.14)                   |                                                                                                                                                         |  |
+| Goodness of fit measures      |                                       |                        |                                         |                                                                                                                                                         |  |
+| Mean Pearson χ2/d.f.          | 1.0313                                | 1.1559                 | 1.0346                                  | 1.0110                                                                                                                                                  |  |
+| Pearson R2                    | 37.48%                                | 37.24%                 | 31.90%                                  | 43.06%<br>0.2296                                                                                                                                        |  |
+| α                             | 0.1941                                | 0.2355                 | 0.2092                                  |                                                                                                                                                         |  |
+
+*Note*: Refer to Eq. [\(1\). T](#page-2-0)he dependent variable is the natural logarithm of yearly total accidents. Parameter estimate (S.E., χ2).
+
+where *E*(*A*) is the predicted accident frequency per zone per year, VKT the zonal vehicle kilometers travelled during peak hours (6:00–9:00 am), *Xi* the other explanatory variables, *a*0, *b*<sup>0</sup> and *bi* is the model parameters
+
+The above model form is consistent with the non-linear relationship between accidents and traffic intensity (measured in this case by VKT) suggested by several other researchers. Separate models are calibrated for total and severe (fatal plus non-fatal injury) accidents per year and, for each, models were estimated for the entire day and for the morning peak period (6:00–9:00 am) to which the VKT and other traffic variable estimates from EMME/2 pertain. The final model parameters and corresponding χ<sup>2</sup> and goodness-of-fit criteria are shown in Tables 2 and 3. This Table is taken from [Hadayeghi et al. \(2003\)](#page-9-0) which discusses the modelling approach and related issues.
+
+The models developed in this study for both 1996 and 2001 include major arterial and minor road data in each zone, but exclude freeway data since the presence of a freeway in a zone was found after exploratory analysis to be generally unrelated to characteristics of the zone.
+
+# **5. Measures of model transferability**
+
+The transferability of a model relates to the application of a model estimated in one context based on adequate data to another context with limited data. Model transferability is based on the idea that the estimated model from a previous study in a different context may provide useful information for estimating parameters appropriate for the application context. "Local" or "estimation" context refers to the context for which the adequate dataset was used in the model estimation. The "transfer" or "application" context refers to the context for which predictions are required, and to which the model from the local context is transferred for application.
+
+Transportation planning models are not perfectly transferable between contexts since the models do not capture the behavior of individuals completely and the data available for the estimation of the unknown parameters are not error free due to a variety of sources such as sampling errors ([Lerman, 1981\).](#page-9-0) As [Lerman \(1981\)](#page-9-0) and [Ben-Akiva \(1981\)](#page-9-0) discuss, it is appropriate to describe the measure of transfer effectiveness on a continuous
+
+Table 3 Results of negative binomial regression for models III and IV
+
+| Parameter                       | Yearly total accident model for<br>morning period (Model III) |                         | Yearly severe accident model for<br>morning period (Model IV) |                        |  |
+|---------------------------------|---------------------------------------------------------------|-------------------------|---------------------------------------------------------------|------------------------|--|
+|                                 | 1996 Model                                                    | 2001 Model              | 1996 Model                                                    | 2001 Model             |  |
+| Intercept (log of α0)           | 1.7851 (0.2909, 37.66)                                        | 0.4328 (0.3127, 1.92)   | 0.2079 (0.0403, 2.7)                                          | 1.0527 (0.4562, 5.32)  |  |
+| Log of VKT                      | 0.2225 (0.0430, 26.73)                                        | 0.3268 (0.0574, 32.47)  | 0.2171 (0.0606, 12.85)                                        | 0.3163 (0.0803, 15.53) |  |
+| Total major road in kilometer   | 0.1171 (0.0201, 34.07)                                        | 0.2127 (0.0202, 111.39) | 0.1316 (0.0264, 24.8)                                         | 0.2528 (0.0267, 89.51) |  |
+| Number of households (× 10−3)   | –                                                             | –                       | 0.0892 (0.0220, 16.43)                                        | 0.0701 (0.0216, 10.51) |  |
+| Number of employements (× 10−3) | 0.0692 (0.0162, 18.18)                                        | 0.0533 (0.0155, 11.80)  | –                                                             | –                      |  |
+| Speed                           | −0.0324 (0.0073, 19.59)                                       | −0.0139 (0.0063, 4.91)  | −0.0213 (0.0102, 4.34)                                        | −0.0141 (0.0087, 2.61) |  |
+| Volume/capacity (V/C)           | −1.1156 (0.3296, 11.46)                                       | −0.7236 (0.3234, 5.01)  | −0.9074 (0.4500, 4.07)                                        | −0.8137 (0.4412, 3.40) |  |
+| Intersection density            | –                                                             | –                       | –                                                             | –                      |  |
+| Total minor road in kilometer   | 0.0108 (0.0051, 4.52)                                         | 0.0006 (0.0049, 0.02)   | –                                                             | –                      |  |
+| Goodness of fit measures        |                                                               |                         |                                                               |                        |  |
+| Mean Pearson χ2/d.f.            | 1.0805                                                        | 1.0802                  | 1.0791                                                        | 0.9659                 |  |
+| Pearson R2                      | 36.56%                                                        | 48.46%                  | 31.57%                                                        | 42.17%                 |  |
+| α                               | 0.1586                                                        | 0.1959                  | 0.1961                                                        | 0.2290                 |  |
+
+*Note*: Refer to Eq. [\(1\). T](#page-2-0)he dependent variable is the natural logarithm of yearly total accidents. Parameter estimate (S.E., χ2).
+
+scale rather to describe transferability as a dichotomous property. Transferability is a property that may exist to a greater or lesser degree depending on the formation of the model and the characteristics of the estimation and application contexts [\(Lerman,](#page-9-0) [1981; Ben-Akiva, 1981\).](#page-9-0)
+
+Measures of model transferability were used to assess the effectiveness of models when they are transferred for use in an application context (2001 in this study) that is different from their calibration context (1996). The indices and test-statistics are all based on log-likelihood measure. These measures, defined below, are largely similar to those used by [Koppelman and Rose](#page-9-0) [\(1983\)](#page-9-0) and [Badoe \(1994\).](#page-9-0)
+
+#### *5.1. Nested likelihood ratio test (LR)*
+
+This measure indicates the statistical similarity of the estimation and application model coefficients. In this test, the null hypothesis is that the application and estimation context can be presented by a single model. Basically this test examines whether the model parameters have remained stable over time and is given by
+
+$$LR = -2[LL_{i \cup j}(\hat{\theta}_{i \cup j}) - LL_{i}(\hat{\theta}_{i}) - LL_{j}(\hat{\theta}_{j})]$$
+(2)
+
+The above equation is χ<sup>2</sup> distributed with degrees of freedom equal to the number of model parameters. In the event of rejection of the null hypothesis, an asymptotic *t*-test can be performed to test which parameter(s) were responsible for the rejection. For this, the *t*-statistic is given as
+
+$$t = (\hat{\theta}_{ki} - \hat{\theta}_{kj}) / \sqrt{(\text{var}(\hat{\theta}_{ki}) + \text{var}(\hat{\theta}_{kj}))}$$
+(3)
+
+where ˆ θik is the estimate of parameter *k* in context *I* and var(ˆ θik) is the variance of the estimate of parameter *k* in context *i*.
+
+#### *5.2. Transfer index (TI)*
+
+Transfer index (TI) is a relative measure that indicates how well a transferred model performs in predicting the application dataset relative to a locally estimated model in the application context and of similar specification. For this particular research, this index compares performance of the transferred and application context models relative to the performance of a 2001 "constant" model that has no independent variables. The index has an upper bound, 1.0, which is attained when the transferred model performs as well as an estimated model on the application data. It has no lower bound. Negative values imply that the transferred model is worse than the local constant model. It is expressed as
+
+$$TI = [LL_j(\hat{\theta}_i) - LL_j(\hat{c})]/[LL_j(\hat{\theta}_j) - LL_j(\hat{c})]$$
+(4)
+
+where LLj(ˆ θi) is the log-likelihood value in context *j* using model parameters estimated from year *i*, LLj(ˆ θj) the loglikelihood value given by application context model *j* and LLj(ˆc) is the log-likelihood value given by constant model estimated in application context *j*.
+
+## **6. Updating 1996 models**
+
+This section provides an approach for updating local models into an application context in a situation when only limited data are available in the application context for calibrating a meaningful model. The updating technique is helpful in applying an "old" estimated model to an application context to predict present conditions with minimum data requirements, consequently cutting the cost of data collection and time. Also, it seems desirable, when transferring a model, to update it by making adjustments based on locally available information. To illustrate such a situation, and to investigate the feasibility of this approach, random subsets of zonal information were selected from the 2001 database. To explore the impact of sample size on updating performance, 10 samples of each of 10, 20, 25, and 35 zones and their corresponding information were randomly selected from the 2001 dataset. However, before using updating procedures, a set of models were estimated using different sample sizes. These models were used to examine whether the coefficients in the estimation and application contexts are similar when limited data are available.
+
+#### *6.1. Updating using a calibration factor*
+
+The updating procedure was recommended by [Harwood et](#page-9-0) [al., 2000](#page-9-0) and a recent paper ([Oh et al., 2003\),](#page-9-0) for application in the Interactive Highway Safety Design Model (IHSDM) for transferring accident prediction models calibrated for one geographic region to apply to another region. In this procedure, [Harwood et al. \(2000\)](#page-9-0) in essence obtained a calibration factor as the total number of accidents observed in a sample from one jurisdiction divided by the sum of the predicted number of accidents in the sample using the model from another jurisdiction. The calibration factor was calculated as follows:
+
+calibration factor = 
+$$\sum_{i=1}^{n} Y_i / \sum_{i=1}^{n} \hat{Y}_i$$
+ (5)
+
+where *Yi* is the number of observed accidents in zone *i* in 2001 for a given sample size and Yˆ<sup>i</sup> is the predicted number of accidents in zone *i* by applying values of the 2001 independent variables (for the same sample size of 2001) to the 1996 model.
+
+Then, the calibration factors estimated for each sample size were simply multiplied by the base models (1996 models) in order to "transfer" the 1996 models for application to 2001 conditions.
+
+### *6.2. Bayesian updating*
+
+Bayesian updating, which was introduced by [Atherton and](#page-9-0) [Ben-Akiva \(1976\),](#page-9-0) is another approach that is adopted for updating the 1996 models using the 2001 sample data. This methodology combines sample information with prior information in order to achieve more accurate updated information. [Atherton and Ben-Akiva \(1976\)](#page-9-0) assumed both prior and posterior (updated) distributions of the parameters to be normally distributed. The updated parameters are expressed as
+
+$$\beta_{\text{updated}} = [(\beta_{96}/\sigma_{96}^2) + (\beta_{01\text{-sample}}/\sigma_{01\text{-sample}}^2)]/[(1/\sigma_{96}^2) + (1/\sigma_{01\text{-sample}}^2)]$$
+(6)
+
+$$\sigma_{\text{updated}} = 1/[(1/\sigma_{96}^2) + (1/\sigma_{01\text{-sample}}^2)]^{0.5}$$
+ (7)
+
+where βupdated is the 2001 updated coefficient, β<sup>96</sup> the coefficient estimated from 1996 model, β01-sample the coefficient estimated from 2001 sample model, σupdated the standard deviation of 2001 updated coefficient, σ<sup>96</sup> the standard deviation of coefficient estimated from 1996 model and σ01-sample is the standard deviation of coefficient estimated from 2001 sample model.
+
+#### *6.3. Measures of goodness of fit*
+
+Measures of goodness of fit are used to assess the performance of the updated models relative to the 1996 models. These measures are similar to those used by [Oh et al. \(2003\).](#page-9-0)
+
+#### *6.3.1. Mean prediction bias (MPB)*
+
+This measure describes the magnitude and direction of the average bias in the subject model. The smaller the absolute value of average prediction bias is, the better the model predicts the observed data. Positive values of MPB indicate that the model over-predicts the observed accident data, and vice versa:
+
+MPB = 
+$$\sum_{i=1}^{n} (\hat{Y}_i - Y_i)/n$$
+ (8)
+
+where Yˆ<sup>i</sup> is the predicted number of accidents at zone *i*, *Yi* the observed number of accidents at zone *i* and *n* is the data sample size.
+
+#### *6.3.2. Mean absolute deviation (MAD)*
+
+This measure provides a measure of the average misprediction of the model. A value close to zero suggests that the model on average predicts the observed data well. It is defined as
+
+$$MAD = \sum_{i=1}^{n} |(\hat{Y}_i - Y_i)|/n$$
+ (9)
+
+# *6.3.3. Mean squared prediction error and mean squared error*
+
+The mean squared prediction error (MSPE) is the sum of squared differences between observed and predicted accidents divided by the sample size. It is defined as
+
+MSPE = 
+$$\sum_{i=1}^{n} (\hat{Y}_i - Y_i)^2 / n$$
+ (10)
+
+The mean squared error (MSE) is the sum of squared differences between observed and predicted accidents divided by the sample size minus the number of model parameters. It is defined as
+
+$$MSE = \sum_{i=1}^{n} (\hat{Y}_i - Y_i)^2 / (n - p)$$
+ (11)
+
+where *p* is the number of model parameters.
+
+A comparison of MSPE and MSE shows potential over or under fitting of the models to the estimation data. A MSPE that is higher than MSE may indicate that models may have been over fit to the estimation data and that some of the observed relationships may have not been correctly shown in the model.
+
+#### **7. Results and discussion**
+
+#### *7.1. Transferability results*
+
+The dependent variable of each developed model is the number of zonal accidents per year. The regression parameters were estimated based on the maximum likelihood method for negative binomial (NB) regression and using PROCGENMOD in SAS statistical software ([SAS Institute Inc., 1999\).](#page-10-0)
+
+# *7.1.1. Yearly model for total accidents*
+
+[Table 2](#page-4-0) presents the parameter estimates and corresponding χ2's for the 1996 model and the 2001 model for total yearly accidents (Model I). A comparison of the parameter estimates in [Table 2](#page-4-0) indicates that the speed becomes insignificant at the 5% level in the 2001 model. The parameter estimates of each variable have identical signs in both models. The dispersion parameters (α) are in the range of 0.19 and 0.23 for the 1996 model and the 2001 model, respectively, which indicates that the data are quite overdispersed relative to the Poisson distribution and that the negative binomial error structure is therefore justified. The goodness of fit measures for each model, also presented in [Table 2,](#page-4-0) show the mean Pearson χ<sup>2</sup> ratios to be close to 1, an indication that the NB regression model fits the data very well. The Pearson *R*<sup>2</sup> values for the 1996 model and the 2001 model are 37.48% and 37.24%, respectively, which indicate that the developed models explain the variation in the dependent variable to a reasonable degree. In addition, a test was done to examine whether keeping the dispersion parameter in the 2001 model the same as the one in the 1996 model would affect the 2001 model performance considerably. It was found that the model parameters and corresponding parameters' signs were almost the same as those of Model I. The total predicted number of accidents from this model for the whole study area is different by less than 1% from the total from Model I, which is quite insignificant in comparison. Therefore, no further investigation was conducted.
+
+Results of tests of parameter equality, which were done by the nested likelihood ratio test, are shown in [Table 4.](#page-7-0) These indicate that 1996 and 2001 model parameters are not statistically equal. Therefore, the null hypothesis can be rejected. The errors in the modelling calibration procedures are most likely to affect transferability of the explanatory variables and the estimated parameters. To investigate whether the rejection of the null hypothesis is because of an individual coefficient or sub-set of coefficients, the asymptotic *t*-test is used to test equality of 1996 and 2001 variable coefficient estimates. Part (b) of [Table 4](#page-7-0) presents the results of this test for the yearly model for total accidents. As it can be seen, only two explanatory variables of VKT and number of households cannot be rejected at the 5% significance level under the null hypothesis of equality in the two contexts. Theoretically, it can be concluded that the
+
+<span id="page-7-0"></span>Table 4 Results from transferring 1996 Models to 2001 dataset
+
+| Transfer measures                                         | Model I | Model II | Model III | Model IV |  |
+|-----------------------------------------------------------|---------|----------|-----------|----------|--|
+| (a) Absolute and relative transfer measures               |         |          |           |          |  |
+| Nested likelihood ratio                                   | 128.6   | 52.4     | 177.4     | 30.84    |  |
+| a (× 105)<br>LL01(θ96)                                    | 3.087   | 0.460    | 0.179     | 0.0116   |  |
+| Transfer index                                            | −0.966  | 0.528    | −0.881    | 0.640    |  |
+| Parameter                                                 | Model I | Model II | Model III | Model IV |  |
+| (b) Statistical test of parameter equality in two context |         |          |           |          |  |
+| Log of VKT                                                | −0.29   | 0.99     | −1.45     | −0.99    |  |
+| Total major road in kilometer                             | −2.71   | −4.13    | −3.35     | −3.23    |  |
+| Number of households (× 10−3)                             | 1.30    | 0.90     | –         | 0.62     |  |
+| Number of employements (× 10−3)                           | –       | –        | 0.71      | –        |  |
+| Speed                                                     | −3.19   | −2.10    | −0.06     | −0.54    |  |
+| Volume/capacity (V/C)                                     | −1.73   | −1.88    | −1.21     | −0.15    |  |
+| Intersection density                                      | 1.65    | 1.41     | –         | –        |  |
+| Total minor road in kilometer                             | –       | –        | 1.44      | –        |  |
+
+<sup>a</sup> LL01(θ96) is the 2001 context log-likelihood using θ<sup>96</sup> from year 1996.
+
+roadway parameters such as major road kilometers and posted speed did not remain stable over time. The value of transfer index, which describes the degree to which the log likelihood of the transferred model exceeds the reference model relative to improvement provided by a model developed in the application context, is found to be −0.966. The negative sign implies that the transferred model is worse than the local reference model.
+
+## *7.1.2. Yearly model for severe accidents*
+
+The model parameters and corresponding χ<sup>2</sup> values for the yearly model of severe accidents (Model II) are shown in[Table 2.](#page-4-0) All the coefficients are close in magnitude and share the same sign in the both the 1996 and 2001 models. The intersection density and *V*/*C* variables are insignificant at the 5% level in the 2001 model but are significant in the 1996 model.
+
+The dispersion parameters (α) of 0.2092, and 0.2226 for the 1996 model and the 2001 model, respectively, again confirm the appropriateness of using negative binomial as opposed to Poisson regression. The mean Pearson χ<sup>2</sup> ratios for the two models are within the acceptable range of 0.8–1.2, another indication that the choice of the negative binomial error structure appears to be appropriate. The Pearson *R*<sup>2</sup> values for the 1996 model and the 2001 model are 31.90% and 43.3%, respectively, which indicate that the developed models explain the variation in the dependent variable to a reasonable degree.
+
+Results of the test of parameter equality in Table 4 indicate that the 1996 and 2001 model parameters are not statistically equal. *V*/*C*, speed, and major road kilometers are the parameters most responsible for the statistical inequality, as shown in part (b) of Table 4. The value of transfer index of 0.528, suggests that the transferred model provides some portion of the information obtained from the 2001 model, irrespective of model specification.
+
+#### *7.1.3. Morning peak period accident models*
+
+For the morning peak period models, accidents between 6:00 and 9:00 am were extracted from the dataset by running a query in Microsoft Access. [Table 3](#page-4-0) shows the parameter estimates for the total accident models (Model III) and severe accident models (Model IV) for the morning peak period. For both Models III and IV, the signs of all variables and constant terms are the same for the 1996 and 2001 models. The coefficients in the 2001 models are largely different from those of the 1996 models, in particular, those with low levels of significance such as total minor road kilometers in the 2001 model. For both Models III and IV, the coefficient of the total major road kilometers variable for the 2001 models is three times as large as that for the 1996 models.
+
+The goodness of fit statistics show mean Pearson χ<sup>2</sup> values close to one, again indicating that the models fit the data reasonably well. The dispersion parameters (α) for all the models are significantly different from zero, again confirming the appropriateness of negative binomial regression. The Pearson *R*<sup>2</sup> values are reasonable for all the models.
+
+The statistical tests of equality of model parameters for the Models III and IV reject the null hypothesis of equality, indicating that some or all model parameters have not remained stable over time. Both Models III and IV exhibit statistically significant equality at the 5% significance level in all the parameters, except major road kilometers. The values of Transfer index are found to be −0.881 and 0.640 for Models III and IV, respectively, suggest that the transferred model provides some portion of the information obtained from the 2001 model, irrespective of model specification. However, the negative value, as explained previously, implies that the transferred model is worse than the local reference model.
+
+# *7.2. Updating results*
+
+As expected, none of the models calibrated with limited data performed reasonably in terms of the magnitude of the coefficient parameters or their level of significance in the application context regardless of the sample size. The magnitude of coefficient parameters for some of the models were the same as the 1996 models but either the parameters were insignificant at the 5% confidence level or the goodness of fit measures were not
+
+<span id="page-8-0"></span>Table 5Goodness of fit measures for predicting 2001 accidents from 1996, 2001 and updated models
+
+| del<br>Mo<br>typ<br>es | GO<br>F             | 20<br>01<br>del<br>Mo | 6<br>199<br>del<br>Mo | dat<br>ing<br>wi<br>th<br>le o<br>f<br>Up<br>sam<br>p<br>10<br>in<br>20<br>01<br>zon<br>es |                                           | dat<br>ing<br>wi<br>th<br>le o<br>f<br>Up<br>sam<br>p<br>20<br>in<br>20<br>01<br>zon<br>es |                                           | dat<br>ing<br>wi<br>th<br>le o<br>f<br>Up<br>sam<br>p<br>25<br>in<br>20<br>01<br>zon<br>es |                                                 | dat<br>ing<br>wi<br>th<br>le o<br>f<br>Up<br>sam<br>p<br>35<br>in<br>20<br>01<br>zon<br>es |                                           |
+|------------------------|---------------------|-----------------------|-----------------------|--------------------------------------------------------------------------------------------|-------------------------------------------|--------------------------------------------------------------------------------------------|-------------------------------------------|--------------------------------------------------------------------------------------------|-------------------------------------------------|--------------------------------------------------------------------------------------------|-------------------------------------------|
+|                        |                     |                       |                       | Re<br>cal<br>ibr<br>d<br>ate<br>199<br>6 m<br>ode<br>l                                     | Ba<br>ian<br>dat<br>yes<br>up<br>e        | Re<br>cal<br>ibr<br>d<br>ate<br>199<br>6 m<br>ode<br>l                                     | Ba<br>ian<br>dat<br>yes<br>up<br>e        | Re<br>cal<br>ibr<br>d<br>ate<br>199<br>6 m<br>ode<br>l                                     | Ba<br>ian<br>dat<br>yes<br>up<br>e              | Re<br>cal<br>ibr<br>d<br>ate<br>199<br>6 m<br>ode<br>l                                     | Ba<br>ian<br>dat<br>yes<br>up<br>e        |
+| Mo<br>del<br>I         | MP<br>B             | 1.1<br>7              | −3<br>7.8<br>0        | −1<br>0.5<br>7 (<br>27<br>.39<br>)                                                         | −3<br>0.0<br>5 (<br>39<br>.78<br>)        | −6<br>.87<br>(<br>15.<br>49<br>)                                                           | −2<br>8.0<br>4 (<br>18.<br>50<br>)        | 1.3<br>1 (<br>17.<br>59<br>)                                                               | −3<br>4.1<br>5 (<br>26<br>.86<br>)              | −2<br>.42<br>(<br>21<br>.94<br>)                                                           | −4<br>0.6<br>4 (<br>23<br>.72<br>)        |
+|                        | MA<br>D<br>MS<br>E  | 59<br>.20<br>67<br>80 | 65<br>.17             | 71<br>.62<br>(<br>4.8<br>6)                                                                | 61<br>.95<br>(<br>16.<br>97<br>)          | 69<br>.73<br>(<br>2.6<br>1)                                                                | 58<br>.85<br>(<br>16.<br>14)              | 71<br>.51<br>(<br>5.3<br>8)                                                                | 65<br>.19<br>(<br>18.<br>26<br>)                | 71<br>.33<br>(<br>5.6<br>1)                                                                | 65<br>.77<br>(<br>16.<br>39)              |
+|                        | MS<br>PE            |                       | 92<br>94              | 95<br>169<br>04<br>(<br>0)                                                                 | 58<br>89<br>93<br>(<br>23<br>)            | 56<br>89<br>(<br>294<br>)                                                                  | 96<br>85<br>350<br>6)<br>(                | 66<br>68<br>90<br>(<br>8)                                                                  | 46<br>83<br>92<br>(<br>00<br>)                  | 05<br>67<br>92<br>(<br>3)                                                                  | 62<br>63)<br>87<br>(<br>44                |
+| del<br>Mo<br>II        | MP<br>B             | 2.3<br>1              | −0<br>.51             | −0<br>.66<br>(<br>8.5<br>9)                                                                | 7.4<br>9 (<br>20<br>.33<br>)              | 0.5<br>4 (<br>3.0<br>5)                                                                    | −2<br>.69<br>(<br>5.4<br>1)               | 0.6<br>2 (<br>2.1<br>2)                                                                    | 0.5<br>3 (<br>1.8<br>0)                         | −1<br>.17<br>(<br>4.7<br>1)                                                                | 3.2<br>1 (<br>6.0<br>0)                   |
+|                        | MA<br>D<br>MS<br>E  | 14<br>.89<br>44<br>9  | 15<br>.36             | 17<br>.68<br>(<br>2.3<br>8)                                                                | 21<br>.45<br>(<br>13.<br>33)              | 16<br>.97<br>(<br>0.5<br>6)                                                                | 15<br>.69<br>(<br>2.1<br>4)               | 16<br>.78<br>(<br>0.4<br>6)                                                                | 13<br>.99<br>(<br>1.8<br>7)                     | 16<br>.85<br>(<br>0.7<br>3)                                                                | 16<br>.58<br>(<br>2.5<br>8)               |
+|                        | MS<br>PE            |                       | 41<br>9               | 54<br>5 (<br>123<br>)                                                                      | 80<br>8 (<br>90<br>2)                     | 48<br>2 (<br>12)                                                                           | 44<br>5 (<br>127<br>)                     | 47<br>6 (<br>4)                                                                            | 51<br>3 (<br>90<br>)                            | 49<br>4 (<br>25<br>)                                                                       | 44<br>1 (<br>180<br>)                     |
+| Mo<br>del<br>III       | MP<br>B<br>MA<br>D  | 2.2<br>8<br>8.1<br>1  | −6<br>.10<br>8.5<br>1 | −1<br>.42<br>(<br>3.5<br>6)<br>8.4<br>0.7                                                  | −6<br>.09<br>(<br>3.5<br>2)<br>7.4<br>3.2 | −0<br>.53<br>(<br>1.7<br>7)<br>8.2<br>0.3                                                  | −6<br>.54<br>(<br>1.7<br>8)<br>8.7<br>1.6 | 1.1<br>6 (<br>1.9<br>0)<br>8.6<br>0.6                                                      | −8<br>.24<br>(<br>2.3<br>8)<br>10<br>.16<br>1.5 | 0.2<br>6 (<br>2.5<br>1)<br>8.4<br>0.7                                                      | −5<br>.87<br>(<br>3.1<br>3)<br>8.2<br>2.1 |
+|                        | MS<br>E             | 13<br>2               |                       | 7 (<br>7)                                                                                  | 9 (<br>1)                                 | 4 (<br>6)                                                                                  | 4 (<br>3)                                 | 2 (<br>3)                                                                                  | (<br>7)                                         | 8 (<br>0)                                                                                  | 1 (<br>0)                                 |
+|                        | MS<br>PE            |                       | 16<br>8               | 15<br>2 (<br>12)                                                                           | 13<br>8 (<br>155<br>)                     | 14<br>0 (<br>4)                                                                            | 14<br>3 (<br>50<br>)                      | 14<br>2 (<br>9)                                                                            | 27<br>8 (<br>149<br>)                           | 14<br>3 (<br>11)                                                                           | 17<br>8 (<br>149<br>)                     |
+| Mo<br>del<br>IV        | MP<br>B             | 0.3<br>9              | −0<br>.52             | 0.2<br>3 (<br>1.1<br>1)                                                                    | −0<br>.76<br>(<br>1.0<br>8)               | −0<br>.04<br>(<br>0.8<br>5)                                                                | −0<br>.28<br>(<br>0.6<br>9)               | 0.1<br>0 (<br>0.6<br>6)                                                                    | −0<br>.48<br>(<br>1.0<br>6)                     | −0<br>.25<br>(<br>0.5<br>8)                                                                | −0<br>.29<br>(<br>0.7<br>9)               |
+|                        | MA<br>D<br>MS<br>PE | 2.3<br>9<br>10<br>.15 | 2.3<br>4              | 2.6<br>5 (<br>0.1<br>6)                                                                    | 2.6<br>6 (<br>0.6<br>2)                   | 2.5<br>4 (<br>0.1<br>8)                                                                    | 2.3<br>2 (<br>0.7<br>5)                   | 2.5<br>3 (<br>0.1<br>3)                                                                    | 2.5<br>0 (<br>0.3<br>8)                         | 2.4<br>7 (<br>0.1<br>0)                                                                    | 2.2<br>3 (<br>0.3<br>9)                   |
+|                        | MS<br>E             |                       | 10<br>.90             | 12<br>.58<br>(<br>1.0<br>9)                                                                | 12<br>.89<br>(<br>7.0<br>6)               | 12<br>.02<br>(<br>0.6<br>5)                                                                | 11<br>.16<br>(<br>7.3<br>5)               | 11<br>.72<br>(<br>0.4<br>2)                                                                | 10<br>.46<br>(<br>3.9<br>8)                     | 11<br>.73<br>(<br>0.4<br>6)                                                                | 9.2<br>2 (<br>3.7<br>9)                   |
+
+*Note*: MPB <sup>=</sup> mean prediction bias, MAD <sup>=</sup> mean absolute deviation, MSPE <sup>=</sup> mean squared prediction error, MSE <sup>=</sup> mean squared error (S.D.).
+
+<span id="page-9-0"></span>satisfactory. This was expected since the sample sizes selected for this study were too small (less than 8% of the total data) to result in a meaningful conclusion. Therefore, it was concluded that the updating procedure is the best approach when limited data are available and no further investigation was conducted regarding developing models with limited data.
+
+[Table 5](#page-8-0) presents the relative goodness of fit measures for predicting the number of accidents in 2001 using the 1996 models, the 2001 models, models that apply a calibration factor to the 1996 model and the Bayesian approach. For the latter two, various sample sizes are used to estimate the "updated" models that are applied.
+
+As expected, the measures indicate that 2001 model fits the 2001 data better than other models. The real value of these results is in the comparison of the performance of the "updated" models among themselves for different sample sizes and with the 1996 model. This comparison shows that in general the recalibrated "updated" models perform better than the updated Bayesian models and the1996 models. However, no clear-cut conclusions can be drawn regarding the performance of updated models in respect to the sample size used for estimating them.
+
+#### **8. Conclusions and summary**
+
+This paper examined the temporal transferability of zonal accident prediction models for the City of Toronto. In so doing, model calibration and related transferability issues were also addressed. The coefficient estimates for the 2001 models are consistent with the 1996 models. Most of the coefficient parameters and constant terms are significant for the 1996 and the 2001 models across all four accident type models (Models I–IV).
+
+The results of nested likelihood ratio test indicate that the model parameters from two urban contexts are not statistically equal and have not remained stable over time. Therefore, from a theoretical perspective, the transfer of macro-level accident prediction models is undesirable. However, relative measures of transferability indicate that the transferred models yield useful information in the application context. Transfer index values show that transferred models capture at least 52% of the variance provided by estimating all the parameters in the 2001 data. The results of the *t*-test for all the models indicate that the VKT and socioeconomic and demographic parameters are stable over time.
+
+In the absence of a complete dataset for calibrating the 2001 model, it looks like the "updated" models, which are developed by applying calibration factors to the 1996 models, perform reasonably and better than the 1996 models and the Bayesian updated models regardless of the sample size used for estimating these factors. The Bayesian approach has not yielded good results in general. The possible problem for this method is that in this method the coefficients in model transferring are heavily affected by the estimation context data since the number of observations is larger in this context. Also, this method assumes that the prior and posterior (updated) distributions of the parameters are normally distributed, which may not be a good assumption.
+
+This research looked only at temporal transferability of zonal accident prediction models. A natural next step for further research would be the investigation of spatial transferability and the combination of temporal and spatial transferability. That research should aim to identify the conditions that make two study areas "similar enough" to justify transferring a model from one to the other.
+
+#### **Acknowledgment**
+
+The authors acknowledge the Natural Sciences and Engineering Research Council of Canada (NSERC) for its support of some of the fundamental research on which this work is based.
+
+# **References**
+
+- Atherton, T.J., Ben-Akiva, M.E., 1976. Transferability and updating of disaggregation travel demand models. Transport. Res. Rec. 610, 12–18.
+- Badoe, D.A., 1994. An investigation into the long range transferability of work-trip discrete mode choice models. PhD Dissertation. Department of Civil Engineering, University of Toronto, Ont., Canada.
+- Badoe, D.A., Miller, E.J., 1995a. Analysis of the temporal transferability of disaggregate work trip mode choice models. Transport. Res. Rec. 1493, 1–11.
+- Badoe, D.A., Miller, E.J., 1995b. Comparison of alternative methods for updating disaggregated logit mode choice models. Transport. Res. Rec. 1493, 90–100.
+- Ben-Akiva, M., 1981. Issues in transferring and updating travel-behaviour models. In: Stopher, P.R., et al. (Eds.), New Horizons in Travel Behaviour. Lexington Books, Lexington, MA, pp. 665–686.
+- Ben-Akiva, M., Bolduc, D., 1987. Approaches to model transferability and updating: the combined transfer estimator. Transport. Res. Rec. 1139, 1–7.
+- EMME/2, 1998. User's Guide Manual Release 9. INRO Consultants Inc.
+- Gunn, H.F., 2001. Spatial and temporal transferability of relationships between travel demand, trip cost and travel time. Transport. Res. Part E: Logist. Transport. Rev. 37 (2/3), 163–189.
+- Gunn, H.F., Ben-Akiva, M.E., Bradley, M.A., 1985. Tests of the scaling approach to transferring disaggregate travel demand models. Transport. Res. Rec. 1037, 21–30.
+- Hadayeghi, A., Shalaby, A.S., Persaud, B., 2003. Macro-level accident prediction models for evaluating safety of urban transportation systems. Transport. Res. Rec. 1840, 87–95.
+- Harwood, D.W., Council, M., Hauer, E., Hughes, W. E., Vogt, A., 2000. Prediction of the expected safety performance of rural two-lane highways. Report FHWA-RD-99-207, FHWA, Department of Transportation.
+- Koppelman, F.S., Rose, G., 1983. Geographic transfer of travel choice models: evaluation and procedure. In: Optimization and Discrete Choice in Urban Systems: Proceedings of the International Symposium on New Directions in Urban Systems Modeling, pp. 272–309.
+- Koppelman, F.S., Wilmot, C.G., 1982. Transferability analysis of dissagregate choice models. Transport. Res. Rec. 895, 18–24.
+- Ladron de Guevara, F., Washington, S.P., Oh, J., 2004. Forecasting crashes at the planning level: simultaneous negative binomial crash model applied in Tucson, Arizona. Transport. Res. Rec. 1897, 191–199.
+- Lerman, S.R., 1981. A comment on interspatial, intraspatial, and temporal transferability. In: Stopher, P.R., et al. (Eds.), New Horizons in Travel Behaviour. Lexington Books, Lexington, MA, pp. 628–632.
+- Lovegrove, G.R., Sayed, T., 2005. Macro-level collision prediction models for evaluating neighbourhood traffic safety. In: Proceedings of the Sixth Transportation Specialty Conference, Canadian Society of Civil Engineering, Toronto, Canada.
+- Oh, J., Lyon, C., Washington, S., Persaud, B., Bared, J., 2003. Validation of FHWA crash models for rural intersection lessons learned. Transport. Res. Rec. 1840, 41–49.
+
+- <span id="page-10-0"></span>Parody, T.E., 1977. An analysis of disaggregate mode choice models in prediction. Transport. Res. Rec. 649, 23–31.
+- National Cooperative Highway Research Program (NCHRP), in press. Incorporating Safety into Long-range Transportation Planning, Project 8-44. National Cooperative Highway Research Program.
+- Persaud, B., Lord, D., Palmisano, J., 2002. Calibration and transferability of accident prediction models for urban intersections. Transport. Res. Rec. 1784, 57–64.
+- SAS Institute Inc., 1999. SAS/STAT® Users Guide, Version 8. SAS Institute Inc., Cary, NC.

Zotero/001_artiklid/Road Safety Indicators/hakkertusesexposureriskroad2002.md

@@ -0,0 +1,821 @@
+---
+category: literaturenote
+citekey: hakkertusesexposureriskroad2002
+title: The uses of exposure and risk in road safety studies
+authors: "Hakkert, A.S.; Braimaister, L."
+year: 2002
+date: 2002-00-00 2002
+url: "https://swov.nl/en/publication/uses-exposure-and-risk-road-safety-studies"
+zotero_key: TE6DS2K6
+zotero_storage: C779GGFD
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Hakkert, A.S."
+status: converted
+---
+# **The uses of exposure and risk in road safety studies**
+
+Prof. dr. A.S. Hakkert & dr. L. Braimaister
+
+R-2002-12
+
+![](_page_2_Picture_0.jpeg)
+
+# **The uses of exposure and risk in road safety studies**
+
+## **Report documentation**
+
+Number: R-2002-12
+
+Title: The uses of exposure and risk in road safety studies
+
+Author(s): Prof. dr. A.S. Hakkert & dr. L. Braimaister
+
+Research theme: Road users: the relation between behaviour, surroundings and
+
+accidents
+
+Theme leader: I.N.L.G. van Schagen
+
+Project number SWOV: 37.230
+
+Keywords: Exposure (human), vehicle mile, application (use), risk, accident,
+
+theory, statistics.
+
+Contents of the project: This report explores the theoretical possibilities of defining
+
+exposure and risk, discusses the problems associated with the use of exposure and risk and gives examples of various safety studies in which use is made of exposure and risk indicators.
+
+Number of pages: 53
+
+Price: L 11,25
+
+Published by: SWOV, Leidschendam, 2002
+
+SWOV Institute for Road Safety Research P.O. Box 1090 2260 BB Leidschendam The Netherlands Telephone +31-703173333 Telefax +31-703201261
+
+## **Summary**
+
+This report explores the theoretical possibilities of defining exposure and risk, discusses the problems associated with the use of exposure and risk and gives examples of various safety studies in which use is made of exposure and risk indicators.
+
+The report sets out with a definition of the three central terms used: accidents, exposure and risk. Each of the three terms is defined and definition problems and problems concerning accuracy of measurement and reporting are described. The conclusion reached is that there are general definitions of exposure and of risk, as used in the field of health prevention and risk analysis, but that in road safety practice, these terms should be defined within the context of the issue studied. For each application, the correct exposure measure should be used. This is sometimes impossible on account of the unavailability of the required information, or the necessity to collect it at great cost. Generally, the more aggregate the exposure measure, the more indirect the variables that are introduced, which casts shadows over the resulting risk calculations.
+
+In the case of transport, the most widely used measure of exposure is the number of kilometres travelled for each travel mode. In some cases, useful additional insight is provided by taking into account the speed of travel, in which case exposure is expressed as the amount of time spent in the traffic system. One of the developments in recent years has been the installation of electronic and telecommunication equipment inside vehicles and along roads. Another development is the increasingly widespread use of mobile telephones. As a result it is becoming easier to collect up-to-date and reliable information on a variety of parameters that could be of importance in the calculation of vehicle exposure and risk. Additional information on the distribution of speeds, types of vehicles and following distances also seems to be a future possibility.
+
+One of the contexts in which the term risk is used, is in comparing risks between different parts of the transport system, different transport modes or even different activities outside the field of transport. The desire is to have various activities exposed to equal risks, so as to establish a fair distribution of risks. After discussing the various problems and pitfalls in using this approach, it is concluded that the desire for equal risks in various segments of the transportation system is not practical. It is more useful to search for ways to make each segment of the transport system as safe as possible, keeping cost-effectiveness considerations into account.
+
+## **Contents**
+
+| 1.   | Introduction                                                       | 7  |
+|------|--------------------------------------------------------------------|----|
+| 2.   | Definitions of exposure and risk                                   | 8  |
+| 2.1. | Definitions of exposure                                            | 8  |
+| 2.2. | Definitions of risk                                                | 8  |
+| 2.3. | The size of the safety problem in terms of risk and exposure       | 9  |
+| 3.   | Quantifying exposure and risk                                      | 11 |
+| 3.1. | Exposure measurement                                               | 11 |
+| 3.2. | Quantifying risk by hours of exposure                              | 11 |
+| 3.3. | The relevant measures of exposure and risk                         | 14 |
+|      | 3.3.1.<br>Aggregate measures of exposure and risk                  | 15 |
+|      | 3.3.2.<br>Traffic volume as a measure of exposure                  | 16 |
+|      | 3.3.3.<br>Problems with quantifying exposure                       | 16 |
+|      | 3.3.4.<br>The possibilities with new information technology (IT)   | 17 |
+|      | 3.3.5.<br>Traffic exposure at intersections                        | 18 |
+|      | 3.3.6.<br>Pedestrian exposure and risk                             | 19 |
+| 3.4. | The use of fN diagrams                                             | 19 |
+| 4.   | Some issues associated with the use of exposure and risk           | 22 |
+| 4.1. | The desire to assess risk levels and to strive for equal risks     | 22 |
+| 4.2. | Over-representation and risk                                       | 23 |
+| 4.3. | Doing without information on risk and exposure - an example        | 24 |
+| 4.4. | National safety levels and targets                                 | 26 |
+| 4.5. | What is an acceptable level of risk?                               | 26 |
+| 4.6. | Risk versus effectiveness                                          | 28 |
+| 5.   | Some examples of the use of risk calculations in a number          |    |
+|      | of road safety applications                                        | 31 |
+| 5.1. | The involvement of lorries in fatal accidents                      | 31 |
+| 5.2. | Drivers involved in accidents while under the influence of alcohol | 33 |
+| 5.3. | The effect of speed on the risk of accident involvement            | 35 |
+| 5.4. | A preliminary study to explore directions for elaborating on risks | 36 |
+| 5.5. | Predictions of traffic safety in 2010 in the Netherlands           | 36 |
+| 5.6. | The involvement of elderly drivers in accidents                    | 37 |
+| 5.7. | The involvement of young drivers in accidents                      | 38 |
+| 5.8. | Road safety in the Netherlands up to 1999                          | 40 |
+| 6.   | A more detailed discussion on risk and exposure in the             |    |
+|      | context of infrastructure                                          | 41 |
+| 6.1. | The relationship between accidents and traffic volumes             | 41 |
+|      | 6.1.1.<br>Safety Performance Functions (SPF)                       | 43 |
+|      | 6.1.2.<br>The relation between aggregate number of collisions      |    |
+|      | and distance travelled                                             | 43 |
+| 6.2. | Hazardous locations                                                | 44 |
+| 7.   | Summary and conclusions                                            | 47 |
+|      | References                                                         | 50 |
+
+## 1. **Introduction**
+
+Many safety studies involve the use of the terms accidents, exposure and risk. One sometimes gets the impression that the use and definition of the terms exposure and risk differ from study to study and are chosen in an adhoc fashion. This report explores the theoretical possibilities of defining exposure and risk, discusses the problems associated with the use of exposure and risk and gives examples of various safety studies in which use is made of exposure and risk indicators. It will be shown that there is no general definition of exposure and of risk and that these terms should be defined within the context of the issue studied. The report also deals with many of the limitations connected with the use of exposure and points out the many pitfalls that lie on the path of the correct use of the term risk in safety studies.
+
+The report sets out the generally used definitions of the terms exposure and risk in chapter two. This is followed by a chapter on the quantification of exposure on both an aggregate level and as applying to the study of road infrastructure. Chapter 4 presents some of the general issues associated with the use of risk and exposure, over-representation, the setting of national targets and risk versus effectiveness. Chapter 5 presents and discusses a variety of examples of safety studies that use risk definitions and some safety studies that were conducted without the use of risk as a denominator to evaluate safety effects. Chapter 6 discusses, in more detail, the relationship between accidents, exposure and risk as they are applied in studies that are concerned with the safety of road infrastructure, including the relationship between accidents and traffic volumes and the issue of risk as applied to the identification of hazardous locations. Chapter 7 presents the conclusions of the report.
+
+# 2. **Definitions of exposure and risk**
+
+Whereas the definition of a road accident is generally well-understood in road safety literature, the definitions of the concepts exposure and risk are much less well-defined. An accident is generally defined, in this context, as an event in which at least one motor-vehicle was involved, that occurred on a public road and which resulted in injury. This definition is used because in most countries computerised systems exist for the recording of such accidents. This is also the definition mostly used by police forces to record an accident. Other road accidents are registered in other databases or are unrecorded. National differences exist in the definitions of accidents that are recorded by the police, and in the definition of accidents with casualties that are to be entered into statistical databases and definitions of the various degrees of severity of injury. International data bases do not always account for such differences in definitions. The international accident data base IRTAD, maintained by the German Federal Road Research Institute (BASt), makes an effort to report on these differences and account for them where possible.
+
+Although many shortcomings can be identified regarding this definition of an accident, at least we are talking about a recordable event which is to a certain degree exact. As to exposure and risk, on the other hand, we first have to agree on a definition and then it remains to be seen if the accepted notions are quantifiable and reasonably exact.
+
+#### 2.1. **Definitions of exposure**
+
+In the context of this report, exposure is meant as exposure to risk. To what extent are certain segments of the population likely to be involved in an accident? The measure of exposure is generally defined as some form of the amount of travel, either by vehicle or on foot. Once the amount of travel is known for certain activities, or road users, and if we know the number of crashes that are associated with that activity or population, the associated risk can be calculated. Risk assessments can be used to improve transport safety and determine public health priorities. …."The various ways of measuring the amount of travel are referred to collectively as 'exposure data' because they measure traveller's exposure to the risk of death or injury." (European Transport Safety Council (ETSC), 1999)
+
+### 2.2. **Definitions of risk**
+
+According to the Oxford Dictionary, risk is a "hazard, chance of bad consequences, loss etc., exposure to mischance". Collins Dictionary describes risk as "a possibility that something unpleasant or undesirable might happen; something that you do which might have unpleasant or undesirable results." In the field of road safety the concept of risk is used as a way to quantify the level of road safety relative to the amount of exposure, as opposed to the absolute level of safety as measured by the absolute number of accidents or casualties. The literature differentiates between various kinds of risk such as personal risk, societal risk, individual risk, group risk. Other differentiations can be added such as objective risk, subjective risk, voluntary risk and compulsory risk. A treatise on the various definitions of risk in the literature, their definitions, meanings and relevance is outside the scope of this document. For our purposes, risk will be used to mean the probability of an accident occurring. Such a definition was proposed in Hauer (1982).
+
+One element which might usefully be added to the definition of risk as a probability is taking into account the severity of the outcome of an event. This distinction is proposed in Haight (1986). Most people would agree that, whereas the probability of death in one game of 'Russian roulette' (onesixth) is considerably less than the probability of heads in the toss of a coin (one-half), Russian roulette is 'riskier' than coin-tossing. This distinction can be achieved when adding to the probability the concept of expected costs. When the cost of the outcome of each probability is multiplied by the probability of each event, risk can be defined as an expectation. Van Poortvliet (1999) quotes a definition of risk that takes both aspects into account, i.e both the probabilistic nature of risk and the severity of the consequences: "A combination of the probability, or frequency, of occurrence of a defined hazard and the magnitude of the consequences of the occurrence."
+
+Reliability engineers use the term risk to express the probability of a welldefined, hazardous event occurring. In many industrial applications the consequences of such events can be estimated by using models that cover the chemical, physical and biological effects of such a hazardous event. Thus risk analysis typically yields a set of likelihoods, accident scenarios and consequences (Van Poortvliet, 1999). The quantitative approach makes it possible to calculate the expected losses during a given period of time, usually expressed in terms of fatalities, casualties and/or material damage. In the field of road safety a similar approach is generally used. This approach is generally preferred over the qualitative approach.
+
+Popular perception associates risk with both the probability of a hazardous event for someone involved in a certain activity and with the severity of the outcome. If an accident occurs, problems generally arise when one tries to compare situations with both different probabilities and different consequences. Therefore, it is often preferable to talk about accident rates (or casualty rates) rather than risk. When comparing accident rates, it is assumed we are talking about situations with outcomes of similar severity, otherwise the situations should not be compared.
+
+#### 2.3. **The size of the safety problem in terms of risk and exposure**
+
+Having defined risk and exposure, it is now possible to define safety and the size of the safety problem in those terms. Figure 1 (Rumar,1999) defines the safety problem - I as a function of exposure, accident risk and injury risk:
+
+I = E x A/E x I/A, where I is the number of people injured, E is exposure, A/E is the probability of an accident (accident risk), I/A is the probability of being injured in an accident (injury risk).
+
+![](_page_11_Figure_0.jpeg)
+
+Figure 1. The safety problem (human injury) illustrated by the volume of the box. From: Rumar (1999).
+
+From this definition it becomes clear that countermeasures to improve road safety can come from activities along any one of the three axes. One can think of measures that reduce exposure, measures that reduce the risk of an accident and measures that reduce the risk of injury. It is also possible to take the time trend element into consideration. The absolute size of the safety problem, expressed in either the number of accidents or the number of casualties (SAFETY) of a certain severity results from multiplying the degree of risk, which has a trend with the exposure, which also has a trend element expressed as:
+
+SAFETY (severity) = RISK (trend) x EXPOSURE (trend).
+
+# 3. **Quantifying exposure and risk**
+
+#### 3.1. **Exposure measurement**
+
+Exposure is generally expressed in a form related to the amount of travel. In most countries traffic data (volumes) are counted for traffic engineering purposes but few countries have systems of counting traffic volumes that enable comprehensive measurement of volumes on various parts of the road system, segregated by type of road user. Some countries have such systems for parts of the road system - motorways, main rural roads, major intersections, etc. Still other countries have only approximate estimates of the amount of travel, conducted on the basis of the known fuel consumption in the country.
+
+Some countries conduct systematic counts at a series of counting sites, chosen to be nationally representative, covering all types of roads and different regions, and classifying counts according to type of vehicle and time of day. Such counts can be combined with detailed information about the road network to provide national and regional estimates of traffic volumes. Even in those countries, such estimates are generally not available on a per-city basis. Also in many cases local roads and minor roads are not included in the counting systems and the amount of travel on such roads cannot be estimated.
+
+For the calculation of exposure by different groups of road users: car drivers, car passengers, public transport passengers, cyclists and pedestrians the information from traffic counts does not suffice. Additional information is generally obtained from national travel surveys, which are conducted in many countries with a frequency of between 5-10 years.
+
+Data from different sources can be combined to calculate disaggregate exposure estimates for different population subgroups.
+
+At this stage in the report, it can be said that for each application, the correct exposure measure should be used. This is sometimes impossible because the required information is not available, or has to be collected at great cost. Generally, the more aggregate the exposure measure, the more intervening variables are introduced that cast shadows over the resulting risk calculations.
+
+#### 3.2. **Quantifying risk by hours of exposure**
+
+A number of authors have suggested that it is useful to calculate risk by hours of exposure to a certain activity (Evans, 1993; ETSC, 1999; Wesemann et al., 1998). In the case of transport, the most widely used measure of exposure is distance travelled, but it is obvious that even for transport the speed at which such travel is conducted might influence the risk. Speeds for various transport modes (walking, cycling, motorised transport) are widely different and therefore it has been suggested to normalise exposure (vehicle kms travelled) by multiplying by speed,
+
+thereby obtaining a risk measure which is expressed as accidents or casualties per hour of exposure.
+
+Collins (1990) provides details on accident fatality rates for various activities in transport and other fields of activity per passenger-hour and per passenger-km travelled (Table 1).
+
+| Transport and other activities | Fatalities per 100 million<br>passenger-hours | Fatalities per 100 million<br>passenger-km |
+|--------------------------------|-----------------------------------------------|--------------------------------------------|
+| Passenger travel by            |                                               |                                            |
+| Bus or coach                   | 1.4                                           | 0.06                                       |
+| Rail                           | 6.0                                           | 0.1                                        |
+| Car                            | 12.4                                          | 0.4                                        |
+| Water                          | 16.0                                          | 0.8                                        |
+| Air                            | 20.0                                          | 0.04                                       |
+| Foot                           | 27.0                                          | 7.0                                        |
+| Pedal cycle                    | 64.0                                          | 4.6                                        |
+| Two wheel motor vehicle        | 342.0                                         | 11.4                                       |
+| Employment                     |                                               |                                            |
+| All work                       | 0.9                                           |                                            |
+| Banking and financing          | 0.17                                          |                                            |
+| Chemical industry              | 1.1                                           |                                            |
+| Construction work              | 4.9                                           |                                            |
+| All railway work               | 5.6                                           |                                            |
+| Extraction of ores             | 13.0                                          |                                            |
+| Being at home                  |                                               |                                            |
+| All ages                       | 2.6                                           |                                            |
+| People under 65                | 1.0                                           |                                            |
+| People 65 and over             | 11.5                                          |                                            |
+
+Table 1. Accidental fatality rates for transport and other activities. In Evans (1993), from Collins (1990).
+
+It can be seen that travel is a relatively high risk activity compared with such other activities as being at work (most kinds of work) and being at home. Large differences can be observed in the risk associated with the various travel modes, with bus and rail travel being relatively safe both in terms of vehicle kilometres travelled and hours exposed, air travel being relatively safe in terms of kilometres travelled but not in terms of hours of travel, due to the high speeds of travel. Travel by vulnerable road users, on foot, pedal cycle and motor-cycle being relatively unsafe both in terms of hours and kilometres of exposure.
+
+Koornstra (in: ETSC,1999) also presents a risk comparison for various modes of travel aggregated for the whole of the European Union (Table 2).
+
+| Travel mode | 108<br>person km |            | 108      | person hours |
+|-------------|------------------|------------|----------|--------------|
+|             | absolute         | normalised | absolute | normalised   |
+| ROAD        |                  |            |          |              |
+| total       | 1.1              | 13.8       | 3.3      | 16.5         |
+| bus/coach   | 0.08             | 1          | 2        | 1            |
+| car         | 0.8              | 10         | 30       | 15           |
+| foot        | 7.5              | 93.8       | 30       | 15           |
+| cycle       | 6.3              | 78.8       | 90       | 45           |
+| mot.c/moped | 16.0             | 200        | 500      | 250          |
+| TRAINS      | 0.04             | 0.5        | 2        | 1            |
+| FERRIES     | 0.33             | 4.2        | 10.5     | 5.3          |
+| PLANES      | 0.08             | 1.0        | 36.5     | 18.3         |
+
+Table 2. Fatality risks over distance and time for travel modes in the EU. Source: ETSC (1999). Normalised rates calculated and added by the authors.
+
+Taking bus and coach travel as a normative unity, it can be seen that risk expressed in terms of person-kilometres travelled is 10 times greater for cars and 100-200 times greater for the vulnerable modes (foot, cycle and motor-cycle/moped), with motor-cycles clearly being the most hazardous. Train and plane travel are also relatively safe. Looking at the same data with risk expressed in terms of person-hours of travel, the picture does not change a great deal. Travel by foot or bicycle becomes slightly less hazardous, because of the low speeds of these modes, whereas air travel becomes significantly more hazardous because of the high speeds of travel (hence the small number of hours per km of travel. Chapter 5 of the report will discuss the usefulness of making such comparisons in more detail.
+
+Wesemann et al. (1998) suggested the use of risk per hour of exposure involved in various activities as a measure of comparison. Having collected data on accidents and fatalities at home, at work, in sports and in traffic from a variety of sources, it might be possible to calculate the risk of injury associated with these activities in terms of exposure in hours. However, for their study the detailed data needed to present such a comparison was not available. Instead, a much coarser comparison was conducted in terms of the number of individuals exposed to each activity (traffic and home). For the case of traffic accidents a calculation was presented comparing the risk of injury and fatality per distance and per hour of travel for various age groups, male and female (Tables 3 and 4).
+
+Comparing results of risk in each age group per kilometre travelled and per hour exposed does not change the picture drastically. The two groups that have the higher risks, by both measures of exposure, are the young (15-24) and the old (65+). When looking at risk per hour of exposure, the risk of the young group increases and the risk of the old group decreases somewhat. This seems to indicate that the young are exposed to the risks at higher speeds.
+
+| Age              | Male   | Female | Total  |
+|------------------|--------|--------|--------|
+| Fatalities       |        |        |        |
+| 15-24            | 0.1520 | 0.0512 | 0.1055 |
+| 25-39            | 0.0676 | 0.0297 | 0.0530 |
+| 40-49            | 0.0478 | 0.0271 | 0.0403 |
+| 50-59            | 0.0577 | 0.0359 | 0.0493 |
+| 60-64            | 0.0821 | 0.0594 | 0.0725 |
+| 65+              | 0.2445 | 0.1675 | 0.2059 |
+| Total            | 0.0884 | 0.0504 | 0.0730 |
+| Hospitalisations |        |        |        |
+| 15-24            | 2.1633 | 1.2094 | 1.7232 |
+| 25-39            | 0.8189 | 0.6016 | 0.7354 |
+| 40-49            | 0.6037 | 0.6072 | 0.6050 |
+| 50-59            | 0.7185 | 0.9057 | 0.7907 |
+| 60-64            | 0.7948 | 1.2733 | 0.9965 |
+| 65+              | 2.0844 | 2.5745 | 2.3300 |
+| Total            | 1.0559 | 1.0023 | 1.0342 |
+
+Table 3. Risk of fatality and hospitalisation per 10<sup>7</sup> traveller kilometres in the Netherlands, 1994-1996. From: Wesemann et al. (1998).
+
+| Age              | Male   | Female | Total  |
+|------------------|--------|--------|--------|
+| Fatalities       |        |        |        |
+| 15-24            | 0.3462 | 0.1077 | 0.2313 |
+| 25-39            | 0.2037 | 0.0620 | 0.1365 |
+| 40-49            | 0.1366 | 0.0493 | 0.0951 |
+| 50-59            | 0.1415 | 0.0603 | 0.1026 |
+| 60-64            | 0.1549 | 0.0734 | 0.1120 |
+| 65+              | 0.3395 | 0.1775 | 0.2474 |
+| Total            | 0.2210 | 0.0880 | 0.1553 |
+| Hospitalisations |        |        |        |
+| 15-24            | 4.9201 | 2.5474 | 3.7801 |
+| 25-39            | 2.4648 | 1.2577 | 1.8934 |
+| 40-49            | 1.7259 | 1.1049 | 1.4310 |
+| 50-59            | 1.7604 | 1.5216 | 1.6462 |
+| 60-64            | 1.4965 | 1.5780 | 1.5393 |
+| 65+              | 2.9030 | 2.7290 | 2.8040 |
+| Total            | 2.6389 | 1.7539 | 2.2021 |
+
+Table 4. Risk of fatality and hospitalisation per 10<sup>6</sup> hours of travel in the Netherlands, 1994-1996. From: Wesemann et al. (1998).
+
+#### 3.3. **The relevant measures of exposure and risk**
+
+In the context of risk calculations it is important to define and select the correct measure of exposure. The selection of what should serve as a 'correct' measure of exposure will depend mainly on the intended use. A large number of exposure variables can be selected, each suitable to a specific problem. In many cases, because of the lack of sufficiently detailed data, or the lack of accurate data, it is not possible to select the correct measure of exposure to suit a particular issue.
+
+#### 3.3.1. Aggregate measures of exposure and risk
+
+In international comparisons and for trend studies on a national level, in many cases, the number of inhabitants or the number of vehicles is selected as the available exposure measure. Risk calculated as the number of fatalities divided by the number of inhabitants can be considered as a measure of mortality, i.e. what is the chance of death per population unit? The advantage of this risk measure is that it uses fairly reliable data which are generally widely available. It is therefore possible to conduct such international comparisons. The same cannot be said when attempting to calculate the risk of injury. Between countries large differences exist in the reporting procedures of road casualties. Various levels of under-reporting of accidents and casualties exist in different countries (Nilsson, 1997). Defining risk as the number of fatalities (or casualties) per number of vehicles can be seen as a proxy to the risk of travel. In this context the number of vehicles is selected as a proxy to the number of vehiclekilometres travelled which is a variable that is much more difficult to obtain reliably in many countries. Obviously, if a certain type of vehicle is considered, one should use only relevant accident figures for the type of vehicle under consideration.
+
+One of the complications associated with this type of comparison is that the two risk measures- fatality per population unit and fatality per vehicle unitbehave very differently over time when comparing different countries. It should also be remembered that both measures are very comprehensive, not differentiating between different segments of the population (by age group, sex, urban or rural, etc.) and different types of vehicles. It is generally found that trends relating to the number of vehicles are a reasonable proxy to the number of vehicle kilometres travelled. In the literature, many studies exist reporting on risk comparisons conducted with these measures of exposure.
+
+Such comparisons are useful in obtaining a very coarse picture of the safety situation, without contributing much to the understanding of the differences found between countries or changes over the years. For such an understanding risk calculations should be conducted on a much more detailed level.
+
+Table 5 presents some examples of risk indicators on an aggregate level and their availability in international databases as well as the US database on fatal accidents (FARS).
+
+|                                             | WHO | IRF | ECMT | IRTAD | EURO<br>STAT | FARS |
+|---------------------------------------------|-----|-----|------|-------|--------------|------|
+| Accidents/inhabitants                       |     |     |      | +     |              |      |
+| Accidents/VehKm                             |     | +   |      | +     |              |      |
+| Accidents/VehKm I/O built-up areas          |     |     |      | +     |              |      |
+| Accidents/VehKm by road class               |     |     |      | +     |              |      |
+| Killed/inhabitants                          | +   |     | +    | +     | +            | +    |
+| Killed/inhabitants by age group             | +   |     |      | +     | +            | +    |
+| Killed/inhabitants by age group & sex       | +   |     |      |       | +            | +    |
+| Killed/licensed drivers                     |     |     |      |       |              | +    |
+| Killed/million vehicles                     |     |     | +    |       |              | +    |
+| Killed/million road users by road user type |     |     | +    |       |              |      |
+| Killed/VehKm                                |     | +   |      | +     |              | +    |
+| Killed/VehKm I/O built-up areas             |     |     |      | +     |              |      |
+| Killed/VehKm by road class                  |     |     |      | +     |              |      |
+| Injured/inhabitants                         |     |     |      | +     | +            | +    |
+| Injured/inhabitants by age group            |     |     |      | +     | +            | +    |
+| Injured/inhabitants by age group & sex      |     |     |      |       | +            | +    |
+| Injured/ licensed drivers                   |     |     |      |       |              | +    |
+| Injured/million vehicles                    |     |     |      |       |              | +    |
+| Injured/Vehkm                               |     |     |      | +     |              |      |
+| Injured/VehKm I/O built-up areas            |     | +   |      | +     |              |      |
+| Injured/VehKm by road class                 |     |     |      | +     |              |      |
+
+Table 5. Available risk indicators. From: DUMAS (1998).
+
+#### 3.3.2. Traffic volume as a measure of exposure
+
+Some form of traffic flow is generally taken as the measure of exposure when calculating risk on a certain part of the road network or for a certain type of vehicle. The most frequently used measure of exposure is the number of vehicle kilometres travelled. Details on the collection of data on vehicle kilometres travelled were given in Section 3.1 of the report. A more detailed discussion on the relations between accidents and traffic volumes will be given in Section 6.1 of the report. Risk levels on various parts of the road network or at various types of intersections are deduced from the ratio between accidents, casualties or fatalities and the relevant exposure measure related to traffic flow.
+
+#### 3.3.3. Problems with quantifying exposure
+
+Most countries collect data on traffic volumes in a systematic way. These would almost certainly include traffic count data that are needed for decisions on building a new road, widening an existing one or re-designing an intersection. Some countries also have aggregated traffic counts for some parts of the road network (generally the major roads). A few countries have aggregated traffic counts which can be converted to reliable numbers of vehicle kilometres travelled, but these are subdivided for various parts of
+
+the urban and non-urban network. When such traffic censuses are available, they can give us the necessary information on exposure, by type of road and type of vehicle, from which detailed risks can be calculated. In many other countries the alternative to a traffic census, is to measure fuel consumption on a national scale and to calculate the amount of travel on the basis of this measurement. Such calculations are open to many different error sources and generally lead to only rough estimates of the number of vehicle kilometres travelled. Such estimates can also not be subdivided into types of road. These are the major difficulties in obtaining exposure measures on a national scale. For more detailed studies, relating to a specific research question, it is generally required to collect special and dedicated exposure data. This is an expensive process and is not always feasible.
+
+Further information can be obtained from travel surveys. Such surveys, conducted either at home or along the road, can supply further information on the number of trips people make, trip purposes, travel modes and details about the travellers. Some countries have such travel surveys and conduct them periodically every 5-10 years. Even in those countries, travel surveys generally only apply to the adult population and generally do not include trips on foot.
+
+The lack of detailed and high quality exposure data is one of the reasons that in so many cases international comparisons are conducted on a percapita or per-vehicle basis.
+
+A recent report by the US Bureau of Transportation Statistics (BTS) discusses transportation data and risk measures (BTS, 2001). It states that:
+
+"More accurate, comprehensive and consistent measures of risk exposure could help our understanding of the relative importance of factors contributing to transportation crashes and improve our analysis of safety trends. Better exposure measures require data not only on the numbers of fatalities, injuries and accidents, but also data that indicate the overall level of transportation activity. Analyses of safety trends for non-motorised modes- bicycling and walking- suffer from the absence of exposure measures (such as hours of exposure to traffic). Moreover, bicyclists and walkers often take trips too short in length to be counted in national travel surveys. Furthermore, trips that begin and end at a residence, without an intermediate stop, are typically not counted, thus excluding much recreational bicycling and walking. Yet, more than 6,400 pedestrians and bicyclists died in 1995 in crashes involving motor-vehicles (15 per cent of all transportation fatalities)" "There is also a need for exposure information on specific populations (e.g children or elderly drivers). For example, inadequate exposure data on children under five years of age makes the evaluation of some transportation risks difficult. Moreover, better exposure measures and incident data are needed for evaluating risks associated with the transportation of hazardous materials…." (BTS, 2001).
+
+#### 3.3.4. The possibilities with new information technology (IT)
+
+### Traffic volumes
+
+One of the developments in recent years is the installation of electronics and telecommunications inside vehicles and along the roads. The increasingly widespread use of mobile phones also presents many new possibilities. As a result it is becoming easier to collect up-to-date and reliable information on a variety of parameters that could be of importance in the calculation of vehicle exposure and risk. The amount of traffic flow can be determined by a number of IT technologies which are becoming cheaper each year. Above-ground vehicle detectors will in time become more prevalent and will become part of an ever-expanding system of traffic monitoring and control. Such systems are already well on their way along
+
+motorways and will probably also be introduced on other types of roads and at intersections. Eventually it should be possible to develop a reliable system of traffic volume measurement on a road network-wide basis. Additional information on the distribution of speeds, types of vehicles and following distances also seems to be a future possibility.
+
+### Traffic conflicts
+
+The study of traffic conflicts or critical situations has been carried out since the 1960's and is sometimes proposed as an intermediate estimator of risk. The argument provided is that at most locations the number of accidents is (fortunately) very small and subject to large chance variations. To be able to form an opinion on the level of safety of such a location, or the assessment of the effect of a safety improvement in a short period of time, it is important to have a method for the establishment of risk. The measurement of conflicts (which can be regarded as a proxy to accidents) is one practical method in this context. Flows can thus be related to conflicts, which are measured over a short period of several days, instead of being related to accidents, for which a number of years of data are needed. One of the drawbacks of the conflict methods (Brundell-Freij & Ekman, 1990) is that conflict measurement is a highly complicated and expensive method of data collection. With the advances in video observation, pattern recognition and automatic image processing it is becoming increasingly feasible to use automatic means of traffic observation to detect critical situations, such as conflicts or even accidents. These techniques are already being applied in systems of incidence detection on some motorways around the world.
+
+#### 3.3.5. Traffic exposure at intersections
+
+On road sections, the number of vehicle kilometres is generally accepted as the relevant measure of exposure. At intersections, the correct measure of exposure is not that obvious, and has been the subject of many discussions.
+
+It seems reasonable to suggest that some combination of the traffic flows on the intersecting roads is needed as the measure of exposure. The simplest combined measure is the sum of the daily traffic volumes. This is also a measure which is generally available, if traffic volumes are measured at all. But many other combinations are possible. The product of the daily flows, the sum of the products of the hourly flows, the sum of the products of the hourly flows- but subdivided into separate turning movements, etc. How does one decide on the correct measure of exposure, assuming that the right data can be made available. From logic, a multiplicative measure of flows seems more logical than an additive one. 999 Vehicles on one road interacting with one vehicle crossing on the other road would result in 1000 interactions according to the additive rule, as would 500 vehicles interacting with 500 crossing vehicles. According to the multiplicative rule, however, the first situation would result in 999 interactions, and the second one in 25,000 interactions, which seems a more reasonable result.
+
+One way of establishing what is reasonable, is to calculate various indices of exposure and rank them for different intersections. The average number of accidents at those intersections could be recorded (over a period of some years) and also ranked. The measure of exposure that produced the highest correlation could be accepted for our purposes- being accident investigations (Mahalel & Hakkert,1976). The product of the daily crossing flows
+
+and the sum of the products of the hourly crossing flows have been used as exposure measures in many accident studies.
+
+This topic becomes even more complicated when one has to consider the correct exposure measure at roundabouts and at signalised intersections, where for obvious reason the logic described above does not hold.
+
+#### 3.3.6. Pedestrian exposure and risk
+
+In a way which is fairly similar to the discussion of risk and exposure at intersections one can consider the definition of risk to pedestrians. First one has to differentiate between pedestrians moving along the sidewalk (pavement) and pedestrians crossing the road. For pedestrians moving along the sidewalk a measure of exposure similar to vehicle kilometres on a road section - but in this case pedestrian kilometres - could be adopted. For pedestrians crossing the road, obviously the measure of exposure is linked to the number of pedestrians crossing the road. However it should also be some measure of the volume of vehicular traffic along the road. In this manner one obtains a measure of exposure which is the product of the traffic flow and the number of crossing pedestrians. Older & Grayson (1976) used PxV as a measure of exposure, P and V being the hourly numbers of pedestrians and vehicles on a certain road section. The risk was then defined by dividing the annual number of pedestrian accidents by the exposure. The reason accidents are counted annually and volumes hourly, is that the exposure has to be measured especially for such risk studies and is generally measured for a few hours only. Hakkert & Bar Ziv (1976) calculated various risk functions for pedestrians crossing under different road conditions- on zebra crossings, near zebra crossings, away from the crossing, at signalised crossings and near intersections. The proposed method seems to be reasonable for comparative purposes, i.e comparing the risk of crossing under different road conditions, but it remains to be proven that this technique by linking annual accident data with hourly volumes is robust enough. It also immediately becomes obvious that such calculations require special data collection efforts because the data needed are not generally available.
+
+Another way of looking at pedestrian injury rates is proposed by Roberts, Norton & Taua (1996). In their study the number of streets crossed per child per day was selected as measure of exposure. Data from 2873 completed questionnaires were classified by number of cars in the family and by socioeconomic background. It was found that children from poor families and children from families without a car crossed a considerably greater number of street, compared with families with one car and even more so, compared with children from families with two cars. The values ranged between 7.4 street crossings per day to 1.7 crossings. As has been discussed above, it seems that this exposure measure is not perfect, not taking into account the pedestrian and vehicular volumes at each crossing, but it certainly provides for a better indication of risk than just calculating the number of accidents per child.
+
+#### 3.4. **The use of fN diagrams**
+
+The results of calculations on risk, especially in high-risk technologies, are often expressed in fN diagrams, which show the expected frequency f of an event that has at least N fatalities as a function of this number of fatalities. Figure 2 shows calculated fN curves for a number of high-risk technologies. This, in principle, enables a comparison of risks for society as a whole.
+
+The graph starts with events with ten or more fatalities per year. Aviation appears highest on the curve with ten events with ten fatalities or more per year in the US. Rail transport comes lower down the scale with one event per two years (0.5). Lower still appear various types of transport of hazardous materials (LPG, chlorine, LNG). Except for rail transport, the order of the curves does not change when taking events with greater numbers of fatalities.
+
+![](_page_21_Figure_3.jpeg)
+
+Figure 2. Calculated fN curves for high risks in the US. In: Van Poortvliet (1999) from Cappola and Hall (1981).
+
+For the various transport modes (road, rail, civil aviation and merchant shipping) the graph from Evans (1994) is illuminating. Figure 3 presents data for Great Britain for the years 1963-1992.
+
+![](_page_22_Figure_0.jpeg)
+
+Figure 3. fN-curves for British transport accidents: 1963-1992. From Evans (1994).
+
+The total number of casualties and fatalities accumulated through the occurrences of a large number of events with only one or two fatalities is much higher for road transport than for the other modes of transport.
+
+Because of the distribution of the number of fatalities per event in road transport in particular, the use of fN curves is not that helpful. The large majority of fatal road accidents incur one fatality or two. Events with many fatalities are rare and almost invariably capture the media's and politicians' attention with the result that actions are considered and taken which would not have occurred, had the same number of fatalities been spread over a large number of events. This points to a lack of rationality in terms of decision-making on the basis of cost-benefit considerations. In the other transport modes the frequency of events with small numbers of fatalities is much lower. Only rail industry has a considerable number of events with one death or more (around one hundred), compared with over five thousand in road transport. The number of large scale disasters with tens to hundreds of fatalities is reversed for the different modes. Merchant shipping had an event with 189 fatalities once in a hundred years, civil aviation had an event with 146 fatalities once in a hundred years. The corresponding figure for rail was 49, and for road transport 32.
+
+# 4. **Some issues associated with the use of exposure and risk**
+
+#### 4.1. **The desire to assess risk levels and to strive for equal risks**
+
+According to a recent ETSC report it is important to collect exposure data for risk assessments that can be used to improve transport safety. ETSC defines the following ways to conduct such assessments:
+
+- "monitoring casualty trends and evaluating policies which have been introduced to improve safety, in order to provide a sound basis for developing new policies
+- comparing the levels of risk of different types of travel ( for example by transport environment or transport mode)
+- setting priorities, that is identifying those transport situations with high levels of risk in order to formulate policies and concentrate resources to reduce risk levels, especially for high severity crashes (ETSC,1999)."
+
+The safety policy for each mode of transport should be similar: research, develop and apply cost-effective countermeasures that have the potential to reduce crash risk and/or severity.
+
+A basic assumption is that it is desirable to strive for equal risk to various categories of road users, including the vulnerable road user, in order to achieve equal risk on various categories of roads. Unequal risks on different categories of roads and for different categories of road users do, however, exist and it is almost impossible to envisage that such risks can be made equal at reasonable economic costs. Similarly, equal risks in various parts of the country, municipalities, regions, and provinces are almost certainly unachievable at reasonable economic costs. Even the calculation of such risks is almost impossible because of the lack of reliable and detailed exposure data on the provincial and municipal levels. If good data is not available, one resorts to the usual levels of aggregation-
+
+national, provincial or municipal- per inhabitant, per kilometre of road, etc.
+
+At the regional and municipal levels there is also the problem of interpretation. Some authorities have high levels of risk but have done a lot of work on road safety, others have done little. What is the correct measure of risk? At a high level of safety, the additional effort required to achieve more safety is much greater (more costly) than at lower levels of safety, and so positive results are more difficult to achieve. A further complication arises from the fact that equal levels of risk are unrealistic if conditions are not equal. Large municipalities generally also have higher population densities and have more congestion than smaller towns. As a result they will experience lower numbers of fatalities and seriously injured victims per capita. Similarly, provinces or regions that are highly urbanised will have higher population densities and experience lower levels of fatalities and serious injuries. This issue will be discussed in further detail in Section 4.6 which considers the cost-benefit considerations of safety measures and in section 6.1 which discusses accident rates and traffic flows.
+
+In this context one should ask whether it is correct to strive for equal risk or for highest benefit-cost returns on safety improvements. Striving for equal risks on various parts of the road network or equal risks to various groups of road users sounds politically correct, but is, in all likelihood, impossible to achieve at reasonable economic costs. Urban roads will have fewer fatalities per kilometre travelled than rural roads, because of the lower speeds of travel. Motorways will have fewer fatalities per vehicle kilometre travelled than other types of rural roads, because of the level of segregation between different modes of transport (no slow vehicles) and the segregation of movements (no at-grade intersections). It would be more fruitful to try and find ways in which each category of road and road user is made safer by applying cost-effective countermeasures.
+
+More generally, and along similar lines of thought, when dealing with risk comparisons between different activities- is it really worthwhile to strive for equal risk levels for citizens in a variety of occupations? Is it reasonable to expect a factory worker to be exposed to the same level of risk as the bank employee? Looking at the risk and exposure data collected in Evans (1993), this clearly does not seem to be the case. In banking and financial employment there are 0.17 fatalities per hundred-million hours of work, compared with a rate of 4.9 in construction work and 13.0 in mining. In transport modes (under certain rough assumptions of travel speed) the risk of a fatality per hundred-million passenger hours is 1.4 in a bus or coach, 12.4 in a car, 27.0 on foot and 342.0 on a motor-cycle. There is no realistic way in which it will be possible to achieve equal risks for these modes. This concept is considered 'unfair' by some. Others on the other hand do not think there is any unfairness (Ekman, 1996). No person who considers riding a motor-cycle would be doing so under the assumption that it will be as safe as riding a bus, or as safe as travelling in a car. What could be asked however, in such circumstances, is what safety measures could be taken that would improve the level of safety of motor-cyclists and would still be considered cost-effective. Some of these measures would also apply to other types of road users. An example of such a measure is making car fronts safer, which would lead to lesser injury severity when crashes with motor-cyclists and other road users occur.
+
+#### 4.2. **Over-representation and risk**
+
+Much of the work on risk in safety studies seems to stem from accepted and established procedures taken from the field of epidemiology as practiced in medical and health related studies. From epidemiology a logic is adopted which seems to be the basis for the use of exposure and the calculation of risk (Hauer, 1995a). It is the search for over-representation in various populations under study that lies at the basis of this approach. Searching for over-representation seems to suggest the following line of thought and action:
+
+[deviation-from-normal] leads to [clue-to-cause] leads to [clue-toremedy]
+
+Many classical epidemiological-medical studies have successfully taken this approach and in some cases led to major achievements.
+
+One underlying requirement in this chain of thought is a further assumption which is the "presumption-of-sameness-except-for-the-suspected-cause" (Hauer,1995a; Ekman, 1996). In medical studies, this might well be a legitimate assumption in many cases. Hauer (1995a) presents the example of John Snow, who established in 1854 that people who live in the same districts, but whose water is supplied by different companies, had vastly different incidences of cholera, many years before the discovery of the vibrio of cholera. From this Snow could infer what the cause of the cholera epidemic was and what the remedy might be.
+
+In transportation safety studies, the presumption of sameness hardly ever holds good. In the comparison of accident and fatality rates for different modes of transport (air travel vs. road travel), rates for different modes of road transport (lorries, cars, motor-cycles, etc.) it is definitely wrong to assume sameness. By itself, over-representation in accidents or fatalities has no discernable logical link to funding, programming, decision or action. The decision-making process should be concerned about establishing the relevant facts, adopt a set of proven countermeasures and base the decision on the cost effectiveness of the preferred solution.
+
+A subject that has received a lot of attention in recent years is the issue of older drivers and their 'over-representation' in accidents or their higherthan-average risk (Hauer,1995a). The motivation behind this seems to be related to the fact that in many countries society is ageing, and that, therefore, something should be done about this over-representation. Indeed, when the rate is calculated 'per mile driven', drivers in the 75+ group have trice the rate of drivers in the 35-40-year-old age group. Yet, when looking at the numbers, 1486 drivers who were 75-80 years old died in the US in 1986, whereas 3203 drivers in the 30-35-year-old age group died in the same year. This indicates that in terms of absolute numbers, the focus should be on the 30-35-year-old group. It seems perfectly clear that the quest for countermeasures should be driven by improving the safety record for the older driver. This could be done by adapting the highway infrastructure to better suit the older driver, or by other means that could be implemented in a cost-effective manner. The knowledge about the higher involvement rate does not help us further in this matter.
+
+#### 4.3. **Doing without information on risk and exposure - an example**
+
+The following study on the effect of seat belts can serve as an example that demonstrates that it is not always necessary to use exposure data and calculate risks to measure the effect of a safety measure. Yet, the example also shows that omission of exposure data may result in inexplicable outcomes or mistaken conclusions.
+
+In 1983 a law requiring the use of safety belts in cars came into force in Great Britain. Subsequently, Professors Harvey and Durbin from the London School of Economics were asked to evaluate the safety effect of this law. Their results are described in Harvey & Durbin (1986). The authors looked at a time series of fatal and seriously injured car drivers and passengers over a period of fifteen years starting in 1969 and continuing till December 1984, two years after the introduction of the law (Figure 4).
+
+![](_page_26_Figure_0.jpeg)
+
+Figure 4. Car drivers killed and seriously injured in the UK, 1969-1984. From: Harvey and Durbin (1986)
+
+Car passengers were further split into front seat passengers who were directly affected by the law and rear seat passengers. Applying statistical structural time-series models the authors were able to model the series for the before period and estimate the effect of the law by comparing the actual fatality and serious injury figures with their estimate of what those figures would have been without the introduction of the law. Their estimate of the effect was a significant reduction of 23 per cent in the numbers killed and seriously injuries for car drivers and 30 per cent for front seat passengers. For rear seat passengers, however, a small increase of 3 per cent was found. For the number of persons killed similar large reductions in the number of fatalities were found for car drivers and front seat passengers, but the number of rear seat passengers killed increased by 27 per cent. The authors could not find a satisfactory explanation for this increase. One explanation put forward was that there was the possibility that after the introduction of the law, which did not affect rear seat passengers, a certain number of passengers transferred from the front seat to the rear seat. It is exactly this type of question and outcome that could have been answered in detail, if at the time a series of exposure measurements had been in place counting, over time, the number of passengers in front seats and in the rear of cars.
+
+Risk can serve as the basis for the selection of research and action themes which should then be justified on the basis of cost-benefit considerations. Decision-making can be carried out on the basis of comparative risk levels, on the basis of accident and injury data (without risk) and according to costbenefit considerations. Each method has advantages and disadvantages and should be selected according to the circumstances. Decision-making on the basis of the numbers of accidents, injuries or fatalies focuses on the size of the problem and prevents us from focusing our efforts on a segment of road safety that might look bad from the point-of-view of risk but which
+
+might not be very significant in terms of the quantities involved. Decisionmaking on the basis of risk tends to lead to actions that are considered fair in terms of the road users involved. If a segment of road, or a certain type of road, or a group of road users has a risk that is higher than the risk for another group or higher than average, this is considered unfair and decision-makers are urged to adopt policies that would lead to an absolute and relative decrease in risk for that group. In practise, such decisions are not reasonable, not possible and almost certainly not economical. Motorcycle riders have higher risks than car occupants, pedestrians have higher risks of dying from an injury than car occupants, regional and local roads have higher levels of risk than motor-ways and so on. The final decision should generally be based on cost-effectiveness considerations because, in the field of road safety, like in any other field, the amount of expenditure is limited and investments should be allocated to those fields that produce the highest accident, casualty and fatality reduction that can be achieved per unit of money invested.
+
+#### 4.4. **National safety levels and targets**
+
+National road safety targets are generally set in terms of numbers of accidents, casualties and fatalities, or percentage reductions to be achieved. This has been the case in The United Kingdom, The Netherlands, Sweden, Australia and many other countries. There are several reasons for this approach. First, in terms of the impact on the population, it is the absolute numbers that count. Focusing on risk might show a reduction in risk which might be the result of an increase in exposure being greater than the increase in accidents or fatalities. This would thus not lead to an absolute reduction in the number of casualties or fatalities. Targets based on risk levels are also undesirable because of the fact that risk levels in general tend to decrease over time because of the whole range of road safety activities and because of inherent changes in the traffic volumes and composition. Once having set targets in terms of numbers, it is however possible to use risk levels to check for changes in certain areas, regions or municipalities. If the number of casualties or fatalities in a region is increasing faster than average, it is worth looking at the exposure and risk to study the differences before reaching conclusions.
+
+When dealing with trend analyses of road safety phenomena, it might be possible to work with the rough accident and injury data for short-term prognosis. On the basis of past trends, when dealing with before-after evaluations, it is generally possible to apply time series models and intervention-type models to predict the short-term effect of a countermeasure using accident and casualty data. In the case of long-term projections, 5-10 years onwards, it is generally necessary to first estimate the underlying trend in exposure data before making a prognosis.
+
+#### 4.5. **What is an acceptable level of risk?**
+
+In a general way, the question of what is an acceptable level of risk in road traffic is virtually impossible to answer. Any limit other than zero risk is arbitrary and disputable. Zero risk, however, may be a valid ambition in theory, but hardly possible, if at all, to realise in practice. Furthermore, when talking about the acceptable level of risk a distinction must be made between various types of risk, for example individual risk versus societal
+
+risk and voluntary risk versus compulsory risk. What risk level is acceptable for an individual road user when he or she makes a particular trip with a particular transport mode? What is the acceptable risk level for society as a whole, in other words how many accidents or casualties per year per unit of exposure are we as a society willing to accept at public roads? And is it reasonable to accept higher risks when these are taken on a voluntary basis (e.g. a recreational bicycle tour) than when taken on a compulsory basis (e.g. for professional drivers or home-work travel). Some of these questions are answered by society, through actions that government and local agencies take on the programs and budgets devoted to road safety. There are no absolute and correct levels, as becomes obvious when looking at the actions taken in different countries, which result in them having different levels of risk and safety. At the individual level there are also no correct and absolute answers. Individual risk varies from person to person, from task to task and can even vary for the same person depending on mood, timepressure and whether the risk is voluntary or compulsory.
+
+An interesting issue related to risk and acceptability considerations is raised by Erskine (1996). Erskine makes the distinction between risks to those who have access to cars and those who do not have access. He points out that, although risks associated with the use of cars, in terms of both the number of casualties and fatalities per vehicle- km travelled, have been decreasing considerably over the years, it is the parts of the population that do not have access to cars, i.e. the poor, the young, the old and women (i.e the nondriving population) that are adversely affected by the increased exposure to motorised travel. In order to come to more socially responsible decisions, the needs of these separate populations should be taken into account.
+
+Along a slightly different line of thought, one could also make the distinction between risks involved as a passenger or driver of a certain type of vehicle, and the risks associated with the use of such a vehicle to other road-users. In this way, lorries are a very safe means of transport to their occupants, because very few fatalities occur to drivers and passengers of lorries, both in absolute numbers and expressed as fatalities per vehicle-km travelled, but they are involved in a proportion of fatal accidents which is considerably larger than their share in vehicle kilometres driven. The same is the case for buses. Cars are also becoming safer and safer for their occupants, because most technological innovation and development has been applied to improve the safety of the occupant: US and European safety regulations, seat belts, air bags, etc. Thus the risk of being injured or killed per vehicle-km travelled continually decreases. However, the issue of injuries and fatalities caused by these vehicle to the vulnerable road-user are only now being addressed and still there are few technological developments that are effective.
+
+Similarly, relevant for the transport of dangerous goods in particular, one could make a distinction between the accident or injury risk for road users and the risk for third parties, for example for those living in the area and being exposed to toxic chemicals following a road accident with a dangerous goods truck. In the Netherlands this distinction is made in the external safety report which is required for the dangerous goods transport industry. As an indicative value, the third party risk in this type of industry should not exceed 10-4 for events resulting in 10 victims and 10-6 for events resulting in 100 victims per year per kilometer route (RIVM, 1998).
+
+### 4.6. **Risk versus effectiveness**
+
+So far the issue of the economic costs associated with countermeasures that aim to change levels of risk and thereby the number of casualties or fatalities in the transport system has not been dealt with in this report. Riskbenefit assessment has been applied in such diverse fields as medicine, public health and environmental protection. The methods used in these fields might also be applied to the field of motor-vehicle safety (Thompson, Graham & Zellner, 2001). Risk-benefit analyses can be categorised according to the situation and may include cases where:
+
+- neither risks nor benefits can be readily quantified;
+- risks can be quantified, but benefits cannot readily be quantified (or vice versa);
+- risks and benefits are quantified in very different terms or units;
+- risks and benefits are quantified in similar terms.
+
+It is this last category which has been focal in medical and pharmaceutical research and which is of great interest to the transport safety field, especially when considering the introduction of road safety devices or other countermeasures. Thompson, Graham & Zellner (2001) discuss a number of quantitative approaches to deal with this issue.
+
+One approach (QALY) calculates the quality-adjusted life years associated with a certain safety measure. QALYs combine information on duration of life and health-related quality of life into a numerical index. This technique is widely used in the medical field to assess new drugs or medical interventions and could be applied in the road safety field as well.
+
+A slightly different technique, called DALY is promoted by the World Bank and the World Health Association (WHO). DALY also combines information on duration of life and quality of life into a numerical index that can be aggregated across people. The DALY approach evaluates health states in terms of a single dimension, degree of disability, whereas QALY addresses a wide range of dimensions of quality of life.
+
+A third approach which has been suggested calculates the Normalised Injury-Fatality Costs (NIC). It is a research-based tool which assumes that all health-state values and preferences are reflected in the actual lifetime costs of medical treatment, ancillary costs (which include costs of permanent partial incapacity) and fatality costs to society. NIC intentionally neglects so-called 'pain and suffering' costs, as these were believed to be subject to enormous variations.
+
+Despontin, Verbeke & Brucker (1997) suggest that decision-makers should approach the issue of road safety countermeasure evaluation in a twodimensional way as illustrated in Table 6.
+
+| Factor contributing to<br>accidents / safety<br>characteristics | Road user                                                                                             | Vehicle                                                                                                                                         | Road                                                                                                                    |
+|-----------------------------------------------------------------|-------------------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------------|
+| Exposure                                                        | - Separation of different road<br>user categories<br>- Changes in number of<br>trips-                 | -<br>Regulation of vehicle<br>traffic                                                                                                           | -<br>Regional planning of<br>traffic control<br>-<br>Road improvements<br>reducing travel distances                     |
+| Risk                                                            | - Improvement of education,<br>information and road user<br>behaviour in relation to<br>traffic rules | -<br>Crash avoidance<br>measures: vehicle speed<br>regulations<br>-<br>Vehicle equipment<br>(studded tires, running<br>lights in daytime, etc.) | -<br>Improvements on roads,<br>streets or traffic network<br>-<br>Speed reduction signs<br>-<br>Road/street maintenance |
+| Accident consequence                                            | - Individual protection<br>equipment (seat belts,<br>helmets)<br>- First aid education                | -<br>Improved crash tolerance                                                                                                                   | -<br>'Soft' road side design<br>-<br>Alarm telephones                                                                   |
+
+Table 6. A classification of road safety measures. In: Despontin, Verbeke & Brucker (1997) from OECD (1981).
+
+Safety measures are classified on the basis of two criteria. The first criterion is related to 'factors contributing to accidents'- the road user, the vehicle and the road. The other criterion relates to the safety characteristic changed, namely exposure, risk or accident consequence. The final decision on which safety countermeasure should be preferred should in each case be taken on a cost-benefit basis.
+
+Elvik (1997) also thinks that cost-benefit analysis will become increasingly important in the field of road safety decision-making. One reason is the limited financial resources that will be available for road safety. Funds should therefore be applied as efficiently as possible. Another reason is that more and more often a choice will have to be made between competing objectives, which could be environmental or based on mobility, energy or safety. Converting conflicting objectives to an economic yardstick is one way of treating problematic decisions.
+
+One underlying dilemma has not been adequately resolved. On the one hand, in terms of accident or fatality risk, the transport system becomes safer as the years go by. In almost all countries for which reliable statistics are available, risks have been decreasing over time. At the same time, exposure, measured in vehicle kilometres travelled has continued to increase in most countries. The result is that in many cases the absolute number of accidents, and in some cases the number of fatalities have continued to increase. As to effectiveness it could be said that the transport system has become more efficient in terms of the number of accidents or fatalities it produces for each unit of exposure, however, in overall terms of safety, the system has become less safe by producing more casualties, or fatalities in some countries, as a result of the increased exposure. To produce further safety improvements it should therefore be attempted to reduce levels of exposure by modes of transport that have high risks associated with them and convert some of that exposure to modes that
+
+have lower risks. Even that approach should be regarded with caution as the following two examples will demonstrate.
+
+The first example is taken from Evans, Frick & Schwing (1989). It is often stated that the most dangerous part of a trip by plane is the trip to the airport. Evans, Frick & Schwing show this not to be the case. When comparing air travel with road travel, the risks look very much worse for road travel. Death rates per billion miles travelled are 0.6 by air, compared to 24 by road. But, there are three reasons why such a comparison is inappropriate. First, the airline rate is 'passenger fatalities per passenger mile', whereas the road rate is 'all fatalities (including occupants, pedestrians, motorcyclists, etc.) per vehicle mile'. Second, road travel that competes with air travel is mostly done on rural freeways, which are among the safest part of the road network. Third, driver and vehicle characteristics vary over a wide range of risks. In their study, Evans, Frick & Schwing make a number of assumptions and show that a 40-year-old, belted, alcohol-free driver of a car which is 700 pounds heavier than average is slightly less likely to be killed in a 600-mile-trip on the rural interstate freeway system than in a regularly scheduled airline trip of the same length. For 300 mile trips, the driving risk is about half that of flying.
+
+In discussing costs and benefits associated with road safety measures two approaches are possible and are in widespread use. The first approach evaluates all the costs associated with the introduction of a certain safety measure and compares these with the benefits achieved from the reduced number of accidents, casualties or fatalities expressed in monetary values. The second approach does not convert the reductions in accidents and casualties into monetary values, but compares the numbers of accidents saved with the costs involved. This is the cost-effectiveness approach. In cost-benefit calculations different values are adopted for accidents, casualties of varying degrees of severity and for fatalities in different countries. Even the theoretical methods adopted in the different countries are not the same. Some are adopting the gross loss of output approach, some the net loss of output approach and some adopt the approach considered more advanced and correct, the willingness-to-pay approach. This approach is considered by some to be un-ethical and therefore to be avoided. It is however wrong to assume that by taking the cost-effectiveness approach such issues can be avoided. If one safety scheme prevents a certain number of casualties of varying degrees of severity and another scheme prevents a different mix of severities at a different cost, the decision still has to be made how to weigh these different mixes. The monetary conversion adopts one particular mix.
+
+# 5. **Some examples of the use of risk calculations in a number of road safety applications**
+
+Having established that there is no universal treatment of exposure and risk in safety studies and evaluations, it is illuminating to present some examples of the way in which exposure and risk should be used in a number of applications. The first examples are taken from the safety literature. These are followed by a number of examples taken from a number of studies conducted by researchers at SWOV.
+
+#### 5.1. **The involvement of lorries in fatal accidents**
+
+It is well known and documented that lorries are involved in a considerable percentage of fatal accidents in most countries. In Sweden, in 1992, out of the total number of fatalities- 759 in the year 1992, 132 fatalities (of which seven lorry occupants) occurred in collisions with lorries and another seven fatalities occurred in single vehicle lorry accidents (Table 7).
+
+| Killed as      | Killed in          |              | In collision with |     |    |                |         |        |       |      |       |     |
+|----------------|--------------------|--------------|-------------------|-----|----|----------------|---------|--------|-------|------|-------|-----|
+|                | single<br>accident | Pass.<br>car | Lorry             | Bus | MC | Mop./<br>cycle | Tractor | Animal | Train | Tram | Other | Sum |
+| Car occupant   | 195                | 133          | 80                | 16  | 1  |                | 3       | 14     | 14    | 10   |       | 466 |
+| Lorry occupant | 7                  | 2            | 7                 |     |    |                | 1       |        |       |      |       | 17  |
+| Bus occupant   | 2                  | 0            | 0                 |     |    |                |         |        |       |      |       | 2   |
+| Motor cyclist  | 19                 | 10           | 1                 |     |    |                |         | 3      |       |      |       | 33  |
+| Mopedist       | 2                  | 7            | 7                 |     | 1  |                |         |        |       |      |       | 17  |
+| Bicyclist      | 9                  | 41           | 20                | 3   | 1  | 2              |         |        |       |      |       | 76  |
+| Pedestrian     | 0                  | 92           | 17                | 7   | 2  | 3              | 5       |        |       | 4    | 8     | 138 |
+| Tractor        | 1                  |              |                   |     |    |                |         |        | 3     |      |       | 4   |
+| Others         | 5                  | 1            |                   |     |    |                |         |        |       |      |       | 6   |
+| All killed     | 240                | 286          | 132               | 26  | 5  | 5              | 9       | 17     | 17    | 14   | 8     | 759 |
+
+Table 7. Fatalities related to vehicles or road users and accident types in Sweden, 1992. From Nilsson, 1997).
+
+From the absolute numbers it would seem that, on the one hand, lorries are a very safe mode to their occupants, with only 14 fatalities per year, compared with 466 killed car occupants, 138 killed pedestrians or 76 killed cyclists. On the other hand 132 fatalities in collisions with lorries is 17% of the total number of fatalities and is a much higher proportion than the number of lorries in the total vehicle population. This seems to lead to a number of possible conclusions.
+
+- It is likely that lorries drive a much greater distance per year compared with other types of vehicles.
+- It is likely that because of their greater mass, once an accident occurs, the outcome will be more severe and the injured person is most likely to be in the other vehicle or to be a vulnerable road user.
+- It is possible that lorry drivers are bad drivers and are therefore involved in a higher proportion of collisions than their share in the vehicle park.
+
+Without additional information on the exposure background of lorries and other types of vehicles it is not possible to come to the correct conclusions. Even with the correct exposure information for the various types of vehicles it is very difficult to obtain reliable exposure information for the vulnerable road users, the pedestrians and the cyclists, and even when that is available, from travel surveys or other means, it generally applies only to the adult population and not to children.
+
+One additional complication manifests itself in the calculation of involvement rates (risks) of lorries. This issue was recently treated by Hauer (2001). Hauer's conclusion is that, for the rates to be calculated correctly, if vehicle exposure is used in the denominator, then the count of vehicles involved in accidents must be in the numerator (and not the number of accidents in which such vehicles are involved). Similarly, if driver exposure is in the denominator then the count of drivers of a certain category in accidents should be in the numerator and not the number of accidents involving that category of driver.
+
+A second issue discussed by Hauer (2001) is the question of over-representation. His question is- if some type of entity is found over-represented in reported accidents of specific severity, may one conclude that entities of this type are in fact more likely to be involved in accidents? His answer is 'no'. If one is dealing with reported accidents of a specific severity, as one almost invariably is, then the over-representation may be caused by a mix of three factors: the probability to be in an accident per unit exposure, the probability of the accidents to be reported and the probability of the accident to be of the specified severity. Without knowing these three probabilities it is not possible to come to the right conclusions. This issue applies to both lorries and vulnerable road users in particular. It applies to lorries, because on account of their greater mass they are more involved in accidents with higher severity and are therefore likely to be over-reported in registered accidents. To calculate the accident risk of lorries, looking at the registered accidents will result in over-representation, which might not be the case if 'accidents with material damage only' had also been taken into account. The issue also applies to the vulnerable road user, especially the bicycle, for which it is known that accidents are greatly under-reported and are therefore not found in the general accident statistics in their representative share.
+
+A recent report by Lyman and Braver (2001) also addresses the issue of large truck safety by analysing various crash involvement rates. The study examined different measures of truck crash fatality risk, to better understand how two main indicators – the public health aspect of large truck crashes and the risk per unit of travel - have changed over a period of 25 years. The study looked at large truck occupant fatalities (trucks over 10,000 pounds), and passenger vehicle occupant fatalities and calculated their rates per 100,000 members of the population, per 10,000 licensed drivers, per 10,000 registered trucks and per 100 million vehicle miles of travel (VMT) for each year over the period 1975-1999. Results show that in 1999, large truck crashes resulted in 3,916 occupant deaths in passenger vehicles and 747 in large trucks. Passenger vehicle occupant deaths in large truck crashes per 100,000 members of the population have increased somewhat since 1975 (1.28 in 1975, 1.44 in 1999). There has been a considerable decrease in occupant deaths per truck VMT since 1975, but the percentage reduction has been greater for occupants of large trucks (67%)
+
+than for occupants of passenger vehicles (43%). However, truck drivers are at an elevated risk of dying relative to their numbers in the workforce. Large truck involvements in fatal crashes per truck VMT decreased by 68% for single-vehicle crashes and by 43% for multiple-vehicle crashes. In contrast, passenger vehicle involvements in fatal crashes (including those that did not involve a large truck) per passenger VMT have decreased less (33% for single-vehicle crashes, 23% for multiple-vehicle crashes).
+
+Both arguments brought forward by safety groups and by the trucking industry are correct in their assertions. Large truck involvement in fatal crashes has dropped substantially when measured per unit of travel, but the public health burden of large truck crashes, as measured by death per 100,000 members of the population, has not improved over time because of the large increase in truck mileage. Countermeasures are needed to better protect both passenger vehicle occupants in collisions with large trucks and the occupants of large trucks.
+
+#### 5.2. **Drivers involved in accidents while under the influence of alcohol**
+
+One of the important topics in road safety and road safety countermeasures is the subject of driving under the influence of alcohol (DUI). Work on this subject began many years ago and has always been high on the agenda of decision-makers and the police. A major mile-stone in this field was the study of Borkenstein et al (1964). The study became known in the technical literature on the subject as the Grand Rapids study. Borkenstein collected information on alcohol levels of drivers involved in accidents and similar information on drivers in the population at large (not involved in accidents). Borkenstein's study involved about 5000 accident-involved drivers and a similar number of drivers as control group. From this information Borkenstein developed a risk function which describes the over-involvement of drinking drivers in accidents as a function of their blood-alcohol level (Figure 5). Without collecting the information on the general driving population, which serves as the exposure measure, it would be impossible to determine the over-involvement.
+
+From the curve it can be seen that at alcohol levels of between 0.5 till 0.79 promille the chance of an accident increases significantly by 35% compared with the zero alcohol level. At 0.8 to 0.99 promille the chances have increased by a factor two, and above that the risk of involvement continues to increase exponentially. A subsequent study, conducted along similar lines in Germany (Kruger, Kazenwadel & Vollrath, 1995) produced similar results. At all levels of alcohol consumption accident-involved drivers were overrepresented. The degree of over-involvement was even somewhat higher than in the original Borkenstein study.
+
+Having collected accident data and exposure data on such a large scale the Borkenstein study was regarded as authoritative and provided the foundation on which many countries established their blood alcohol limits. A few countries, including the Netherlands, adopted a 50 g/100 l limit, more countries adopted an 80 g/100 l limit and the US initially adopted a limit of 100 g/100 l (Mathijssen and Twisk, 2001). Countries in the East-European block still maintain a zero-tolerance limit, as does Iceland. The Scandinavian countries maintain a 20 g/100 l limit.
+
+![](_page_35_Figure_0.jpeg)
+
+Figure 5. The relationship between Blood Alcohol Concentration (BAC) and accident risk. From: Mathijssen & Twisk (2001) after Borkenstein et al., 1964.
+
+One initial by-product of the original Borkenstein study was that an underinvolvement in accidents seemed to occur at low levels of alcohol consumption, between 10 g/100 l and 40 g/100 l. This phenomenon became known as the Grand Rapids Dip (Hurst, Harte & Frith, 1994). Subsequent reanalysis of the original data by Hurst and others has shown that this dip was a result of the statistical composition of the samples and when this is taken into account correctly, the dip does not exist.
+
+The use of exposure data in this example was crucial in determining the effect of alcohol in accident-involved accidents. Without the exposure data it would not have been possible to measure the extent of over-involvement and decision-makers would not have had the scientific basis on which to adopt countermeasures, which in this case were the alcohol limits. Through the adoption of this course of action the number of drivers who continue to drive while under the influence has decreased considerably in most countries.
+
+A similar approach is now essential in determining the correct course of action regarding regulations on the use of drugs and medication. Without extensive surveys on the use of these substances by the driving population at large, in comparison to the driving population involved in accidents, it will not be possible to come to scientifically based regulations. Without the scientific basis, it will be difficult to persuade the public and the decisionmakers to follow the right course of action.
+
+#### 5.3. **The effect of speed on the risk of accident involvement**
+
+Another issue which is of major importance in road safety research is the effect of speed on the risk of being involved in an accident. Because of the extensiveness of this topic it will not be treated in detail, but some points which bear upon the issue of risk will be discussed. Speed is not generally mentioned as a major contributor to accidents, when looking at the possible causes of accidents detailed in accident reports filled out by the police who arrive at the scene of an accident. This absence is sometimes taken to mean that speed is not a major contributing factor to accidents. The reason that speed is not mentioned in such reports is that it is almost impossible to determine the speed of a vehicle involved in a crash without an in-depth investigation, which is not normally conducted by the police. As a result police officers tend to stick to the manoeuvres and actions, prior to the accident, that they can prove and report; such actions as not giving way, changing lanes, not stopping for a stop sign etc.
+
+Research has shown, however, that speed is indeed a major contributory factor in accidents and should be dealt with as a topic of great importance. But speed in itself is not all that needs to be known in order to investigate the subject. Of importance are also the speed in relation to the speed of other vehicles using the road,this is generally called the speed differential and was described in detail by Solomon (1964). Another factor of major importance is the variation in speed (speed variance), which was described in detail by Finch et al. (1994). Both these variables indicate that, in order to study the importance of speed on accident involvement, one has to have further details on the speed of the traffic at large and not just of speeds of vehicles involved in crashes. The information on traffic speed should be regarded as a measure of exposure.
+
+Finch et al. (1994) reviewed data from a variety of sources that seems to indicate that the relationship between accident involvement rate and deviation from the mean traffic speed is U-shaped. Taylor, Lynam & Baruya (2000) referring to studies of Maycock et al. (1998) and Quimby et al. (1999) report different results. The increase in risk at higher speeds than average was established but the increase at lower speeds than average could not be determined. In this case the relevant exposure measure is the speed of vehicles travelling along a stretch of road. Not only the mean speed of such vehicles is of importance but also the speed variance. Once such information is known it becomes easier to determine the accident potential of such a road or to consider the possible effects of countermeasures that are contemplated.
+
+It seems that speed and speed distribution are significant factors in establishing the risk of an accident and in determining the severity of such an accident. The implication of this for accident and risk studies is, that speed information should be taken into account. As in many cases speed information is not available, great care should be taken when calculating and comparing risks on various parts of the road and transport system. It seems wise to avoid comparing risks between parts of the transport system that have very different speed distributions or between modes with widely different speeds, as much as possible.
+
+#### 5.4. **A preliminary study to explore directions for elaborating on risks**
+
+In 1998 the Dutch Ministry of Transport decided to reinforce its road safety policy by promoting the use of risk indicators. Roszbach (1998) prepared a report which was intended as a preliminary study on the subject exploring possible directions for the quantification of risks. Roszbach dealt with the definitions of exposure and risk, compared various types of risk and discussed problems associated with the quantification of risk in various applications. Roszbach points out that, in terms of policy setting, working with risk indices on a provincial and municipal level can be counterproductive. It might be better to work with absolute numbers of accidents, casualties and fatalities. Increased mobility leads to more vehicle kilometres (increased exposure). Accident rates (risks) decrease with increased exposure and as a result those provinces and municipalities with larger increases in mobility might be rewarded for improved performance. Another point made by Roszbach, as indeed also made in the present report, is that exposure data should reflect the problem being studied. The report also points out that for the assessment of infrastrucure safety it is better to work with accident rates as a measure of traffic volumes. In this report this has been termed Safety Performance Functions (SPF). This approach is to be preferred over working with average accident rates per type of road section and intersection (index numbers). Roszbach suggests a step-by-step approach to safety decision-making:
+
+- Determining that something has to be done. On this level it is often possible to work with the raw accident and casualty data;
+- Determining where something should be done. In some cases exposure data will already be required at this level;
+- Determining what should be done. At this level we generally need comparative and normalised data.
+
+#### 5.5. **Predictions of traffic safety in 2010 in the Netherlands**
+
+A report by Commandeur & Koornstra (2001) analyses road safety developments in the Netherlands during the period 1948-1998 at a nationally aggregated level. The purpose was to find models that could describe developments of mobility (exposure) and fatality and casualty risk as accurately and as simply as possible. Having developed fairly accurate models, prognoses were made of future developments in mobility, casualty risk, number of road fatalities and hospitalisations up to the year 2010. The models developed presented a fairly accurate presentation of the trends in past years. Yearly deviations from observed mobility and risk were studied. It was found that there is a relation between the deviations for mobility and fatal risk figures. It appears that mobility deviations show a strong, negative trend relation with the risk deviations ten to eleven years later. The models developed seemed to provide a strong basis for the prognoses of mobilty, casualties and fatalities produced in the report. One of the drawbacks of these models is related to the relationship between mobility and risk deviations. It seems to be that they are very country-specific and have to be carefully calibrated on local data. Negative trend correlations in different countries varied between four and twenty years and in some countries the correlations were positive instead of negative (Commandeur, 2002).
+
+#### 5.6. **The involvement of elderly drivers in accidents**
+
+One of the issues which has received attention in recent years is the involvement of elderly drivers in crashes. This interest is partly the result of various misconceptions about elderly drivers, and partly of the growing number of the elderly in the driving population in motorised societies. It seems therefore relevant to have a clear picture on this issue, based on data and research findings. A recent SWOV report by Davidse (2000) dealt with this issue. The report consists of three main parts. First, an analysis is conducted of the present situation in the Netherlands, based on crash data, population data and travel distance data. Secondly, a prognosis is made of future trends and implications and finally an inventory and evaluation of measures to possibly lower the risks for elderly drivers are presented. The present report will deal only with the first part of the study. Davidse calculated the risks, the number of casualties and the accident involvement of elderly car drivers and compared them with various other age groups. The elderly were defined as being 65 years old and older,and were sub-divided into two groups: aged 65-74 and aged 75 and older. Other age groups were 18-24, 25-29, 30-39, 40-49, 50-59 and 60-64 years old. Rates were defined as car driver casualties, or fatalities or accidents per thousand million (billion) vehicle kilometres travelled. In most rates, a U-shaped curve was established of risk against age groups, with increased rates for the young and old drivers. The main conclusions drawn by Davidse were that:
+
+- The 75-year-olds and older had the highest chance of being killed in a road accident (death risk). The 18-24 year olds came second. The 75-year-olds and older came second in victim risk (the chance of being killed or injured, regardless of the severity). The 18-24 years old had the highest victim risk.
+- The death risk of those 75-year-olds and older is high because they are (physically) more vulnerable than other age groups. On the other hand, however, the death risk of the (male) 18-24 years old is high because they are more often involved in accidents than other age groups. The elderly car driver will, therefore, profit from an improvement of secondary safety more than other car drivers.
+- A reduction of the number of accidents involving elderly car drivers (primary safety) can be achieved by designing measures that are specifically aimed at the types of accidents in which the elderly more often appear to be the 'guilty party'. These are especially accidents resulting from not giving the right of way and accidents involving leftturns at intersections.
+
+A recent study by Li, Braver & Chen (2001) came to very similar conclusions. Using national US data systems, the role of fragility (susceptibility to injury) versus excessive crash involvement in the increased fatality risk of older drivers per vehicle mile of travel (VMT) was estimated. For each age and gender group, deaths per driver involved in a crash (a marker of fragility) and drivers involved in crashes per VMT (a marker of excessive crash involvement) were computed. Compared with drivers aged 30-59, those younger than 20 and those aged 75 or older both had much higher driver death rates per VMT. The highest death rates per mile driven, 13-fold increases, were observed among drivers aged 80 or older, who also had the highest death rates per crash. Among older drivers, marked excesses in crash involvement did not begin until age 75, but explained no more than
+
+about 30-45% of the elevated risk in this group of drivers; excessive crashes explained the risk among drivers aged 60-74 to a lesser extent. In contrast, crash overinvolvement was a major factor contributing to the high risk of death among drivers younger than 20, accounting for more than 95% of their elevated death rates per VMT. These findings suggest that measures to improve the protection of older vehicle occupants in crashes will be more effective than measures directed towards reducing the involvement of older drivers in crashes.
+
+The above two studies demonstrate the usefulness of rates in the analysis of a problem. By focussing separately on the crash involvement rates and the fatality rates per vehicle kilometre for drivers of different age groups, the correct countermeasure approach could be identified. Some of these results could have been identified by just working with the fatality and casualty numbers, by calculating the number of fatalities per hundred crashes for each age group, but this would not have taken into account the relative amounts of travel for each age group.
+
+#### 5.7. **The involvement of young drivers in accidents**
+
+One example of the necessity to work with both the accident numbers and risk rates is demonstrated in Twisk (2000). In the Netherlands, in the period between 1985 and 1997 the number of young drivers involved in serious accidents dropped considerably. The number of drivers in the age group 20-24 involved in serious accidents decreased from 2829 in 1985 to 1502 in 1997, a decrease of 47%. The number of drivers in the age group 18-19 decreased from 682 in 1985 to 343 in 1997, a decrease of 50%. The number of accident-involved drivers in other age groups decreased by a much smaller percentage: from 9082 in 1985 to 8050 in 1997, a decrease of less than 12%. Figure 6 shows the absolute number of young drivers involved in serious accidents for 18-19 and 20-24 year old male and female drivers. Figure 7 shows the number of vehicle kilometres driven by these groups and Figure 8 shows the accident risk developments.
+
+It immediately becomes clear that the large reduction in the number of young drivers involved in serious crashes is the result of reduced exposure. In the report by Twisk the reduction in mileage is further traced to demographic and other changes that occurred over the period concerned, which had a major effect on the age group studied. An economic recession affected the number of licensed young drivers, and free public transport travel passes were issued to young students.
+
+The study demonstrates the need to work in an ad-hoc fashion, on both the raw casualty data and on exposure and risk rates to come to a comprehensive investigation of a certain phenomenon.
+
+![](_page_40_Figure_0.jpeg)
+
+Figure 6: Absolute number of young drivers involved in serious accidents by age group, sex and year in the Netherlands. From: Twisk, 2000.
+
+![](_page_40_Figure_2.jpeg)
+
+Figure 7: Mileage (kms) by age group, sex and year in the Netherlands. From: Twisk, 2000.
+
+![](_page_41_Figure_0.jpeg)
+
+Figure 8. Accident risk of young drivers versus total driver group by sex and year in the Netherlands. From: Twisk, 2000.
+
+#### 5.8. **Road safety in the Netherlands up to 1999**
+
+Van Schagen (2000) published a report which describes the extent and nature of road safety in the Netherlands over a period of 15 years and which served as the basis for prognoses on the developments in road safety to be expected in the future (the year 2010). Trends, backgrounds and explanations of the data were presented. A number of developments which are known to have a strong relation with safety, were studied in detail. These included certain driver behaviours-speeding, drinking-and-driving and the use of seat belts. Developments in certain driver groups were also analysed separately: moped riders, lorries, older car drivers and cyclists. It was possible to reach conclusions about the prognoses after making separate prognoses about the trends in vehicle kilometres and the casualty and fatality rates. Having established the development in exposure and in casualty rates, it became possible to predict the future number of casualties and fatalities in the year 2010. On the basis of these predictions conclusions were reached on the possibility of reaching the strategic targets the government had set for that year.
+
+# 6. **A more detailed discussion on risk and exposure in the context of infrastructure**
+
+#### 6.1. **The relationship between accidents and traffic volumes**
+
+One of the main reasons for defining and using exposure in studies related to infrastructure is to equalise for differences in intensity of use, so as to make comparisons meaningful. The unit of study might be a number of road sections or a number of intersections. Accident rate (Ar) is thus defined as:
+
+#### Ar = Expected number of accidents over a certain period of time Amount of exposure over a period of time
+
+The number of accidents is related to the exposure and generally increases with an increase in exposure. The shape of the curve that relates the two entities is nowadays termed a 'safety performance function' (Hauer, 1995a; Ekman, 1996; Persaud et al, 1999). It has by now been well- established that for road sections the shape of the safety performance function (SPF) is not a linear one. Figure 9 gives an example of a hypothetical SPF for a road. Accidents vary according to the amount of exposure in a non-linear way. In this context the exposure is generally expressed as traffic volume (AADT) or as vehicle kilometres travelled. When discussing intersection safety the relevant measure of exposure would be the number of vehicles entering the intersection, or some measure thereof. In that case too, the relationship is normally not linear.
+
+![](_page_42_Figure_5.jpeg)
+
+Figure 9. Safety performance function and accident rates. From: Hauer (1995a).
+
+It can thus be stated that division by exposure does the job of equalising only if the SPF is a straight line( through the origin). If the SPF is non-linear, the accident rate is of limited use, or incorrect. For the passing motorist a lower rate would still mean that per travel unit the chance of an accident is lower, hence we could talk of a lower risk. From the point of view of those who wish to consider the effect of a road safety intervention (a change to a road section or intersection) it would be impossible to relate the change in the accident rate to the intervention. One would first have to know the shape of the SPF before one could make an intelligent statement. A drop in accident rate does not necessarily mean that the intervention improved safety. The rate of accidents or fatalities per million vehicle kilometres travelled (million VKT), which is a widely used index of comparison, generally decreases over time for a variety of reasons. Vehicle kilometres travelled are generally defined as the distance travelled per vehicle multiplied by the number of vehicles. Each additional vehicle-km causes a smaller increase in the number of accidents. The decreasing slopes of curves OA and OB (see Figure 10) represent this fact. The ratio of accidents per vehicle-km is given by the angle a. The meaning of an improvement in safety is that, for each amount of travel, the resulting number of accidents is reduced. Therefore, curve OB describes an improvement over curve OA. For each amount of travel, the resulting number of accidents is smaller. It can be seen from Figure 10 that the change in accident rate by itself does not indicate a change in safety. Point C, having a smaller slope than D (a<b) is worse than D from the overall safety point of view, but the fact that the accident rate is lower means that for a passing motorist the risk is lower.
+
+![](_page_43_Figure_1.jpeg)
+
+Figure 10. The relationship between number of accidents and cumulative vehicle kilometres travelled. From: Hakkert, Livneh & Mahalel, 1976.
+
+Similarly, when taking this line of argument to other fields of safety intervention it can also be concluded that there is little ground for assuming that the number of aviation fatalities is linearly related to the number of flighthours, or that the number of truck accidents is proportional to truck-miles travelled or to ton-miles carried.
+
+The next part of this section will present findings from some studies that indicate the non-linearity of the relationship between accidents and traffic flow.
+
+#### 6.1.1. Safety Performance Functions (SPF)
+
+The detailed relationships between accidents and traffic volumes have been discussed extensively in the literature and during the past 5-10 years have become known as Safety performance Functions (SPF) in the North-American literature (Hauer, 1995a; Persaud et al.1999). A safety performance function (SPF) is the relationship between the traffic volume (AADT) on a highway and the safety of that highway, defined as the number of accidents (defined per unit of time and highway length). SPFs have been calibrated for a number of locations, mainly on rural highways, both twolane rural roads and freeways.
+
+Persaud et al. (1999) present a number of SPFs developed as part of their work on rural roads in Ontario, Canada. For a set of 197 rural four-legged signalised intersections in Ontario, during the years 1988-1993, the SPF was estimated as:
+
+```
+No. of accidents/year = 0.0005334 x (total entering AADT) exp 0.8776
+```
+
+For two-lane paved rural highways in Ontario, during the years 1988-1993, a set of 28,000 0.5 km long sections was selected. Only accidents with injury were included in the study. The SPF developed was:
+
+```
+No. of injury accidents/year = 0.000637 x (section length) x [exp(-0.08215)
+X (total road width)] x (AADT) exp 0.8993
+```
+
+It can be seen that for both the intersection and road section models the exponents for traffic volume were 0.8776 and 0.8993, indicating a nonlinear relationship between accidents and traffic volume.
+
+For 4-lane freeways in Canada, Persaud et al. (1999) calibrated the following SPF for injury accidents:
+
+```
+Injury accidents = (Section length) (0.0000537) (AADT) exp (1.01786)
+```
+
+In this case it is shown that the relationship is almost linear.
+
+#### 6.1.2. The relation between aggregate number of collisions and distance travelled
+
+Smeed (1974) studied the relationship between the aggregate number of collisions between certain types of vehicles and the distance travelled by these vehicles. He calculated and plotted the product of the distances travelled by vehicles of certain categories (cars, lorries, motor-cycles, pedalcycles) against the number of two-vehicle accidents of those categories. He concludes that "it seems that we must, to a large extent, give up the hypothesis that the number of two-vehicle accidents is generally proportional to the product of the distances travelled by the types of vehicles involved."
+
+As, however, traffic volumes vary greatly during different parts of the day, it might be possible to look for a relationship by clustering accidents and traffic volumes into groups with similar traffic volumes. The number of nonpedestrian single vehicle accidents was plotted against the estimated distances travelled by the vehicles in these hours. The numbers of two-vehicle accidents of various categories in the various hours was plotted against the functions of the distances travelled in these hours. It was found that, to a considerable extent, the number of single vehicle accidents involving any type of vehicle during any daylight hour was found to be proportional to the distance travelled by vehicles of that type during those hours. The number of collisions between any two categories of vehicles during any daylight hour was found to be proportional to the product of the distances travelled by each type of vehicle during that hour. Significant discrepancies were found, however, between the expected number of accidents and the actual numbers, indicating that other factors, not taken into consideration, might also have an influence. It could also be the case that significant errors exist in the estimation of distances travelled for each type of vehicle.
+
+In terms of safety studies related to infrastructure it seems advisable to develop and take account of Safety Performance Functions before making comparisons. The development and use of these SPFs is becoming widespread in many countries and the errors introduced by not accounting for differences in traffic volumes seem no longer warranted.
+
+#### 6.2. **Hazardous locations**
+
+The subject of hazardous locations plays a central role in safety management of road infrastructure. At each level of administration responsible for the construction and maintenance of roads - municipalities, provincial and central government - a certain number of locations (road sections and intersections) can be identified that have a 'higher than expected' number of accidents. Once identified, countermeasures can be proposed for such locations and the locations can be improved. Many models exist in the literature that deal with the identification of such locations, generally termed black-spots or hazardous locations. Intrinsic to all the more advanced methods is that they incorporate a procedure for the estimation of the number of accidents that should be expected at a location of the type under consideration. Hauer (1995b) provides a review of many of the models that have been proposed in the past (see also Mahalel, Hakkert & Prashker, 1982). The first task at hand is always to devise a method that will identify a limited number of locations (sections and/or intersections) for further consideration, to be chosen from the many thousands of such locations a road authority has under its jurisdiction.
+
+The final aim of this procedure should be the engineering treatment of a number of locations that will bring about a reduction in the number of accidents at the lowest costs possible. The locations do not therefore have to be hazardous or have a considerable concentration of accidents in order to qualify. In that sense the terms 'hazardous location' or 'black-spot' are not neutral words and Hauer suggested the cumbersome term "sites with promise" (Hauer, 1995b). The more recently proposed procedures for identifying sites for safety investigation are based on the Experimental Bayes (EB) technique. This essentially aims to smooth out the random fluctuation in accident data by specifying the safety of a site as an estimate of its longterm mean (m) instead of its short-term accident count (Persaud, B., Lyon, C. & Nguyen, T. (1999). The latter propose an improvement to the established methods by basing the selection of sites not only on the knowledge of the long-term mean but also on the calculated number of 'treatable' accidents.
+
+There are a number of selection principles to choose a limited number of locations for further consideration out of a long list of locations. Basically the selection principle can be the number of accidents or the accident rate, being the number of accidents divided by the exposure. Variations on these two principles are also possible. Therefore, the methods proposed in the literature can be divided into selection according to the highest number of accidents or selection according to the highest rate. Again, some methods suggest a combination of the two. In deciding which is the better method, it is important to ask what the purpose of the process is. Hauer(1995b) cites three possible motives:
+
+#### - Motive 1.
+
+Economic efficiency: identify sites at which remedial action would prove cost-effective.
+
+#### - Motive 2.
+
+Professional and institutional responsibility: recognise and rectify sites that are deficient either because of how they were built or because they have deteriorated while in use.
+
+#### - Motive 3.
+
+Fairness: identify sites that pose an unacceptably high hazard to the user.
+
+For motives 1 and 2 the selection should be based on the number of accidents or the changes in the number of accidents. For motive 3 the selection should be based on the accident rate. The list resulting from a selection on the basis of motive 3, i.e. risk to the road user, will generally result in a list which includes locations with a small number of accidents and low volumes of traffic, because these are the sites that have high accident rates. Such sites will generally have low rates of return and are not to be recommended on the basis of cost-benefit considerations.
+
+Table 8, taken from Hauer (1995b), describes five different selection criteria of which the first three are based on efficiency and rely on the number of accidents, and the last two assign priority on the basis of fairness and selection according to accident rate.
+
+To conclude this discussion of methods for the identification of hazardous locations it has been shown that methods based on the number of accidents would generally provide a more cost-effective approach, resulting in the higher number of accidents being saved per invested unit of money. Methods based on selection according to accident rate, seem to introduce a measure of fairness into the system, but on the whole will not result in a cost-effective allocation of budgets and will not result in achieving a significant reduction in accidents. If the methods are to be made more reactive to a deterioration in the safety situation, methods have been suggested which are a variation on the accident number method.
+
+| Selection<br>criteria |                                      | Consideration                                                                                                                | Rank by                                 | Comment                                                                                                                                                                                                                                                                                                                                   |                                 |
+|-----------------------|--------------------------------------|------------------------------------------------------------------------------------------------------------------------------|-----------------------------------------|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|---------------------------------|
+| A                     |                                      | Measures that reduce the probability<br>and/or severity of accidents (eg.<br>illumination, slope reduction)                  | F<br>(best done<br>by target<br>accid.) | These are usually high-exposure sites and tend to be near<br>the top of the list year after year. Once they are 'identified'<br>and examined, one can decide whether a safety<br>improvement is appropriate. Sites of this kind may not<br>need to be monitored and repeatedly identified.                                                |                                 |
+| B                     | E<br>F<br>F<br>I<br>C<br>I           | Deficiency since site opened to traffic<br>(or since reconstruction) that can<br>perhaps be cheaply rectified                | ûF/1F<br>(Total<br>accid.)              | Such sites should be identified as soon as possible. The<br>entire accident history (since opening or reconstruction)<br>seems relevant to identification. However, once a site is<br>shown to be normal, continued monitoring seems<br>pointless.                                                                                        | R<br>E<br>S<br>P<br>O<br>N<br>S |
+| C                     | E<br>N<br>C<br>Y                     | Deterioration of safety while in use that<br>perhaps can easily be rectified                                                 | Size of<br>jump in F                    | In this case periodic monitoring of all sites is required. It<br>may turn out that jumps in F are statistically difficult to<br>spot. It seems sensible to monitor for deterioration by<br>means other than accident occurrence.                                                                                                          | I<br>B<br>I<br>L<br>I<br>T<br>Y |
+| D                     | F<br>A<br>I<br>R<br>N<br>E<br>S<br>S | It is not acceptable to have sites where<br>the risk to road users is too large                                              | R                                       | These are often low-exposure sites. At such sites, a high<br>R may be a natural consequence of lower design<br>standards. It will take a long time to have enough accidents<br>to estimate R reliably. Use of accumulated accidents may<br>be necessary. Once such sites are labelled 'normal' further<br>monitoring may not be required. |                                 |
+| E                     | T<br>O<br>U<br>S<br>E<br>R           | It is not acceptable to have sites where<br>the risk to a road user is considerably<br>larger than at sites of similar class | ûR/1R                                   | For the reason listed above ûR is also difficult to estimate<br>reliably and use of the accumulated accident history and<br>clues other than accidents are important. Once such sites<br>are labelled 'normal' further monitoring may not be<br>required.                                                                                 |                                 |
+
+Table 8. Motives, considerations and ranking criteria. From: Hauer, 1995b.
+
+# 7. **Summary and conclusions**
+
+The report sets out with a definition of the three central terms used: accidents, exposure and risk. Each of the three terms is defined and problems of definition, accuracy of measurement and reporting are described.
+
+The conclusion reached is that there are general definitions of exposure and risk as used in the health prevention and risk analysis fields, but that in road safety practice these terms should be defined within the context of the issue studied. In the context of this report, exposure is meant as exposure to risk. To what extent are certain segments of the population, or certain types of vehicles likely to be involved in an accident? The measure of exposure is generally defined as some form of the amount of travel, either by vehicle or on foot. In the report, risk is used to mean the probability of an accident occurring, weighted in some way by the severity of the accident outcome. In many cases it would be better, and more neutral, to refer to rates and not to risks.
+
+As was discussed in the report, for each application, the correct exposure measure should be used. This is sometimes impossible because the required information is not available, or has to be collected at great cost. Generally, the more aggregate the exposure measure, the more indirect variables are introduced, which casts shadows over the resulting risk calculations.
+
+In the case of transport, the most widely used measure of exposure is the amount of travel for each travel mode. In some cases, useful additional insight is provided by taking into account the speed of travel, in which case exposure is expressed as the amount of time spent in the traffic system. Due to the very different conditions and circumstances, it is considered to be of little use to extend this comparison of risk per unit of time to other activities beyond the field of transport. Within the field of transport, transport modes can usefully be compared, up to a certain point, by risk per unit of travel or per unit of time. Taking bus and coach travel as a normative unity, it was shown that risk expressed in terms of person kilometres travelled is 10 times greater for cars and 100-200 times greater for the vulnerable modes (foot, bicycle and motor-cycle), with motor-cycles clearly being the most hazardous. Train and plane travel are also relatively safe. Looking at the same data, with risk expressed in terms of person hours of travel, does not change the picture greatly. Travel by foot and travel by bicycle become slightly less hazardous, because of the low speeds of these modes, whereas air travel becomes significantly more hazardous because of the high speeds (hence the small number of hours per kilometre).
+
+The lack of detailed and high quality exposure data is one of the reasons that in many cases international comparisons are conducted on a per capita or per vehicle basis, and deal mostly with fatalities because of the inaccuracy of injury reporting. One of the developments in recent years is the installation of electronics and telecommunications inside vehicles and along roads. The increasingly widespread use of mobile phones also presents many new possibilities. As a result it is becoming easier to collect up-to-date and reliable information on a variety of parameters that could be of importance in the calculation of vehicle exposure and risk. Additional information on the distribution of speeds, types of vehicles and following distances also seems to be a future possibility.
+
+On road sections, the number of vehicle kilometres is generally accepted as the relevant measure of exposure. At intersections, the correct measure of exposure is not that obvious, and has been the subject of many discussions. The product of the daily crossing flows and the sum of the products of the hourly crossing flows have been used as exposure measures in many accident studies. The definition of risk to pedestrians can be discussed in a way which is fairly similar to the discussion of risk and exposure at intersections. In this manner one obtains a measure of exposure which is the product of the traffic flow and the crossing number of pedestrians. The risk is then defined by dividing the annual number of pedestrian accidents by the exposure.
+
+One of the contexts in which risk is used, is in comparing risks between different parts of the transport system, different transport modes or even different activities outside the field of transport. The desire is to make various activities exposed to equal risks, so as to make risk distribution 'fair'. After discussing the various problems and pitfalls with this approach, it is concluded that the desire for equal risks in various segments of the transportation system is not practical. It is more useful to search for ways to make each segment of the transport system as safe as possible, keeping cost-effectiveness considerations into account.
+
+The report also describes studies in which useful safety evaluations were conducted without having to resort to risk and exposure measures. On a more general level, risk can serve as the basis for the selection of research and action themes, which should then be justified on the basis of costbenefit considerations. Decision-making can be carried out on the basis of comparative risk levels, on the basis of accident and injury data (without risk) and according to cost-benefit considerations. Each method has advantages and disadvantages and should be selected according to the circumstances. Decision-making on the basis of the number of accidents, injuries or fatalities focuses on the size of the problem, and prevents us from focussing our efforts on a segment of road safety that might look bad from the point of view of risk, but which might not be not very significant in terms of the quantities involved. Decision-making on the basis of risk tends to lead to actions that are considered fair in terms of the road users involved. It has been demonstrated in the report that this is actually not practical or reasonable. The final decision should generally be based on cost-effectiveness considerations because, in the field of road safety, like in any other field, the amount of expenditure is limited and investments should be allocated to those fields that produce the highest accident, casualty and fatality reductions that can be achieved per unit of money invested.
+
+In the context of national levels of safety and the setting of road safety targets, it was shown that targets are generally set in numbers of casualties and fatalities. This seems preferable over setting targets regarding risks, which can become intertwined with changes in exposure (mobility), which will affect the risk. When comparing risks and safety between different transport modes, different regions of a country or different countries,
+
+however, it seems that comparisons will have to be made on the basis of accidents and fatalities per vehicle kilometre travelled and in some cases, per passenger hour. In other cases a risk per passenger trip may be more meaningful.
+
+In chapter five, the report describes a number of studies from the major fields of road safety research and activity which deal with the issue of risk. The fields covered include the subject of the involvement of lorries in fatal accidents; driving under the influence of alcohol; the influence of speed in crashes and crash severity; risk used in assessing national safety levels; accident, injury and fatality trends; target setting; and elderly and young drivers.
+
+Chapter six discusses the relationship between accidents and traffic volumes in more detail. This is an issue of central importance when dealing with safety of the road infrastructure and has to be understood before making risk calculations. It was demonstrated that this relationship has to be determined first, in the form of a 'safety performance function (SPF)', before risks on various network sections can be compared. Many studies have demonstrated that the relationship between accidents and traffic volumes is not a linear one, which complicates matters greatly. When considering safety in the context of road infrastructure, there is a need to take information on traffic volumes into account. The report has documented that gross comparisons per vehicle kilometre travelled will lead to errors. Nowadays, enough information is available to develop safety performance functions (SPFs) for various segments of the road system to make meaningful comparisons. SPFs for intersections and for pedestrian and bicycle travel should also be developed.
+
+The report concludes with the issue of hazardous locations and establishes that locations had better be identified on the basis of the expected number of accidents, based on models developed for similar locations, rather than on the basis of accident rates (risks).
+
+## **References**
+
+Borkenstein, R.F., Crowther, R.F., Shumate, R.P., Ziel, W.B. & Zylman, R. (1964). The role of the drinking driver in traffic accidents. Research Report. Department of Police Administration, Indiana University, Bloomington, Indiana, USA.
+
+Brundell-Freij, K. & Ekman, L. (1990). Safety and exposure- aspects of the relationship with special attention to vulnerable road users. Research report. Dept. of Traffic Planning and Engineering, University of Lund, Sweden.
+
+BTS (2001). Transportation data and risk measures. Bureau of Transportation Statistics, www.bts.org, USA
+
+Commandeur, J.J.F. & Koornstra, M.J. (2001). Prognoses voor de verkeersveiligheid in 2010 (Predictions of road safety in 2010). SWOV report R-2001-9. SWOV, Leidschendam, the Netherlands
+
+Commandeur, J.J.F. (2002, in preparation) Algemene en periodieke trends in de ontwikkeling van de verkeersveiligheid in acht ontwikkelde landen (General and periodical trends in road safety development in eight developed countries). SWOV, Leidschendam, the Netherlands.
+
+Davidse, R.J. (2000). Ouderen achter het stuur (The elderly at the wheel). SWOV report D-2000-5. SWOV, Leidschendam, the Netherlands.
+
+Despontin, M., Verbeke, A. & Brucker, K. de (1997). The economic evaluation of road safety in the European Union. In: Proceedings of the European Seminar on Cost-Effectiveness of Road Safety Work and Measures. November 1997, Luxembourg.
+
+DUMAS (1998) Accident Investigation; final report Work Package 4. Prepared by DENCO (GR), University of Brescia (I), INRETS (F), TRL (UK), AUTh (GR). Developing Urban Management And Safety (DUMAS).
+
+Ekman, L. (1996). On the treatment of flow in traffic safety analysis. Bulletin 136, Institute of Traffic Studies, Technical University of Lund, Sweden
+
+Elvik, R. (1997). Cost-benefit analysis of road safety measures is more important and relevant than ever before. In: Proceedings of the European Seminar on Cost-Effectiveness of Road Safety Work and Measures. November 1997, Luxembourg.
+
+Erskine, A. (1996). The burden of risk: who dies because of cars? Social Policy and Administration, 30(2), 143-157.
+
+ETSC (1999). Exposure data for travel risk assessment: current practice and future needs in the EU. European Transport Safety Council, Brussels, Belgium
+
+Evans, A. (1993). Older road users - the role of government and the professions. In: Proceedings of road safety conference. AA Foundation for Road Safety Research, London.
+
+Evans, A.W. (1994). Multiple-fatality transport accidents 1946-1992. In: Department of Transport (ed.) Transport Statistics Great Britain 1994, pp. 17-29. HMSO.
+
+Evans, L., Frick, M.C. & Schwing, R.C. (1989). Is it safer to fly or drive? –A problem in risk communication. General Motors Research report, General Motors Research Laboratories, Michigan, USA.
+
+Finch, D.J., Kompfner, P., Lockwood, C.R. & Maycock, G. (1994). Speed, speed limits and accidents. Project Report 58, S211G/RB, Transport Research Laboratory, Crowthorne, Great Britain
+
+Haight, F.A. (1986). Risk, especially risk of traffic accident. Accident Analysis and Prevention, 18, pp. 359-366.
+
+Harvey, A.C. & Durbin, J. (1986). The effects of seat belt legislation on British road casualties: a case study in structural time series modelling. Journal of the Royal Statistical Society, 149, pp.187-227.
+
+Hauer, E. (1982). Traffic conflicts and exposure. Accident Analysis and Prevention, 14, pp. 359-364.
+
+Hauer, E. (1995a). On exposure and accident rate. Traffic Engineering + Control, 36(3), pp.134-138.
+
+Hauer, E. (1995b). Identification of sites with promise. TRB Record 1542. Transportation Research Board, Washington DC, USA.
+
+Hauer, E. (2001). Computing and interpreting accident rates for vehicle types or driver groups. TRB meeting paper, TRB annual meeting, Washington DC, January 2001, USA
+
+Hakkert, A.S. & Bar-Ziv, J. (1976). Accident risk and delays at mid-block pedestrian crossings. International Forum Series (Vol. 2): Traffic, transportation and urban planning. George Godwin Publishers, London, pp. 165-174.
+
+Hakkert, A.S., Livneh, M. & Mahalel, D. (1976). Levels of Safety in Accident Studies - A Safety Technology Index. In: Proceedings of the 8th Australian Road Research Conference, August 1976, Session 27, pp. 1-6.
+
+Hurst, P.M., Harte, D. & Frith, W.J. (1994). The Grand Rapids Dip Revisited. Accident Analysis and Prevention, 26, pp. 647-654.
+
+Kruger, H.P., Kazenwadel, J. & Vollrath, M. (1995). Grand Rapids Effects revisited: Accidents, Alcohol and Risk. In: Proceedings of the 13th International Conference on Alcohol, Drugs and Traffic Safety, ICADTS. Li, G., Braver, E.R. & Chen, L.H. (2001). Exploring the high driver death rates per vehicle-mile of travel in older drivers: fragility versus excessive crash involvement. IIHS research report. The Insurance Institute for Highway Safety, Arlington, USA
+
+Lyman, S. & Braver, E.R. (2001). Occupant deaths in large truck crashes in the United States: 25 years of experience. IIHS research report. The Insurance Institute for Highway Safety, Arlington, USA.
+
+Mahalel, D. & Hakkert, A.S. (1976). Estimating the number of accidents at intersections from a knowledge of the traffic flows on the approaches. Journal of Accident Analysis and Prevention, Volume 10, no. 1, pp 69-79.
+
+Mahalel, D., Hakkert, A.S. & Prashker, J.N. (1982). A system for the allocation of safety resources on a road network. Accident Analysis & Prevention, 14 (1), pp. 45-56.
+
+Mathijssen, M.P.M. & Twisk, D.A.M. (2001). Opname en afbraak van alcohol in het menselijk lichaam. (Demonstration of the metabolism of alcohol in the human body). R-2001-19. SWOV, Leidschendam.
+
+Nilsson, G. (1997). Methods and necessity of exposure data in relation to accidents and injury statistics. IRTAD Special report. Swedish Road and Transport Research Institute, Linkoping, Sweden
+
+Older, S.J. & Grayson, G.B. (1976). An international comparison of pedestrian risk in four cities. In: Proceedings of the International Conference on Pedestrian Safety. Michlol, Technion, Haifa, Israel
+
+Persaud, B. et al. (1999). Applying the science of highway safety to effect highway improvements- a multi-disciplinary approach. In: Proceedings of the Canadian Multi-disciplinary Road Safety Conference XI, Quebec, Canada, April 1999.
+
+Persaud, B., Lyon, C. & Nguyen, T. (1999). Empirical Bayes procedure for ranking sites for safety investigation by potential for safety improvement. TRB record 1665, Transportation Research Board, Washington DC, USA.
+
+Poortvliet, A. van (1999). Risks, disasters and management - A comparative study of three passenger transport systems. Ph.D. thesis, Technical University Delft, the Netherlands.
+
+RIVM (1998) Milieubalans 1998. From: http://www.milieubalans.rivm.nl
+
+Roberts, I., Norton, R. & Taua, B. (1996). Child pedestrian exposure rates: the importance of exposure to risk relating to socioeconomic and ethnic differences in Auckland, New Zealand. Journal of Epidemiol.Community Health, 50, pp. 162-165.
+
+Roszbach, R. (1998). Risicobenaderingen in wegverkeer. (Risk approximations in road traffic). SWOV report R-98-13. SWOV, Leidschendam, the Netherlands.
+
+Rumar, K. (1999). Road safety and benchmarking. In: Proceedings of the Paris Conference on Transport Benchmarking. November 1999, Paris, France.
+
+Schagen, I.N.L.G. van (2000). De verkeersonveiligheid in Nederland tot en met 1999. (Road safety in the Netherlands up to 1999). SWOV report D-2000-15. SWOV, Leidschendam, the Netherlands.
+
+Smeed, R.J. (1974). The frequency of road accidents. Zeitschrift für Verkehrssicherheit, 20(2), pp. 95-108.
+
+Solomon, D. (1964). Accidents on main rural highways related to speed, driver and vehicle. Bureau of Public Roads, Dept. of Commerce, USA.
+
+Taylor, M.C., Lynam, D.A. & Baruya, A. (2000) The effects of drivers' speed on the frequency of road accidents. TRL report 421. Transport Research Laboratory, Crowthorne (UK).
+
+Thompson, K.M., Graham, J.D. & Zellner, J.W. (2001). Risk-benefit analysis methods for vehicle safety devices. In: Proceedings of the 17th International Conference on the Enhanced Safety of Vehicles, June 2001, the Netherlands.
+
+Twisk, D.A.M. (2000). Why did the accident involvement of young (male) drivers drop by about 50 percent? In: Proceedings of the 10th seminar on behavioural research in road safety. Esher, Surrey, 3-5 April 2000, pp. 109- 117.
+
+Wesemann, P., Bos, J.M.J., Hertog, P.C. den, Adriaanse, M.C. & Blankendaal, A.C.M. (1998). Onveiligheid van wonen, verkeer, arbeid en sport. (Safety in the sectors and social activities: domestic, traffic, industry and sport). SWOV report R-98-70, SWOV, Leidschendam, the Netherlands

Zotero/001_artiklid/Road Safety Indicators/hareruepidemiologyroadtrafficaccidents2022.md

@@ -0,0 +1,405 @@
+---
+category: literaturenote
+citekey: hareruepidemiologyroadtrafficaccidents2022
+title: "The epidemiology of road traffic accidents and associated factors among drivers in Dilla Town, Southern Ethiopia"
+authors: "Hareru, Habtamu Endashaw; Negassa, Belay; Kassa Abebe, Reta; Ashenafi, Eden; Zenebe, Getachew Assefa; Debela, Berhanu Gidisa; Ashuro, Zemachu; Eshete Soboksa, Negasa"
+year: 2022
+date: 2022-00-00 2022
+doi: 10.3389/fpubh.2022.1007308
+publication: Frontiers in Public Health
+url: "https://doi.org/10.3389/fpubh.2022.1007308"
+zotero_key: S6PNG8QA
+zotero_storage: ZPGBBZPU
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Hareru, Habtamu Endashaw"
+status: converted
+---
+TYPE [Original Research](https://www.frontiersin.org/journals/public-health#editorial-board) PUBLISHED [10 November 2022](https://www.frontiersin.org/journals/public-health#editorial-board) DOI [10.3389/fpubh.2022.1007308](https://doi.org/10.3389/fpubh.2022.1007308)
+
+![](_page_0_Picture_2.jpeg)
+
+#### OPEN ACCESS
+
+Grace Sembajwe, Northwell Health, United States
+
+REVIEWED BY Chalachew Kassaw, Dilla University, Ethiopia Khosro Sadeghniiat, Tehran University of Medical Sciences, Iran Artur Petrov, Tyumen Industrial University, Russia
+
+Habtamu Endashaw Hareru [habteenda@gmail.com](mailto:habteenda@gmail.com)
+
+This article was submitted to Occupational Health and Safety, a section of the journal Frontiers in Public Health
+
+RECEIVED 30 July 2022 ACCEPTED 26 September 2022 PUBLISHED 10 November 2022
+
+Hareru HE, Negassa B, Kassa Abebe R, Ashenafi E, Zenebe GA, Debela BG, Ashuro Z and Eshete Soboksa N (2022) The epidemiology of road trac accidents and associated factors among drivers in Dilla Town, Southern Ethiopia.
+
+*Front. Public Health* 10:1007308. doi: [10.3389/fpubh.2022.1007308](https://doi.org/10.3389/fpubh.2022.1007308)
+
+© 2022 Hareru, Negassa, Kassa Abebe, Ashenafi, Zenebe, Debela, Ashuro and Eshete Soboksa. This is an open-access article distributed under the terms of the [Creative Commons](http://creativecommons.org/licenses/by/4.0/) [Attribution License \(CC BY\).](http://creativecommons.org/licenses/by/4.0/) The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
+
+# The epidemiology of road trac [accidents and associated factors](https://www.frontiersin.org/articles/10.3389/fpubh.2022.1007308/full) among drivers in Dilla Town, Southern Ethiopia
+
+Habtamu Endashaw Hareru<sup>1</sup>\*, Belay Negassa<sup>2</sup> , Reta Kassa Abebe<sup>1</sup> , Eden Ashenafi<sup>3</sup> , Getachew Assefa Zenebe<sup>1</sup> , Berhanu Gidisa Debela<sup>1</sup> , Zemachu Ashuro<sup>2</sup> and Negasa Eshete Soboksa<sup>2</sup>
+
+<sup>1</sup>School of Public Health, College of Medicine and Health Science, Dilla University, Dilla, Ethiopia, <sup>2</sup>Department of Environmental Health, College of Medicine and Health Science, Dilla University, Dilla, Ethiopia, <sup>3</sup>Department of Reproductive Health, College of Medicine and Health Science, Dilla University, Dilla, Ethiopia
+
+Background: Data on the magnitude of road trac accidents (RTAs) were mostly obtained through police records and hospital registration data. However, insucient data reporting masked the gravity of the problem, and little attention was paid to the magnitude and correlation of road trac accidents from the driver's perspective. Therefore, this study aimed to assess the prevalence of RTA and related factors among drivers.
+
+Methods: A community-based cross-sectional study involving 316 drivers was conducted in Southern Ethiopia. The participants were chosen using a systematic random sample technique, and the data were obtained using an interview-administered structured questionnaire. To analyze the data, SPSS software (version 20) was employed. In addition to descriptive statistics, binary logistic regression analysis was also employed to find factors connected to trac accidents. RTA factors were considered statistically significant if they had a *P*-value of 0.05 or below in the multivariate analysis.
+
+Result: The RTA among drivers was 126 (39.9%) (95% confidence interval (CI): 34.2–45.6%) in the previous year. The following factors were associated with RTA: vehicle maintenance (AOR = 0.11, 95% CI: 0.09, 0.96), media utilization (AOR = 0.38, 95% CI: 0.18, 0.65), participation in driving-related training (AOR = 0.73, 95% CI: 0.28, 0.91), punishment for prior trac violations (AOR = 0.56, 95% CI: 0.47, 0.83), and risky driving behavior (AOR = 7.89, 95% CI: 3.22, 12.38).
+
+Conclusion: Two-fifths of the drivers were involved in a trac accident. Risky driving behaviors, vehicle maintenance, media usage, attending driving-related training in the previous 2 years, and prior experience with trac police punishment or warning were all strongly linked to road trac accidents. In light of these statistics, the Federal Ministry of Transport of Ethiopia and
+
+other stakeholders should support making it mandatory for drivers to check their vehicles' safety, provide them with safety training, raise awareness about vehicle maintenance and risky driving behaviors, and enforce strict penalties for trac violations.
+
+associated factors, drivers, epidemiology, road trac accidents, vehicles, risky driving behavior
+
+## Introduction
+
+A road traffic accident (RTA) refers to an incident on a public road or street that injures or kills one or more people and involves at least one vehicle in motion [\(1\)](#page-10-0). As a major threat to public health and economic growth, RTA is a worldwide issue of grave importance. RTA is the eighth largest cause of mortality worldwide. The World Health Organization (WHO) expects RTAs to rise to become the fifth-greatest cause of mortality by 2030. These accidents are one of the top three causes of death for people aged 5 to 44 years and the leading cause of death for young people aged 5 to 29 years [\(2\)](#page-10-1).
+
+RTAs claim the lives of about 3,700 every day and roughly 1.35 million people each year. These RTA fatalities involved cars, buses, motorbikes, bicycles, trucks, and pedestrians. The high proportional burden is borne by pedestrians, cyclists, and motorcyclists, particularly those living in developing countries [\(3](#page-10-2)[–5\)](#page-10-3). Despite a minor decrease in road traffic deaths over the last decade, the current global fatality rate of 16.4 per 100,000 is staggeringly high [\(6\)](#page-10-4). Around 50 million people are injured every year due to non-fatal injuries sustained in RTAs, posing a severe threat to global public health [\(2\)](#page-10-1).
+
+According to the WHO, despite having <60% of the world's registered vehicles, low and middle-income countries account for approximately 93% of deaths and 90% of disabilityadjusted life years (DALYs) lost annually in road traffic crashes. Projections show that these figures will continue to rise over the next 20 years, with Africa having the highest fatality rate [\(5,](#page-10-3) [7\)](#page-10-5).
+
+An RTA is not only a tragedy that kills people but also a disaster that harms people, property, and the environment. The financial burden of these injuries falls heaviest on families, who must deal with both immediate and long-term costs, such as medical bills and lost savings. Road traffic injuries have a high financial impact nationally, especially in developing nations [\(7\)](#page-10-5). Therefore, more effective control measures for the causes of road
+
+Abbreviations: AOR, Adjusted Odd Ratio; CI, confidence interval; DALY, disability-adjusted life years; GDP, gross domestic product; IRB, institutional review board; LMIC, lower- and middle-income countries; RTA, road trac accidents; SNNPR, South Nation Nationalities of People Republic; WHO, the World Health Organization.
+
+accidents are required if we are to achieve a reduction in road deaths and injuries by the year 2030 [\(8\)](#page-10-6).
+
+Economic losses due to road traffic fatalities and injuries are estimated to be 3% of global GDP and 5% of GDP in low- and middle-income nations (LMICs) [\(4\)](#page-10-7). Africa has the highest rate of RTA fatalities per capita, with walkers and other vulnerable road users bearing the brunt of the burden. RTAs are now a major issue in Sub-Saharan Africa (SSA), accounting for roughly one-third of all regional deaths [\(9\)](#page-10-8).
+
+Despite the rising toll of RTAs, road safety is still a neglected concern in many developing countries. The health sector has been hesitant to acknowledge it as a priority public health concern [\(10,](#page-10-9) [11\)](#page-10-10). A substantial amount of evidence implies that RTAs are easily avoided, and many high-income countries have successfully reduced the occurrence of these accidents through proven and cost-effective approaches [\(12\)](#page-10-11). While there has been progress toward worldwide road safety metrics, it is sluggish, and RTA rates remain disproportionately high in LMICs [\(6\)](#page-10-4). The WHO's global status report on road safety in 2018 predicts that SDG target 3.6 will not be met until 2020 [\(5\)](#page-10-3). According to the Third Global Ministerial Conference on Road Safety in 2020, approximately 1.35 million people are killed or injured on the world's roads each year [\(13\)](#page-10-12).
+
+Moreover, in September 2020, the United Nations General Assembly declared the "Decade of Action for Road Safety 2021– 2030," with a new goal of a 50% reduction in RTA-related deaths and injuries by 2030, emphasizing the importance of a holistic approach to road safety and calling for continued improvements in road and vehicle design, as well as enhanced laws and law enforcement, and timely, life-saving emergency care for the injured [\(14\)](#page-10-13). These alarming statistics highlight the importance of updating and enhancing road traffic accident data, which will aid policymakers in adopting evidence-based policies and programs for road safety.
+
+In previous studies, the frequency of RTAs was reported to be 8% in Brazil [\(15\)](#page-10-14), 30 % in Vietnam [\(16\)](#page-10-15), 63.6% in another Brazilian study [\(17\)](#page-10-16), 41.4% in an Indian study [\(18\)](#page-10-17), and 51.50% in an Iranian study [\(19\)](#page-10-0). RTA prevalence varies between sub-Saharan African studies, ranging from 74.0% in a study from Ghana to 87.5% in one from Nigeria [\(20,](#page-10-18) [21\)](#page-10-19). In Ethiopian studies, the RTA varies from 23.5 to 62.5% [\(22](#page-10-20)[–28\)](#page-10-21).
+
+According to a global status assessment on road safety, Ethiopia has at least 114 deaths per 10,000 vehicles per year,
+
+compared to only 10 in the United Kingdom and Ireland and 60 in 39 Sub-Saharan African nations [\(23\)](#page-10-22). A number of studies have also indicated that Ethiopia has one of the world's highest fatality rates per vehicle [\(23,](#page-10-22) [24,](#page-10-23) [29](#page-10-24)[–32\)](#page-10-25). According to the World Health Organization (WHO) estimate, in 2013, the prevalence of road traffic fatalities in Ethiopia was 25.3 per 100,000 population, and the rate is among the highest in the world [\(25\)](#page-10-26).
+
+Although Ethiopia has a higher rate of road traffic morbidity and mortality, data on the scale of road traffic accidents were largely acquired through police records and hospital registration data. However, inadequate data reporting obscured the scale of the problem, and little concern was expressed [\(23,](#page-10-22) [30\)](#page-10-27). Furthermore, even though road crashes are increasing, the government and other stakeholders have given little attention to the issue due to a lack of data on its magnitude and contributing factors from the driver's perspective. Therefore, the objective of this study was to determine the epidemiology of RTA and its associated factors among drivers in Dilla, Southern Ethiopia. Moreover, understanding drivers' perspectives on road traffic accidents can help in the development of policies and strategies for community-based intervention to reduce road traffic accident fatalities. In addition, it can also be used as a sole input to the literature.
+
+## Materials and methods
+
+## Study area and period
+
+The study was conducted in Dilla Town, Gedeo Zone, Southern Ethiopia. Dilla town is one of the two city administrations established in 1904 in the Gedeo Zone. It is 377 km from the capital city, Addis Ababa, and has one main road passing through the town from Addis Ababa to Moyale. The total population in Dilla is 94,189; of which 46,058 (49.9%) are males, and 48,131 (50.1%) are females. During the data collection period, the estimated number of vehicles serving public transportation in Dilla town was 1,266. Bajaj 502, taxis 15, and 159 are private cars (automobiles), 85 are government vehicles, and 505 are motorcycles. There were 304 registered and 673 unregistered vehicles (Source: Dilla Town Administrative Road Transport Office). The study was conducted from 18 October 2021, to 20 November 2021.
+
+## Study design
+
+A community-based cross-sectional study design was conducted to assess the epidemiology of road traffic accidents and associated factors among drivers in Dilla Town, Southern Ethiopia.
+
+## Population
+
+The survey included everyone in Dilla town who drove a privately or government-owned vehicle (automobile, taxi, or Bajaj). All drivers of the selected vehicles outside Dilla, those with <12 months of driving experience before the data collection period, and those who were either unwell or unavailable throughout the study period were excluded.
+
+## Sample size determination
+
+The sample size was calculated using the single-population proportion formula, with prevalence estimates from previous research conducted in Adama city of 56.4 % [\(27\)](#page-10-28), a margin of error of 5%, and a two-sided 95% confidence interval (Zα/2 = 1.96). After attaining the correction formula because the total number of vehicles was <10,000 and by including a 10% non-response rate, the appropriate sample size for this study was 319.
+
+## Sampling techniques
+
+During the study period, there were 1,266 vehicles in the town, of which 15,502, 159, 85, and 505 were taxis, Bajaj, private cars, automobiles, governmental cars, and motorcycles, respectively. A sampling frame of all cars was received from the Dilla Town Road and Transport Bureau, and the computed sample size was proportionally allocated. Individual drivers were also selected using systematic random sampling using their board numbers.
+
+## Study variables
+
+The dependent variable in the study was a road traffic accident. The independent variables included sociodemographic variables (sex, age, marital status, educational status, family size, monthly income, and additional job), personnel (drivers) behavior (chewing khat, habits of alcohol drinking, talking on mobile phones while driving, and driving at speed), vehicle-related factors (service year of the vehicle (perceived vehicle age, vehicle maintenance, and frequency of vehicle health check), and failed brake or mechanical road-related conditions (congested, high traffic roads, damaged roads, and others), driving-related training in the past 2 years, years of holding a driving license, any prior violation of traffic rules, mass media utilization, perceived availability of sufficient traffic signs, and driving experience.
+
+## Operational definition
+
+#### Road trac accident
+
+In this study, a road traffic accident is defined as a collision between cars, between vehicles and pedestrians, between vehicles and animals, or between vehicles and a fixed object that occurred within the previous 12 months. The questions were close-ended. If the driver self-reported RTA experience at least once in the last 12 months, the answer was a "yes." Otherwise, the answer was a "no."
+
+#### Risky driving behavior
+
+If a respondent had engaged in any of the four risky driving behaviors in the last 12 months (i.e., driving after drinking alcohol, chewing khat, driving at high speeds, mobile phoning, or receiving while driving), the answer was a "yes." Otherwise, the answer was a "no."
+
+## Data collection instrument and procedure
+
+An interview-administered structured questionnaire was used to collect data related to the study objectives. Questions about road traffic accidents and related factors were adapted from previous research [\(27,](#page-10-28) [32–](#page-10-25)[37\)](#page-11-0). The data collection tools contained four sections: the first section included the sociodemographic characteristics of the participants; the second section identified the magnitude of road traffic accidents and risky driving behavior of the participants; the third section assessed vehicle-related factors; and the fourth section assessed road-related conditions. Fourth-year public health officers collected the data. After explaining the goal of the study, the interviewers approached the selected study participants at their station for Bajaj and taxi drivers, at their residence or fuel station for automobile drivers, and at their institution for government office cars using information obtained from the city's road and transport office.
+
+## Data quality control
+
+To assure the quality of the data, the questionnaire that was primarily written in English was translated into the local language (Gede'ogna) and then translated back to ensure that it was translated accurately. 1 week before the actual data collection, the instrument was pretested on 5% of drivers from areas other than the research area (Wonago town), and the questionnaire was modified accordingly. 2 days of training were given to the data collectors and supervisors about the research question, sampling technique, data handling, ethical conduct, and quality of data collection. The questionnaires were checked for accuracy and completeness every day by the primary investigator, and modifications were made if errors were discovered in the collected data.
+
+## Data processing and analysis
+
+The data were entered into EPI Info version 7 and analyzed using SPSS version 20 statistical software. Categorical variables were described using frequency or percentage, whereas continuous variables were described using mean and standard deviation. Binary logistic regression analyses were used to determine the influence of each independent variable on road traffic accidents. Those variables with a p-value of < 0.25 on bivariate analysis were chosen for multivariate analysis. Variables with a p-value of < 0.05 in the multivariate analysis were considered significant factors linked with road traffic accidents. Finally, the adjusted odds ratio (AOR) and 95% confidence interval (CI) for statistically significant variables were computed and interpreted.
+
+## Ethical consideration
+
+Ethical approval was secured from the Institutional Review Board of Dilla University (IRB-DU), College of Medicine and Health Science. Permission letters were also secured from Dilla town road and transport offices. Before data collection, information about the purpose of the study and the right not to participate was given to the participants. Oral consent was obtained from all the study participants, and the information collected from the participants was kept confidential.
+
+## Results
+
+## Socio-demographic characteristics
+
+In the current study, out of 319 participants, 316 were included, with a 99% response rate. The majority of the participants (94.6%) were males, with a mean standard deviation (SD) age of 31 (12.6) years and 174 (55%) in the age group of greater than 31 years. More than half (52.2%) of the participants were not married, 142 (45%) had secondary educational status, and 243 (76.9%) had no additional work [\(Table 1\)](#page-4-0).
+
+## Vehicle and driver related variables
+
+Of 316 respondents, ∼124 (39.2%) were driving their own car. Nearly 80% of the vehicles were Bajaj and motor bicycles. The majority of the respondents (64.6%) had between 1 and 4 years of driving experience. Of these, 201 (63.6%) had had the
+
+license for 1 to 4 years. Over half of the participants received drivers related training in the last 2 years.
+
+Moreover, 182 (57.6%) of the participants reported that their vehicles were served for about 1–4 years, and 216 (68.4%) of the participants informed us that their vehicles were checked every 6 months. About three-fourths (75.9%) of the participants also reported that their vehicles had some kind of service in the last year [\(Table 2\)](#page-5-0).
+
+## Risky driving behaviors of the participants
+
+A respondent's risky driving behavior was evaluated when they engaged in one of four behaviors: driving after consuming alcohol, chatting on a cell phone while driving, chewing khat while driving, and/or driving at high speeds. A total of 98 (31.0 %) of the 316 drivers in this study engaged in risky driving behaviors [\(Figure 1\)](#page-5-1). Driving at high speeds, chewing khat while driving, driving after drinking, and the habit of talking on the phone while driving accounted for 24.5, 35.7, 13.3, and 26.5% of the 98 risky driving behaviors, respectively.
+
+## The magnitude of road trac accidents
+
+The magnitude of RTA among drivers in the past year in Dilla Town was 126 (39.9%) [95% confidence interval (CI): 34.2–45.6%] [\(Figure 2\)](#page-6-0).
+
+A total of 42 (33.3%) of 126 participants who had an RTA cited pedestrian concerns as the leading cause of incidents that occurred during the previous 12 months; 39 (30.9%) cited excessive speed; 19 (15%) cited a lack of road signs/zebras or tags; and 20 (15%) cited a breach of traffic rules [\(Figure 3\)](#page-6-1).
+
+## Factors associated with road trac accidents
+
+Variables that have a p-value of < 0.25 in the bivariate logistic regression analysis and are eligible for multivariate logistic regression analysis include gender, educational status, additional job, average monthly income, driving experience, listening to mass media, attending recent driving-related training, receiving any kind of service to a vehicle in the past year, prior punishment by traffic police, availability of sufficient traffic signs or symbols, and risky driving behavior.
+
+Finally, independent variables significantly associated with road traffic accidents at p-value < 0.05 in multivariate logistic regression were those who received any kind of service to a vehicle in the past year, listened to mass media, received drivingrelated training in the last 2 years, prior punishment by traffic police; and risky driving behavior.
+
+<span id="page-4-0"></span>TABLE 1 Socio-demographic characteristics of the participants in Dilla Town, Gedeo Zone, Southern Ethiopia, 2021 (n = 316).
+
+| Variables            |                     | Frequency | Percent<br>(%) |
+|----------------------|---------------------|-----------|----------------|
+| Age (in years)       | <31                 | 142       | 45%            |
+|                      | ≥31                 | 174       | 55%            |
+| Sex                  | Male                | 299       | 94.6           |
+|                      | Female              | 17        | 5.4            |
+| Educational status   | No formal education | 19        | 6              |
+|                      | Primary             | 77        | 24.4           |
+|                      | Secondary           | 142       | 45             |
+|                      | College and above   | 78        | 24.6           |
+| Marital status       | Not married         | 165       | 52.2           |
+|                      | Married             | 151       | 47.2           |
+| Family size (number) | ≤5                  | 101       | 32.0           |
+|                      | >5                  | 215       | 68.0           |
+| Monthly income (ETB) | <1,000              | 26        | 8.2            |
+|                      | 1,000–2,000         | 107       | 33.9           |
+|                      | 2,000–3,000         | 100       | 31.6           |
+|                      | >3,000              | 83        | 26.3           |
+| Any additional work  | Yes                 | 73        | 23.1           |
+|                      | No                  | 243       | 76.9           |
+|                      |                     |           |                |
+
+ETB, Ethiopian Birr.
+
+RTA risk was 89 % lower in vehicles that had maintenance in the previous year than in vehicles that did not [AOR = 0.11, 95% CI: 0.09, 0.96, P-value = 0.001]. Drivers who had received driving-related training in the previous 2 years were 23% less likely to be involved in an RTA than those who did not [AOR= 0.73, 95%CI: 0.28, 0.91, P-value = 0.033]. Drivers who had been charged or warned by traffic cops in the previous year were 44% less likely to have RTA than those who had not [AOR = 0.56, 95% CI: 0.47, 0.83, P-value = 0.041]. RTA was 62 % less likely among drivers who could follow mass media than among those who could not [AOR = 0.38, 95% CI: 0.18, 0.65, P-value = 0.003].
+
+Moreover, drivers who engaged in risky driving behaviors were nearly eight times more likely to be involved in an RTA than those who did not [AOR = 7.89, 95%CI: 3.22, 12.38, P-value = 0.029] [\(Table 3\)](#page-8-0).
+
+## Discussion
+
+In this study, the magnitude of road traffic accidents (RTA) among drivers in the past year in Dilla Town was 126 (39.9%) (95% CI: 34.2–45.6%). This finding is consistent with previous studies conducted in India, in which 41.4% of road traffic accidents occurred among drivers [\(18\)](#page-10-17). Another Ethiopian study conducted in Amhara National Regional State reported 34.5% of RTA among passenger vehicles [\(23\)](#page-10-22). However, this study's
+
+<span id="page-5-0"></span>TABLE 2 Drivers and Vehicle-related characteristics of the participants in Dilla Town, Gedeo Zone, Southern Ethiopia, 2021 (n = 316).
+
+**Variables Frequency Percentage (%)** Vehicle type Bajaj 125 39.6 Motor bicycle 125 39.6 Taxi 4 1.3 Minibus 32 10.1 Mid bus 8 2.5 Public service 22 7 Total 316 100 Driver relation with the car Owner 124 39.2 Recruited 128 40.5 Relative to owner 41 13 Governmental 20 6.3 Other 3 0.9 Total 316 100 The driving experience of the drivers <1 year 64 20.3 1–4 years 204 64.6 5–10 years 48 15.1 Total 316 100 Years of holding a driving license <1year 39 12.3 1–4 years 201 63.6 4–10 years 70 22.2 Above 10 years 6 1.9 Total 316 100 Presence of sufficient zebra line Yes 54 20.7 No 262 82.3 Total 316 100 Have you attained any drivers related training in the last 2 years Yes 165 52.2 No 151 47.8 Total 316 100 For how long has this vehicle served? (in years) <1 67 21.2 1–4 182 57.6 5–10 63 19.9 >10 4 1.3 Total 316 100 Any kind of service in the last year for your vehicles Yes 272 86.1 No 44 13.9 Total 316 100 How frequently do your vehicle healthy checked? Every 6 month 216 68.4 Every year 73 23.1 Every 2 years 19 6.0 ≥3 years 8 2.5 Total 316 100 Any kind of service in the last year for your vehicles Yes 240 75.9 No 76 24.1
+
+(Continued)
+
+TABLE 2 (Continued)
+
+| Variables                            |               | Frequency Percentage |       |  |
+|--------------------------------------|---------------|----------------------|-------|--|
+|                                      |               |                      | (%)   |  |
+|                                      | Total         | 316                  | 100   |  |
+| Working hours                        | 8 h and less  | 107                  | 33.9  |  |
+|                                      | More than 8 h | 209                  | 66.1  |  |
+|                                      | Total         | 316                  | 100   |  |
+| Did you have a habit of following    | Yes           | 140                  | 44.3% |  |
+| mass media                           | No            | 176                  | 55.7% |  |
+|                                      | Total         | 316                  | 100   |  |
+| Perceived availability of sufficient | Yes           | 81                   | 25.6% |  |
+| traffic signs or symbol              | No            | 235                  | 74.4% |  |
+|                                      | Total         | 316                  | 100   |  |
+| Perceived road condition             | Good          | 43                   | 13.6% |  |
+|                                      | Poor          | 273                  | 86.4% |  |
+|                                      | Total         | 316                  | 100   |  |
+|                                      |               |                      |       |  |
+
+![](_page_5_Figure_6.jpeg)
+
+<span id="page-5-1"></span>finding is not consistent with the study conducted among motorcycle taxi drivers in 32 regions of Caicó; the Rio Grande do Norte, Brazil, which found that 63.6% of them were involved in at least one motorcycle accident [\(17\)](#page-10-16). In Southern Nigeria, (87.5%) of commercial motorcyclists were involved in road traffic accidents the year before the study [\(21\)](#page-10-19). In Ghana, 74.0% were reported to have been involved in crashes in the past year before the study [\(20\)](#page-10-18). A systematic review, GIS, and metaanalysis study in Iran showed that the prevalence of road traffic accidents was 51.50% [\(19\)](#page-10-0), and a study was conducted in Adama (54.7%) [\(29\)](#page-10-24). Moreover, it is not consistent with a hospital-based study conducted in Wolaita Zone, SNNPR, Ethiopia, in which 62.5% of trauma victims were due to road traffic accidents [\(30\)](#page-10-27). This might be due to the underreporting of accidents in the study settings.
+
+![](_page_6_Figure_1.jpeg)
+
+<span id="page-6-0"></span>![](_page_6_Figure_2.jpeg)
+
+<span id="page-6-1"></span>The prevalence rate found by this study was also higher than that in a study conducted in Brazil, where the prevalence of road traffic accidents among motorcyclists in the previous 12 months was 8% [\(15\)](#page-10-14). Studies conducted in Vietnam showed that 22.7% and 30% of RTA occurred among taxi drivers and motorcyclists over 1 year, respectively [\(16,](#page-10-15) [31,](#page-10-29) [38\)](#page-11-1); in Mekelle, Ethiopia, 26.4%
+
+of taxi drivers reported involvement in a road traffic crash within the past 3 years [\(32\)](#page-10-25). A study conducted in Chuko town, Sidama, Ethiopia, reported a lifetime prevalence of RTA of 23.55% [\(26\)](#page-10-30). In another study conducted in Mekelle, Ethiopia, where 26.4 % of taxi drivers reported involvement in a road traffic crash over the preceding 3 years [\(39\)](#page-11-2), a study conducted in East Wollega, Western Ethiopia, reported a lifetime prevalence of RTA of 33% [31], and a similar study conducted in Jigjiga, Ethiopia discovered the prevalence of RTA in the previous three years was 32.8% [\(27\)](#page-10-28). The main differences between current research and previous studies in road traffic accidents could be attributed to a lack of safety knowledge; risky driving practices by drivers, such as drinking alcohol and chewing khat while driving [\(33\)](#page-10-31); differences in the study period; the recent increase in the number of vehicles; inadequately experienced drivers; and the approval from drivers who were not adequately trained. Additional factors that may contribute to the high rate of RTA in the research area include a lack of formal training, mobile phone use while driving, dense traffic, failure to prioritize pedestrians, and a lack of awareness campaigns on road safety regulations. Sixtysix percent of drivers in the research setting had more than 8 h of work each day, suggesting that fatigue (which impairs decisionmaking) from working long hours may be linked to the high rate of road traffic accidents observed.
+
+According to this study, drivers who follow social media are less likely to be involved in a car accident than those who do not. This study was consistent with a previous study conducted in Mekelle town, northern Ethiopia [\(32\)](#page-10-25). Therefore, research done in India and Kenya found that media coverage of road traffic accidents can be used as a preventive measure by documenting the experiences of those hurt in RTAs and informing evidence-based solutions; emphasizing the systems approach, which encourages government action [\(34\)](#page-10-32) and encourages people to share information and opinions about road safety through social media [\(35\)](#page-11-3), which can change individual behavior by raising awareness and increasing understanding of the issue.
+
+This study found that prior traffic police punishment was less likely to be associated with road traffic accidents among taxi drivers, which is consistent with previous research conducted in Mekelle, northern Ethiopia [\(32\)](#page-10-25), Hanoi, Vietnam [\(36\)](#page-11-4), and India [\(37\)](#page-11-0), which found that improper enforcement of rules and regulations pertaining to road traffic causes accidents. Penalties for traffic violations may thereby reduce the risk of traffic violations and accidents by motivating drivers to observe traffic laws and improve their understanding of road safety. As a result, both the recent and earlier studies may have implications for altering the kind, degree, and scope of enforcement of traffic violation penalties used generally in Ethiopia, as well as the study's specific context. This may not always be the case. According to a study from China, driving experience significantly impacts how frequently violations occur and accidents happen [\(40\)](#page-11-5).
+
+The prevalence of RTA in this study was lower in vehicles that had been maintained in the preceding year than in vehicles that had not, which is similar to a study conducted in Ghana [\(41\)](#page-11-6) and Nigeria [\(42\)](#page-11-7). Because of this, well-maintained cars not only have a longer lifespan but also improve road safety for all parties involved, including drivers, passengers, and other road users. Accidents and injuries can result from failing to keep a vehicle in excellent operating condition, and a well-designed and maintained vehicle is less likely to be involved in accidents.
+
+According to the current study, drivers who received driving-related training in the previous 2 years were 23 % less likely to be involved in an RTA than those who did not. A study on the efficacy of pre-and post-license driver training found that post-license training programs help improve skills considered relevant to crash prevention, such as hazard perception and advanced vehicle control skills [\(43\)](#page-11-8). Therefore, it is recommended that drivers receive the necessary drivingrelated training to improve their skills and learn about safety measures to prevent traffic accidents and behavioral change in drivers toward risky driving behaviors. Training also enables drivers who had insufficient training during their license examinations to obtain the necessary knowledge.
+
+Moreover, drivers who engaged in risky driving behaviors were more likely to be involved in an RTA than those who did not. This finding is supported by research conducted in Mekelle City, where drivers who consumed alcohol were more likely to be involved in an RTA than those who did not [\(39\)](#page-11-2), as well as a study conducted in Pakistan, Korea [\(38\)](#page-11-1), and Tehran, Iran [\(44\)](#page-11-9). In addition, a study conducted in Hanoi, Vietnam, found that using a cell phone while riding in a taxi was linked with road traffic accidents [\(45\)](#page-11-10). As a result, to successfully minimize the magnitude of road traffic incidents, a proper countermeasure against drivers who engage in risky driving behaviors should be devised. Furthermore, to reduce road traffic accidents, alcohol intake tests, substance use, driver belt use, driver health status, driving speed, and vehicle maintenance status should be checked regularly.
+
+## Limitations of the study
+
+The study's limitations were that taxi drivers are a highly mobile community, which can contribute to discrepancies in the selection of study participants. The study only included living drivers, which can lead to an underestimation of the significance of the conclusion. Underestimation of the prevalence could also be due to social desirability bias. The current study also failed to evaluate drivers' perceptions of risky behavior and other experiences with it. Additionally, some demographic data were not collected, including height, weight, and BMI, which could be used to assess the risk of obstructive sleep apnea and determine whether a participant was at fault or injured in an accident.
+
+<span id="page-8-0"></span>TABLE 3 Factors associated with road trac accidents among vehicle drivers in Dilla Town, Gedeo Zone, Southern Ethiopia, 2021 (n = 316).
+
+| Variables                                              |     | Prevalence of RTA |                    | AOR [95%CI]       | P-value |
+|--------------------------------------------------------|-----|-------------------|--------------------|-------------------|---------|
+|                                                        | Yes | No                |                    |                   |         |
+| Age (years)                                            |     |                   |                    |                   |         |
+| <31                                                    | 62  | 80                | 1.33 [0.85, 2.09]  |                   |         |
+| ≥31                                                    | 64  | 110               | 1                  |                   |         |
+| Sex                                                    |     |                   |                    |                   |         |
+| Male                                                   | 119 | 180               | 0.94 [0.35, 2.55]  | 0.63 [0.69, 4.55] | 0.89    |
+| Female                                                 | 7   | 10                | 1                  | 1                 |         |
+| Educational status                                     |     |                   |                    |                   |         |
+| No formal education                                    | 8   | 11                | 1.54 [0.55, 4.31]  | 1.14 [0.23, 2.33] | 0.07    |
+| Primary school                                         | 37  | 40                | 1.96 [1.02, 3.77]  | 2.34 [0.79, 4.67] | 0.09    |
+| Secondary school                                       | 56  | 86                | 1.38 [0.77, 2.47]  | 1.61 [0.55, 3.42] | 0.17    |
+| College and above                                      | 25  | 53                | 1                  | 1                 |         |
+| Marital status                                         |     |                   |                    |                   |         |
+| Not married                                            | 69  | 96                | 1.96 [0.75, 1.86]  |                   |         |
+| Married                                                | 57  | 94                | 1                  |                   |         |
+| Family size                                            |     |                   |                    |                   |         |
+| Less than or equal to 5                                | 45  | 56                | 0.75 [0.46,1.21]   |                   |         |
+| Greater than 5                                         | 81  | 134               | 1                  |                   |         |
+| Additional job(s)                                      |     |                   |                    |                   |         |
+| Yes                                                    | 26  | 47                | 0.79 [0.46, 1.36]  | 2.45 [0.49, 2.38] | 0.628   |
+| No                                                     | 100 | 143               | 1                  | 1                 |         |
+| Average monthly income                                 |     |                   |                    |                   |         |
+| Less than 1,000                                        | 7   | 19                | 2.22 [0.86,5.75]   | 1.11 [0.92,6.65]  | 0.667   |
+| 1,000–2,000                                            | 32  | 62                | 1.58 [0.88,2.83]   | 4.56 [0.13,8.34]  | 0.083   |
+| 2,000–3,000                                            | 42  | 54                | 1.0 5 [0.59,1.85]  | 2.09 [0.93,7.34]  | 0.467   |
+| Greater than 3,000                                     | 45  | 55                | 1                  | 1                 |         |
+| Did you listen to mass media                           |     |                   |                    |                   |         |
+| Yes                                                    | 31  | 109               | [0.15, 0.41]       | 0.38 [0.18,0.65]  | 0.003** |
+| No                                                     | 95  | 81                | 1                  | 1                 |         |
+| Vehicle maintenance                                    |     |                   |                    |                   |         |
+| Yes                                                    | 21  | 55                | 0.49 [0.28,0.86]*  | 0.11 [0.09,0.96]* | 0.001** |
+| No                                                     | 105 | 135               | 1                  | 1                 |         |
+| Driving experience                                     |     |                   |                    |                   |         |
+| Less than one year                                     | 45  | 19                | 3.05 [1.39,6.66]   | 4.81 [0.89,7.86]  | 0.178   |
+| 1–4 years                                              | 60  | 144               | 0.54 [0.28,1.02]   | 0.78 [0.14, 2.61] | 0.073   |
+| 5–10 years                                             | 21  | 27                | 1                  | 1                 |         |
+| Working hours                                          |     |                   |                    |                   |         |
+| 8 h and less                                           | 42  | 65                | 0.96 [0.60, 1.55]  |                   |         |
+| Greater than 8 h                                       | 84  | 125               | 1                  |                   |         |
+| Attending driving-related training in the last 2 years |     |                   |                    |                   |         |
+| Yes                                                    | 38  | 127               | 0.21 [0.13, 0.35]* | 0.73 [0.28, 0.91] | 0.033** |
+| No                                                     | 88  | 63                | 1                  | 1                 |         |
+| Presence of sufficient zebra line                      |     |                   |                    |                   |         |
+| Yes                                                    | 21  | 33                | 0.95 [0.52, 1.73]  | 0.58 [0.22, 1.50] | 0.289   |
+| No                                                     | 105 | 157               | 1                  | 1                 |         |
+
+(Continued)
+
+TABLE 3 (Continued)
+
+| Variables                                          | Prevalence of RTA |     | COR [95% CI]        | AOR [95%CI]        | P-value |
+|----------------------------------------------------|-------------------|-----|---------------------|--------------------|---------|
+|                                                    | Yes               | No  |                     |                    |         |
+| Prior punishment by traffic polices                |                   |     |                     |                    |         |
+| Yes                                                | 30                | 91  | 0.34 [0.21, 0.56]*  | 0.56 [0.47, 0.83]  | 0.041** |
+| No                                                 | 96                | 99  | 1                   | 1                  |         |
+| Perceived road condition                           |                   |     |                     |                    |         |
+| Good                                               | 16                | 27  | 0.88 [0.45, 1.71]   | 0.53 [0.19, 1.47]  | 0.09    |
+| Poor                                               | 110               | 163 | 1                   | 1                  |         |
+| Availability of sufficient traffic signs or symbol |                   |     |                     |                    |         |
+| Yes                                                | 27                | 54  | 0.69 [0.40, 1.17]*  | 0.45 [0.98,3.07]   | 0.065   |
+| No                                                 | 99                | 136 | 1                   | 1                  |         |
+| Risky driving behaviors                            |                   |     |                     |                    |         |
+| Yes                                                | 75                | 23  | 10.68 [6.08, 18.74] | 7.89 [3.22, 12.38] | 0.029*  |
+| No                                                 | 51                | 167 | 1                   | 1                  |         |
+
+AOR, Adjusted Odds Ratio; COR, Crude Odds Ratio; \*p-value < 0.25 in the bivariate analysis; \*\*p-value ≤ 0.05 in the multivariate analysis; Not married- Single, widowed, and divorced.
+
+## Conclusion
+
+In this study, two-fifths of the drivers in Dilla town had a traffic accident in the previous year (2020) preceding the data collection period. Factors such as the experience of punishment or warning by traffic police in the last year, vehicle servicing, listening to mass media, attending drivingrelated training in the previous 2 years, and risky driving behaviors were all significantly associated with road traffic accidents. As a result, the Federal Ministry of Transport of Ethiopia and other stakeholders should be encouraged to advise drivers on vehicle safety and driving-related safety training, enhance their capacity to listen to the media, and raise awareness about vehicle maintenance and unsafe driving behaviors. Traffic officials should inspect the safety of the vehicles and advise all car owners to verify their vehicles' service. Additionally, it is recommended that drivers who disobey traffic laws and exhibit unsafe driving behaviors should be made aware of their actions and punished. Additional prospective and qualitative research is required to provide detailed accident characteristics and taxi driver behavior and perception.
+
+## Data availability statement
+
+The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author/s.
+
+## Ethics statement
+
+The studies involving human participants were reviewed and approved by the Institutional Review Board of Dilla University (IRB-DU), College of Medicine and Health Science. The patients/participants provided their written informed consent to participate in this study.
+
+## Author contributions
+
+HH and BN were involved in the conceptualization, design, analysis, interpretation, and drafting of the manuscript and the critical revision of intellectual content. RK, EA, GZ, ZA, and NE were involved in the study's design, data analysis, data interpretation, and manuscript drafting and revision. All authors agree to be accountable for all aspects of the work and have approved the final version of the manuscript to be published.
+
+## Acknowledgments
+
+The authors would like to thank the study participants, Dilla University, Dilla Town Transport Office, and Dilla University staff, for their encouragement and constructive comments that contributed to the success of this manuscript. We would also like to thank the 4th-year public health students for participating in data collection and as supervisors.
+
+## Conflict of interest
+
+The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
+
+The reviewer CK declared a shared affiliation with the author(s) to the handling editor at the time of review.
+
+## Publisher's note
+
+All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
+
+## References
+
+- <span id="page-10-0"></span>1. Peden M, Scurfield R, Sleet D, Mohan D, Hyder AA, Jarawan E, et al. (editors). World Report on Road Traffic Injury Prevention. Geneva: World Health Organization (2004).
+- <span id="page-10-1"></span>2. Organization WH. World Health Organization Road Traffic Injuries. Available online at: <https://wwwwhoint/news-room/fact-sheets/detail/road-traffic-injuries> (accessed July 14, 2019).
+- <span id="page-10-2"></span>3. Rana JS, Khan SS, Lloyd-Jones DM, Sidney S. Changes in mortality in top 10 causes of death from 2011 to 2018. J Gen Intern Med. (2021) 36:2517–8. doi: [10.1007/s11606-020-06070-z](https://doi.org/10.1007/s11606-020-06070-z)
+- <span id="page-10-7"></span>4. STA G. Global status report on road safety. Injury Prevention. (2015) 15:286. doi: [10.1136/ip.2009.023697](https://doi.org/10.1136/ip.2009.023697)
+- <span id="page-10-3"></span>5. World Health Organization. Global Status Report on Road Safety 2018. Geneva: World Health Organization (2018).
+- <span id="page-10-4"></span>6. Bari I, Rosen H, Paichadze N, Peden M, Gonzalez JM, Hyder A. 0073 Global road safety: analysis of global status reports on road safety. Inj Prevent. (2021) 27(Suppl 3):A20. doi: [10.1136/injuryprev-2021-SAVIR.50](https://doi.org/10.1136/injuryprev-2021-SAVIR.50)
+- <span id="page-10-5"></span>7. Peden M, Sminkey L. World health organization dedicates world health day to road safety. Inj Prevent. (2004) 10:67. doi: [10.1136/ip.2004.005405](https://doi.org/10.1136/ip.2004.005405)
+- <span id="page-10-6"></span>8. Sagamiko T, Mbare N. Modelling road traffic accidents counts in Tanzania: a poisson regression approach. Tanz J Sci. (2021) 47:308–14.
+- <span id="page-10-8"></span>9. Marquez PV, Farrington JL. The Challenge of Non-Communicable Diseases and Road Traffic Injuries in Sub-Saharan Africa: An Overview. Washington, DC: World Bank (2013). Available online at: [https://openknowledge.worldbank.org/handle/](https://openknowledge.worldbank.org/handle/10986/16451) [10986/16451](https://openknowledge.worldbank.org/handle/10986/16451)
+- <span id="page-10-9"></span>10. World Health Organization. Global Status Report on Road Safety 2015. Geneva: World Health Organization (2015).
+- <span id="page-10-10"></span>11. Watkins K. Safe and Sustainable Roads: An Agenda for Rio+ 20. (2018). Available online at: [https://www.gtkp.com/assets/uploads/20120518-](https://www.gtkp.com/assets/uploads/20120518-161154-4751-Rio_20_Report_lr.pdf) [161154-4751-Rio\\_20\\_Report\\_lr.pdf](https://www.gtkp.com/assets/uploads/20120518-161154-4751-Rio_20_Report_lr.pdf)
+- <span id="page-10-11"></span>12. Elvik R. The stability of long-term trends in the number of traffic fatalities in a sample of highly motorised countries. Acc Anal Prevent. (2010) 42:245–60. doi: [10.1016/j.aap.2009.08.002](https://doi.org/10.1016/j.aap.2009.08.002)
+- <span id="page-10-12"></span>13. WHO. Declaration S. Third Global Ministerial Conference on Road Safety: Achieving Global Goals 2030. Stockholm: World Health Organization (2020).
+- <span id="page-10-13"></span>14. UN.ESCAP. Road Safety: Saving Lives Beyond 2020 in the Asia-Pacific Region. (2020). Retrieved from: <https://hdl.handle.net/20.500.12870/2881>
+- <span id="page-10-14"></span>15. Seerig LM, Bacchieri G, Nascimento GG, Barros AJ, Demarco FF. Use of motorcycle in Brazil: users profile, prevalence of use and traffic accidents occurrence—a population-based study. Ciência Saúde Coletiva. (2016) 21:3703–10. doi: [10.1590/1413-812320152112.28212015](https://doi.org/10.1590/1413-812320152112.28212015)
+- <span id="page-10-15"></span>16. Nguyen-Phuoc DQ, Nguyen HA, De Gruyter C, Su DN, Nguyen VH. Exploring the prevalence and factors associated with self-reported traffic crashes among app-based motorcycle taxis in Vietnam. Transport policy. (2019) 81:68–74. doi: [10.1016/j.tranpol.2019.06.006](https://doi.org/10.1016/j.tranpol.2019.06.006)
+- <span id="page-10-16"></span>17. Almeida GC, Medeiros FdCDd, Pinto LO, Moura JMBdO, Lima KC. Prevalence and factors associated with traffic accidents involving motorcycle taxis. Revista brasileira de enfermagem. (2016) 69:382–8. doi: [10.1590/0034-7167.2016690223i](https://doi.org/10.1590/0034-7167.2016690223i)
+- <span id="page-10-17"></span>18. Ashish T, Devang R. Prevalence of road traffic accidents and driving practices among young drivers. Healthline, J Indian Assoc Prevent Soc Med. (2011) 2:72–5. Available online at: <http://www.iapsmgc.org/OA17V2I2.pdf>
+
+- 19. Najimi-Varzaneh A, Gholami Fesharaki M. Prevalence of road traffic accidents in Iran: a systematic review, GIS and meta-analysis. Iran Red Crescent Med J. (2018) 20:e83852. doi: [10.5812/ircmj.83852](https://doi.org/10.5812/ircmj.83852)
+- <span id="page-10-18"></span>20. Konlan KD, Doat AR, Mohammed I, Amoah RM, Saah JA, Konlan KD, et al. Prevalence and pattern of road traffic accidents among commercial motorcyclists in the Central Tongu District, Ghana. Sci World J. (2020) 2020:3718. doi: [10.1155/2020/9493718](https://doi.org/10.1155/2020/9493718)
+- <span id="page-10-19"></span>21. Johnson OE. Prevalence and pattern of road traffic accidents among commercial motorcyclists in a city in Southern Nigeria. Educ Res. (2012) 3:537– 542. Available online at: <http://www.interesjournals.org/ER>
+- <span id="page-10-20"></span>22. Asefa NG, Ingale L, Shumey A, Yang H. Prevalence and factors associated with road traffic crash among taxi drivers in Mekelle Town, Northern Ethiopia, 2014: A cross sectional study. PLoS ONE. (2015) 10:e0118675. doi: [10.1371/journal.pone.0118675](https://doi.org/10.1371/journal.pone.0118675)
+- <span id="page-10-22"></span>23. Mekonnen FH, Teshager S. Road traffic accident: the neglected health problem in Amhara National Regional State, Ethiopia. Ethiop J Health Dev. (2014) 28:3–10. doi: [10.4314/EJHD.V28I1](https://doi.org/10.4314/EJHD.V28I1)
+- <span id="page-10-23"></span>24. Hailemichael F, Suleiman M, Pauolos W. Magnitude and outcomes of road traffic accidents at Hospitals in Wolaita Zone, SNNPR, Ethiopia. BMC Res Notes. (2015) 8:135. doi: [10.1186/s13104-015-1094-z](https://doi.org/10.1186/s13104-015-1094-z)
+- <span id="page-10-26"></span>25. Violence W, Prevention I. Organization, WH: Global Status Report on Road Safety 2015. Geneva: World Health Organization (2015).
+- <span id="page-10-30"></span>26. Tegegne KT. Assessment of prevalence of road traffic accident and it's associated factors in Chuko Town, Southern Ethiopia. South Asian Res J Pharm Sci. (2019) 1:91–9. doi: [10.36346/SARJPS.2019.v01i03.003](https://doi.org/10.36346/SARJPS.2019.v01i03.003)
+- <span id="page-10-28"></span>27. Wordofa A. Prevalence and factors associated with road traffic accident among drivers of adama town, oromia regional state, Ethiopia (Doctoral dissertation). Addis Ababa University.
+- <span id="page-10-21"></span>28. Hussen M, Hashi A, Seifu W, Omer M. Prevalence of road traffic accidents, and associated factors among taxi drivers in Jigjiga Town, Ethiopian Somali Regional State, Eastern Ethiopia. Prevalence. (2020) 8. doi: [10.21522/TIJPH.2013.08.01.Art018](https://doi.org/10.21522/TIJPH.2013.08.01.Art018)
+- <span id="page-10-24"></span>29. Getahun KA. Time series modeling of road traffic accidents in Amhara Region. J Big Data. (2021) 8:102. doi: [10.1186/s40537-021-00493-z](https://doi.org/10.1186/s40537-021-00493-z)
+- <span id="page-10-27"></span>30. Abegaz T, Berhane Y, Worku A, Assrat A, Assefa A. Road traffic deaths and injuries are under-reported in ethiopia: a capture-recapture method. PLoS ONE. (2014) 9:e103001. doi: [10.1371/journal.pone.0103001](https://doi.org/10.1371/journal.pone.0103001)
+- <span id="page-10-29"></span>31. Seid M, Azazh A, Enquselassie F, Yisma E. Injury characteristics and outcome of road traffic accident among victims at Adult Emergency Department of Tikur Anbessa specialized hospital, Addis Ababa, Ethiopia: a prospective hospital based study. BMC Emerg Med[. \(2015\) 15:10. doi: 10.1186/s12873-015-](https://doi.org/10.1186/s12873-015-0035-4) 0035-4
+- <span id="page-10-25"></span>32. Asefa F, Assefa D, Tesfaye G. Magnitude of, trends in, and associated factors of road traffic collision in central Ethiopia. BMC Public Health. (2014) 14:1072. doi: [10.1186/1471-2458-14-1072](https://doi.org/10.1186/1471-2458-14-1072)
+- <span id="page-10-31"></span>33. Deresse E, Komicha MA, Lema T, Abdulkadir S, Roba KT. Road traffic accident and management outcome among in Adama Hospital Medical College, Central Ethiopia. Pan Afr Med J[. \(2021\) 38:190. doi: 10.11604/pamj.2021.38.190.](https://doi.org/10.11604/pamj.2021.38.190.11650) 11650
+- <span id="page-10-32"></span>34. Jha N, Srinivasa DK, Roy G, Jagdish S, Minocha RK. Epidemiological study of road traffic accident cases: A study from South India. Indian J Community Med. (2004) 29:20–4. doi: [10.4103/0970-0218.62568](https://doi.org/10.4103/0970-0218.62568)
+
+- 35. Tiruneh BT, Dachew BA, Bifftu BB. Incidence of road traffic injury and associated factors among patients visiting the emergency department of Tikur Anbessa specialized teaching hospital, Addis Ababa, Ethiopia. Emerg Med Int. (2014) 2014:439818. doi: [10.1155/2014/439818](https://doi.org/10.1155/2014/439818)
+- <span id="page-11-4"></span>36. Abegaz T, Gebremedhin S. Magnitude of road traffic accident related injuries and fatalities in Ethiopia. PLoS ONE. (2019) 14:e0202240. doi: [10.1371/journal.pone.0202240](https://doi.org/10.1371/journal.pone.0202240)
+- <span id="page-11-0"></span>37. Hassen A, Godesso A, Abebe, L, Girma E. Risky driving behaviors for road traffic accident among drivers in Mekele city, Northern Ethiopia. BMC Res Notes. (2011) 4:535. doi: [10.1186/1756-0500-4-535](https://doi.org/10.1186/1756-0500-4-535)
+- <span id="page-11-1"></span>38. Youn Kw L, Yim Sh. Human factors involved in traffic accidents and unsafe driving behaviors of taxi drivers. Korea: kjoem. (2019) 18:307–17. doi: [10.35371/kjoem.2006.18.4.307](https://doi.org/10.35371/kjoem.2006.18.4.307)
+- <span id="page-11-2"></span>39. Woldu AB, Desta AA, Woldearegay TW. Magnitude and determinants of road traffic accidents in Northern Ethiopia: a cross-sectional study. BMJ Open. (2020) 10:e034133. doi: [10.1136/bmjopen-2019-034133](https://doi.org/10.1136/bmjopen-2019-034133)
+
+- <span id="page-11-5"></span><span id="page-11-3"></span>40. Liu Z, Wu H, Li R. Effects of the penalty mechanism against traffic violations in China: a joint frailty model of recurrent violations and a terminal accident. Acc Anal Prevent. (2020) 141:105547. doi: [10.1016/j.aap.2020.105547](https://doi.org/10.1016/j.aap.2020.105547)
+- <span id="page-11-6"></span>41. Oduro SD. Brake failure and its effect on road traffic accident in Kumasi Metropolis, Ghana. Int J Sci Technol. (2012) 1:448–54.
+- <span id="page-11-7"></span>42. Bun E. Road traffic accidents in Nigeria: a publichealth problem. Afrimedic journal. (2012) 3:34–5. Available online at: [https://www.sciencedirect.com/science/](https://www.sciencedirect.com/science/article/pii/S0925753512001786) [article/pii/S0925753512001786](https://www.sciencedirect.com/science/article/pii/S0925753512001786)
+- <span id="page-11-8"></span>43. Beanland V, Goode N, Salmon PM, Lenné MG. Is there a case for driver training? A review of the efficacy of pre- and post-licence driver training. Saf Sci. (2013) 51:127–37. doi: [10.1016/j.ssci.2012.06.021](https://doi.org/10.1016/j.ssci.2012.06.021)
+- <span id="page-11-9"></span>44. Shams M, Rahimi-Movaghar V. Risky driving behaviors in Tehran, Iran. Traffic Inj Prev. (2009) 10:91–4. doi: [10.1080/15389580802492280](https://doi.org/10.1080/15389580802492280)
+- <span id="page-11-10"></span>45. Truong LT, Nguyen HTT. Mobile phone related crashes among motorcycle taxi drivers. Acc Anal Prevent. [\(2019\) 132:105288. doi: 10.1016/j.aap.2019.](https://doi.org/10.1016/j.aap.2019.105288) 105288

Zotero/001_artiklid/Road Safety Indicators/haueroverdispersionmodellingaccidentsroad2001.md

@@ -0,0 +1,233 @@
+---
+category: literaturenote
+citekey: haueroverdispersionmodellingaccidentsroad2001
+title: Overdispersion in modelling accidents on road sections and in Empirical Bayes estimation
+authors: "Hauer, Ezra"
+year: 2001
+date: 2001-00-00 2001
+doi: 10.1016/s0001-4575(00)00094-4
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/s0001-4575(00)00094-4"
+zotero_key: 3JZ98TUI
+zotero_storage: 2ATV9SP8
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Hauer, Ezra"
+status: converted
+---
+Accident Analysis and Prevention 33 (2001) 799–808
+
+![](_page_0_Picture_2.jpeg)
+
+www.elsevier.com/locate/aap
+
+# Overdispersion in modelling accidents on road sections and in Empirical Bayes estimation
+
+# Ezra Hauer \*
+
+35 *Merton Street*, *Apartment* 1706, *Toronto*, *Ont*., *Canada M*5*S* 3*G*<sup>4</sup>
+
+Received 28 September 2000; received in revised form 08 October 2000; accepted 30 October 2000
+
+#### **Abstract**
+
+In multivariate statistical models of road safety one usually finds that the accident counts are 'overdispersed'. The extent of the overdispersion is itself subject to estimation. It is shown that the assumption one makes about the nature of overdispersion will affect the maximum likelihood estimates of model parameters. If one assumes that the same overdispersion parameter applies to all road sections in the data base, then, the maximum likelihood estimate of parameters will be unduly influenced by very short road sections and insufficiently influenced by long road sections. The same assumption about the overdispersion parameter also leads to an inconsistency when one estimates the safety of a road section by the Empirical Bayes method. A way to avoid both problems is to estimate an overdispersion parameter () that applies to a unit length of road, and to set the overdispersion parameter for a road section of length *L* to *L*. How this would change the estimates of regression parameters for road section models now in use requires examination. Safety estimation by the Empirical Bayes method is altered substantially. © 2001 Elsevier Science Ltd. All rights reserved.
+
+*Keywords*: Empirical Bayes; Overdispersion; Negative binomial; Multivariate model
+
+# **1. Introduction**
+
+Entities have accident counts and traits. An entity can be a road section with traits such as length, traffic flow, number of lanes; entities can be drivers, with gender, age, annual distance travelled as traits; they can be grade crossings, with train traffic, vehicle traffic, sight distance, protective device used as traits; entities can be cities, with road density, number of inhabitants as traits, and so on. Often statistical regression models are formulated to express the expected accident count of an entity as a function of its traits. After the unknown model parameters are estimated, one usually finds that the accident counts are 'overdispersed'. That is, that the differences between the accident counts and model predictions, are larger than what would be consistent with the assumption that accident counts are Poisson distributed. This led researchers to use the Negative Binomial (NB) distribution to represent the distribution of accident counts.
+
+The NB distribution arises when accident counts for entity *i* are Poisson distributed with a mean *i* while is random variable drawn from a gamma distribution with mean=1 and variance=1/ (see, e.g. Guo 1996). We will call the 'overdispersion parameter'. Let *Yi* be the count of accidents on entity *i*. By the NB distribution,
+
+$$P(Y_i = y_i) = \frac{\Gamma(y_i + \phi)}{\Gamma(\phi)y_i!} \left(\frac{\phi}{\eta_i + \phi}\right)^{\phi} \left(\frac{\eta_i}{\eta_i + \phi}\right)^{y_i}.$$
+ (1)
+
+The value of the overdispersion parameter is estimated along with all other unknown model parameters.
+
+The overdispersion parameter features in two uses to which the results of statistical regression modelling are put. First, when the results of modelling are used to estimate what is the mean accident frequency of entities with given traits, the overdispersion parameter serves to indicate how widely the accidents counts are distributed around the estimated mean. Second, when the results of modelling are used to estimate the safety of a specific entity for which the traits and the accident record are known by the Empirical Bayes (EB) method (see, e.g.
+
+<sup>\*</sup> Tel.: +1-416-483-4452; fax: +1-416-483-4415. *E*-*mail address*: ezra.hauer@utoronto.ca (E. Hauer).
+
+Hauer, 1997), the overdispersion parameter determines the relative weights given to the model prediction and the accident record.
+
+When fitting models to accident counts it is often assumed that one overdispersion parameter is common to all entities. Thus, e.g. a single common overdispersion parameter is used by Maycock and Hall (1984)to model accidents at roundabouts, by Hausman et al. (1984) to model patent counts; Hauer et al. (1989) to model accidents at urban signalized intersections; by Bonneson and McCoy (1993) to model accidents at stop-controlled rural intersections; by Guo (1996) to model quarterly counts of surgical procedures; by Miaou (1996) to model truck accidents; by Vogt and Bared (1998) to model accidents on rural road segments and intersections, and by many others.
+
+To use an overdispersion parameter that is common to all entities may be found limiting. There is no reason to expect that the gamma distribution from which is sampled is the same for all the entities that make up the data for the regression, or to all the entities to which the predictions may be thought to apply. One way to remove this limitation is to assume that *i*=*i* (see, e.g. Cameron and Trivedi, 1998 and Maher and Summersgill, 1996). When the parameter is set to 0, the standard NB model is obtained; when 0 is chosen then the variance of the gamma distribution from which is sampled decreases as *<sup>i</sup>* increases. Miaou and Lum (1993) modelled crashes on rural interstate highways using =1. Using the data from Maycock and Hall (1984) for four-arm roundabouts, Maher and Summersgill (1996) report that =0.25 maximized the likelihood of the data. Yet another possibility is to assume that itself is a function of the various covariates and unknown parameters. Heydecker and Wu (1999) used this approach when modelling accidents at three-arm junctions. The attraction of such generalizations is that they can better represent the reality of specific data sets.
+
+My experience with accident prediction models for road sections which differ in length leads me to believe that the choice of the expression for overdispersion is not only a matter of getting a better fit at the price of additional parameters in the model. The issue is not only empirical but also conceptual. In Section 2I show why problems of parameter estimation may arise when the same value of is assumed to apply to road sections that differ in length. In Section 3I discuss the general mechanism by which overdispersion arises in accident modelling. This understanding leads to a suggestion of a remedy to the aforementioned problem. The remedy is discussed in Section 4. In Section 5 I argue that the use of a single and common dispersion parameter also leads to a logical blemish in the Empirical Bayes Estimation of expected accident frequency and that the remedy in Section 4 removes the blemish.
+
+Table 1 Data for two road sections
+
+| Section | Traits         |         | Accident counts        |
+|---------|----------------|---------|------------------------|
+|         | L, length (km) | X, AADT | Y, accidents in a year |
+| 1       | 0.5            | 500     | 1                      |
+| 2       | 10.0           | 500     | 40                     |
+
+#### **2. A problem of influence**
+
+In this section a numerical example is used to illustrate how two alternative assumptions about the distribution of accident counts influence parameter estimates. In this example a parameter is estimated once assuming that accident counts are Poisson distributed and next using the NB distribution assumption. The data in this example are for two road sections with the traits and accident counts given in Table 1. Take, e.g. the model equation *<sup>i</sup>*=*Li Xi* 0.5. In this *<sup>i</sup>* is the expected number of accidents per year for road section *i*, *Li* is the length of road section *i*, and *Xi* is its Annual Average Daily Traffic (AADT). The parameter is to be estimated by the method of maximum likelihood.
+
+#### <sup>2</sup>.1. *Poisson regression*
+
+Assume first that the accident counts (*Y*) comes from a Poisson distribution. The Poisson log-likelihood functions with respect to for section 1, section 2, both sections on the same graph, and the sum for both sections together are shown in Figs. 1 and 2 Fig. 3 Fig. 4.
+
+The maximum in Fig. 1 is at 1=1/(0.5×5000.5)= 0.089, in Fig. 2 at 2=40/(10×5000.5)=0.179. Fig. 3 serves to show that the variation in log-likelihood for section 1 is very small compared to the variation in the log-likelihood for section 2. This is why the much longer road section 2 with its many accidents dominates the joint log-likelihood function in Fig. 4. As a result the maximum for the sum of two log-likelihoods is at <sup>1</sup> and 2=0.175, very close to the maximum for road section 2.
+
+That a data point with many accidents should have more influence on the estimate of than a data point with a few accidents is in accord with intuition and can be explained from basic principles. For road section *i*, is its mean number of accidents divided by *Li Xi* 0.5. When is estimated, the accident count replaces the mean number of accidents. If the accident count is Poisson distributed, the variance of this estimate of is the mean*i*/(*Li Xi* 0.5) <sup>2</sup>=/(*Li Xi* 0.5). Thus, whatever the value of , since *L*1/*L*2=1/20, the variance of <sup>1</sup> is 20 times the variance of 2. In combining two estimates of
+
+![](_page_2_Figure_2.jpeg)
+
+different precision, the combined estimate is most precise if we use a linear combination of <sup>1</sup> and <sup>2</sup> with the inverse of their variances as weights. That is, 1&2= (0.089/20+0.179/1)/(1/20+1/1)=0.175. In this, the weight of 1=0.089 is 1/21 and the weight of 2= 0.179 is 1/1.05.
+
+### <sup>2</sup>.2. *Negatie binomial regression*
+
+Assume now that accident counts (*Y*) come from the negative binomial distribution in Eq. (1) and that it is known that =1. The negative binomial log-likelihood functions with respect to for section 1, section 2, both sections on the same graph, and the sum for both sections together are shown in Fig. 5 Fig. 6 Fig. 7 Fig. 8. The maxima in Fig. 5 and Fig. 6 are, as before, at 1=0.089 and 2=0.179. However, as is evident from the comparison of Fig. 3 and Fig. 7, the log-likelihood function for road section 2 is now much flatter. Therefore, when the log-likelihoods for sections 1 and 2 (Fig. 8) are added, the 1 and 2 shifts a considerable way from <sup>2</sup> in the direction of 1. It is now 0.144. The mutual position of the estimates is depicted in Fig. 9.
+
+![](_page_2_Figure_6.jpeg)
+
+![](_page_2_Figure_7.jpeg)
+
+It is commonly believed that the estimates of model parameters do not change much when accident counts are assumed to be NB instead of Poisson distributed; the belief is that alternative assumptions about the distribution of accident counts affect mainly the precision with which the model parameters are estimated. In this numerical example, in which there are large differences in road section length, the maximum likelihood parameter estimate does depend on what is assumed to be the distribution of accident counts.
+
+The issue is now clear. We expect that road section 2, which is 20 times longer than road section 1 and has 40 times as many accidents, will always exert a strong pull on the maximum likelihood estimate 1&2. It is disturbing to find that assuming accident counts to be NB distributed with a common , considerably weakens the influence of what should be the dominant data point. Why is this so?
+
+The transparent explanation is again provided by examining the relative precision of the estimates <sup>1</sup> and 2. Under the Poisson assumption the variance of an estimate of was mean/(*LX*0.5) 2 . Under the NB assumption the variance of an estimate of is [mean× (1+mean/)]/(*LX*0.5) . If we take mean1=0.144×
+
+![](_page_2_Figure_11.jpeg)
+
+![](_page_3_Figure_2.jpeg)
+
+0.5×5000.5=1.6 accidents per year and mean2=20× 1.6=32 accidents per year, then the ratio of their variances is [1.6(1+1.6)/*L*<sup>1</sup> 2 ]/[(32(1+32)/(20*L*)1 ]= 4.16/2.64-1.5. Combining the two *i*'s we now obtain 1 and 2=(0.089/4.16+0.179/2.64)/(1/4.16+1/2.64)= 0.144. Whereas under the Poisson assumption the variance of <sup>1</sup> was 20 times that of 2, under the NB assumption it is only 1.5 times.
+
+The root of the problem in the numerical example is the innocuous but commonly made assumption, that the overdispersion parameter is the same for road sections 1 and 2. Recall that the NB distribution arises when *<sup>i</sup>* is multiplied by a random variable drawn from a gamma distribution with mean=1 and variance=1/*<sup>i</sup>* . This amounts to assuming that the variance of accident counts is *i*(1+*<sup>i</sup>*/*i*). For road section 2, under the Poisson assumption, the variance of accident counts was the same as the mean (32); using =1 in the NB assumption, the variance of accident counts is 33 times the mean [32(1+32/1)=1056]. The reason for the dramatic increase in the variance of the accident counts of road section 2 is that its mean (32) is
+
+![](_page_3_Figure_5.jpeg)
+
+large compared to the magnitude of the overdispersion parameter .
+
+The issue can be now stated generally:
+
+The lesser the variance of the (accident) counts for a data point (a road section), the larger is its influence on the estimates of model parameters. When counts are assumed to be NB distributed with a common overdispersion parameter , then, entities with a mean that is large in comparison to the magnitude of will suffer a dramatic increase in the variance of their (accident) counts. This brings about a correspondingly large decline in their influence over the most likely value of the model parameters, as compared to their influence when counts are assumed to be Poisson distributed. But entities with large means (e.g. long road sections and road sections with a lot of traffic) are precisely those entities that should influence parameter estimation most.
+
+Miaou and Lum (1993) provide essentially the same argument when they note that '… in estimating the
+
+![](_page_3_Figure_10.jpeg)
+
+![](_page_3_Figure_11.jpeg)
+
+![](_page_4_Figure_2.jpeg)
+
+Fig. 9. Comparison of estimates.
+
+(regression) coefficients for road sections with the same AADT the linear-regression models put considerably more weight on accidents observed on short road sections than those observed on long sections…' (p. 698). They argue in favour of using the Poisson models (where the variance of accident counts is the mean) as opposed to the Normal and Log-Normal models (which they, for some reason, call linear-regression models) in which the assumption is that the variance is proportional to the square of the road section length or the square of the mean.
+
+That under the NB error structure with common , road section 2 does not dominate the estimate of should raise eyebrows. To suggest a sensible remedy, it is necessary to be clear about why overdispersion arises in our circumstance.
+
+## **3. Why are accident counts overdispersed?**
+
+The usual circumstance of statistical road safety modelling is that we have data for many entities (say, road sections) and we wish to fit a model to this data set. For each entity we know the count of accidents that have been reported to the police. In addition, for each entity we know some traits that have been measured or estimated with varying degrees of accuracy. The measured traits are often averages over the road section length (e.g. roadside hazard rating) or averages over time (e.g. Annual Average Daily Traffic — AADT). We then choose to include some such measured traits in the model equation as covariates. These will be called the '*represented traits*'. In addition, there are the measured traits that are not represented in the model, some measurable but unmeasured traits, and perhaps some altogether unmeasurable traits. These are the '*unrepresented traits*'. Both represented and unrepresented traits are linked to accident occurrence.
+
+If for a specific entity, say entity '*a*' it was possible to keep constant all its traits — both represented and unrepresented — and then to repeat the same time period over and over, each time recording the count of accidents, then, the average of the counts would converge on the period mean for this specific entity — *a*. Consider now a different entity, say entity '*b*', with the same represented traits as entity '*a*'. The mean for entity '*b*' (*b*) will likely differ from the mean for entity '*a*' in spite of the fact that their represented traits are identical. This difference has several reasons. First, the measured traits are subject to estimation error. Thus, e.g. AADT is estimated on the basis of few days of traffic counting that may have been carried out in some other year at a nearby location. Therefore identical estimates of AADTs do not mean that, in fact, road sections '*a*' and '*b*' served the same traffic. Second, the same average may represent very different distributions. If road section '*a*' carries 100 vehicles at night and 900 during daytime while road section '*b*' serves 500 vehicles at night and in daytime, '*a*' and '*b*' will have the same AADT but different accident means. Third, entities '*a*' and '*b*' are sure to differ in unrepresented traits; one road may lead to a pub, the other may be popular with deer. For all these reasons, if we imagine a large group of entities (*a*, *b*, *c*, …) with the same represented traits, they must be thought to have different means (*a*, *b*, *c*, …). These means form a distribution with as a mean (when *E*[]=1) and with -2 / as variance (when VAR[]=1/). The roˆle of statistical modelling is to provide an estimate as a function of represented traits.
+
+We are interested in how can be expressed as a function of the represented traits for two common purposes. First, when we wish to ascertain how a certain trait (say, lane width or grade) is associated with (perhaps affects) the expected accident frequency. The second purpose is when we wish to estimate the expected accident frequency of a specific site by combining the prediction for sites with identical represented traits with the accident history of the site. In both instances (using the common statistical parlance) this means that we wish to estimate a random effects model and this is the framework of discussion in this paper. (Were a time series of accident counts and traits available for each entity in a data set, it would also be possible to estimate a parameter *<sup>i</sup>* , *i*= *a*, *b*, *c*, … doing so might be of interest if, e.g. one tried to examine why some entities in the data set seem to have fewer or more accidents than is predicted by their represented traits. In this case we might be interested in a fixed effects model.)
+
+Unless VAR[]=0, - (being the mean of the *i* , in the imagined group of entities *i*=*a*, *b*, *c*, … that have identical represented traits) will not be the same as the of the entity in the data. On this basis it is possible to further clarify the nature of overdispersion. Recall that the definition of 'variance' is the expected value of the squared difference between a random variable and its mean. The expected value of the squared differences between a random variable and any fixed value other than the mean, is larger than the variance. Overdispersion is judged by the magnitude of the squared difference between accident counts and model predictions —
+
+the squared residuals. But model predictions are estimates of the for an imagined population of entities with identical represented traits; model predictions are not estimates of the *<sup>i</sup>* of the entities for which we happen to have the accident counts. Since the squared residuals are computed with respect to a value other than the mean of the entities for which we have data, we must expect the residuals to be larger than the variance of the counts for the entities in the data set. Thus, overdispersion is the necessary consequence of the fact that the means of the entities that serve as data are not what the model aims to predict. The root cause of overdispersion is that entities with the same represented traits have different means.
+
+In the numerical example we noted that the influence of a data point diminishes as the variance of its accident counts increases and that the NB distribution implies that the variance of the accident counts for entity *i* is *i*×(1−*i*/*i*). Therefore, the central question is: how does the VAR[]=1/*<sup>i</sup>* change as a function of the value of represented traits. This question is answered in a very specific way if it is assumed that is always drawn at random from a Gamma distribution with mean 1 and variance 1/. In this case, since *<sup>i</sup>* is assumed to be , the same for all entities, the variance of the means in an imagined population of entities with identical represented traits is proportional to -. Thus, e.g. the use of =1 for both road sections in the numerical example implies that roads with the same represented traits as section 2 (10 mile long road sections with AADT=500) have a VAR[-] that is four hundred times that of roads with the same represented traits as road section 1 (0.5 mile long road sections with AADT=500). Should one expect such a large increase in the variance that is due to traits not represented in the model simply because one road section is much longer than the other road section?
+
+The condition under which this would be true will be examined next. However, before doing so, it is important to comment briefly on why in available data sets some road sections are long and others short. Miaou and Lum (1993)(pp. 697–699) provide a good discussion of this issue. Usually road sections in state or provincial inventories are delimited by intersections. In principle, the aim is to define the road sections so that the road in-between the end-points is of one class (e.g. two-lane, four-lane divided) and is fairly homogeneous in traffic. Additional considerations having to do with the history of major reconstructions also apply. When so defined, road sections are usually not homogeneous in features such as grade, curvature, roadside hazard and the like. In some data bases (e.g. the HSIS — the Highway Safety Information System) an attempt is made to define road sections that are homogeneous in all (or most) measured traits.
+
+#### **4. A remedy**
+
+Road section 2 of the numerical example can be thought to consist of 20 subsections like road section 1. Therefore -2=20-1. With this, under the usual NB assumption, the variance of accidents counts on road section 2 would be 20-1(1+20-<sup>1</sup>/). (The usual NB assumption is that the value of , drawn from the Gamma distribution with mean=1 and variance=1/, applies to the entire road section 2.)
+
+An alternative expression for the variance of accident counts on road section 2 is obtained when one assumes that for each of the 20 constituent subsections a separate is selected from a Gamma distribution with mean 1 and variance 1/. Under this assumption the variance of accident counts on each such sub-section is -1(1+ -1/). If the 20 subsection accident counts are statistically independent, the variance of their sum will be 20-1(1+-<sup>1</sup>/). We now have two different expressions for the variance of accident counts on road section 2 and, obviously, 20-1(1+20-1/)20-1(1+-<sup>1</sup>/).
+
+The reason for the noted difference can be stated in tangible terms. The expression 20-1(1+20-<sup>1</sup>/) has been obtained by assuming that all 20 constituent subsections have the same . That is, all 20 subsections of which road section 2 is made up have the very same 'errors in variables', the same 'distribution that underlies averages', and the same 'unrepresented traits'. In contrast, the expression 20-1(1+20-<sup>1</sup>/) is obtained by assuming that for each of the 20 constituent subsections a new is selected at random from the Gamma distribution with mean=1 and variance=1/. This implies that the 20 constituent sub-sections differ amongst each other in their 'errors in variables', 'distributions underlying averages' and in 'unrepresented traits' in the same manner as any two road sections with the same represented traits would, even when they were not stringed out one behind the other on the same road. In short, under the first assumption the 20 values of are perfectly correlated while under the second assumption they are uncorrelated.
+
+The assumption about the 's being either perfectly correlated or entirely uncorrelated brackets reality. Subsections of the same road will still differ in many respects. Therefore, there is no reason to think that the 's of all the subsections are the same. However, subsections with the same represented traits when stringed out consecutively along the same road are likely to differ less than would subsections that are not spatially proximal. Therefore the 's of subsections along the same road are likely to be correlated, if only because they are used by the same drivers under the same climatic conditions.
+
+A general expression for the variance of accident counts when the 's of subsections are not correlated can be obtained as follows. A road section of length *L*
+
+![](_page_6_Figure_2.jpeg)
+
+can be thought to consists of *L*/*l* subsections, all subsections having the same covariate values, and each subsection being *l* units long. If the entire road section has a mean accident frequency -, each subsection has mean accident frequency =-/(*L*/*l*). As before a mixing variable is drawn randomly from a distribution with mean=1, but now we assume that its variance depends on the length of the subsection *l*. That is, VAR[]=1/(*l*). Using the NB distribution, for each subsection of length *l*, VAR[Accident count for a subsection]=(1+/(*l*)). Let *Y* be the sum of accident counts for the *L*/*l* subsections. Since the subsection counts are statistically independent, VAR[*Y*]= (*L*/*l*)[1+(*L*/*l*)/((*l*)*L*/*l*)]=-[1+-/((*l*)*L*/*l*)].
+
+The sum of independent NB random variables is not NB distributed. However, we can use the result about VAR[*Y*] as a motivation to postulate an alternative negative binomial model for the count of accidents *Y* for a road of length *L*. The alternative assumption is that the NB distribution arises when for the road section of length *L* is multiplied by a random variable drawn from a gamma distribution with mean=1 and variance=1/[(*l*)*L*/*l*].
+
+![](_page_6_Figure_5.jpeg)
+
+To examine the implication of this alternative assumption on parameter estimation, I now return to the numerical example. Assume that (*l*=1km)=1. Then, for road section 1, VAR[]=1/[(*l*=1 km)×0.5/1]= 1/0.5=2 and for section 2, VAR[]=1/[(*l*=1 km)×10/1]=1/10. The corresponding four log-likelihood functions are in Figs. 10–13.The maxima in Figs. 10 and 11 are, as before, at 1=0.089 and 2=0.179. As is evident from the comparison of Figs. 10 and 11, the much larger influence of the log-likelihood function for the accident-rich road section 2 has been restored. Therefore, when the two log-likelihoods are added, the 1 and 2 (=0.175) is again very close to 2.
+
+When it comes to parameter estimation by maximizing a likelihood function then, instead of using in Eq. (1) a that is common to all entities, for entity *i* we used *i*=(*l*)*Li*/*l*. The question is what subsection length *l* should be used? Suppose that using a subsection length *l*=1 km we obtain the maximum likelihood estimate ML(*l*=1) of (*l*=1). This determines the values of all the *i*'s. Naturally, the same values of *<sup>i</sup>* maximize the likelihood function if subsections were, say, 2 km long. This implies that ML(*l*=1)=
+
+![](_page_6_Figure_10.jpeg)
+
+ML(*l*=2)/2. That is, that the ratio ML(*l*)/*l* is constant. Therefore we can choose *l*=1 unit of length, say 1 mile or 1 km and the numerical values of *i*=(1 mile)×*Li*(measured in miles) or *i*=(1 km)× *Li*(measured in km) will be the same. Using the same argument, namely, that there is a unique set of values *<sup>i</sup>* that maximize the likelihood function, we can establish the relationship between ML(1 mile) and ML(1 km); ML(1 mile)×*L*(measured in miles)=ML(1 mile)×0.622×L(measured in km). Therefore ML(1 km)=0.622×ML(1 mile).
+
+The gist of the argument in this section was that reality is somewhere between assuming that all subsections of a road section have precisely the same represented and unrepresented traits, and, alternatively, assuming that these traits vary from sub-section to sub-section of a certain road just as they vary between any subsections of the road system that have the same represented traits. The mathematical equivalent of these two bracketing assumptions is to either use a constant that does not depend on road length or to use *i*=×*Li* . (Both options can be represented algebraically by *i*=×*Li* with =0 or =1. A device for establishing a compromise that reflects reality might be to use a value of from within the (0, 1) range).
+
+# **5. Implication for EB estimation**
+
+In EB estimation the model prediction *<sup>i</sup>* and the accident count *Yi* are combined as follows (see, e.g. Hauer, 1997):
+
+Estimate of 
+$$\eta_i \theta = \text{weight}_i \times (1 - \text{weight}_i) \times (Y_i)$$
+. (2)
+
+In this, *<sup>i</sup>* is the expected number of accidents for entities that have the same represented traits as entity *i* and
+
+weight<sub>i</sub> = 
+$$\frac{1}{1 + \text{VAR}[\eta_i \theta]/E[\eta_i \theta]} = \frac{1}{1 + (\eta_i^2/\phi_i)/\eta_i}$$
+$$= \frac{1}{1 + \eta_i/\phi_i}.$$
+ (3)
+
+The assumption that *i*=*L*<sup>0</sup>= was seen earlier to lead to potential distortions in parameter estimation. The same assumption may lead to an inconsistency when the safety of a road section is to be estimated by the EB method.
+
+To show the potential for inconsistency, consider a road that has along its entire length the same represented traits. The model for this kind of road predicts accidents in a period of specified length. Assume that we have in that period accident counts *y*<sup>1</sup> and *y*<sup>2</sup> on the first and the second half of this road. Since for the whole road weightwhole=(1+-/)−<sup>1</sup> , the estimate of for the road is weightwhole-+(1−weightwhole) (*y*1+ *y*2).
+
+Consider now the first half of this road on which *y*<sup>1</sup> accidents were counted and ask what is to be the estimate of the expected accidents there. One extreme option is to estimate as if both halves of the road had the same . To estimate in this manner reflects the belief that not only do the two halves of the road have the same represented traits, as postulated, but they are also have the same 'errors in variables', the same 'distributions that underlie averages', and the same 'unrepresented traits'. With these assumptions the estimate for the first half of the road would be half the estimate for the whole road. This amounts to asserting that not only is the count of accidents on the second half of the road (*y*2) relevant to estimating what is true about the first half of the road, but that it is just as important as the accident count on the first half of the road (*y*1). One may ask how far can this belief be stretched? How can one judge whether physically separate road segments with the same (or even similar) represented traits have the same , in which case their accident counts should be used jointly to estimate the expected number of accidents on any one of these segments? In any case, if the belief that the same applies to both halves of the road is correct, then the sum of EB estimates for the two halves equals the EB estimate for the whole road. With the assumption of a common the EB estimates are consistent.
+
+The other extreme option is to estimate as if the two road halves differed in their unrepresented traits, in variable errors and in the nature of averages. That is, even though the two halves have the same represented traits, their 's could differ in the same manner as 's differ amongst roads that are not physically contiguous; as if two values of were sampled independently from the same distribution, one for each road half. With this, for the first half of the road the EB estimate is : weighthalf×-/2+(1−weighthalf)×(*y*1), and for the second half of the road the EB estimate is: weighthalf× -/2+(1−weighthalf)×(*y*2). The sum of the two estimates is : weighthalf×-+(1−weighthalf)×(*y*1+*y*2). As was shown earlier, the EB estimate for the road as a whole is weightwhole-+(1−weightwhole) (*y*1+*y*2). We would like the EB estimator be such that, the sum of the estimates for the two halves be the same as the estimate for the whole. The only general way to satisfy this equality is to make weighthalf=weightwhole.
+
+The assumption *i*=*L*<sup>0</sup>= implies that weighthalf=(1+(-/2)/)−<sup>1</sup> and weightwhole=(1+-)−<sup>1</sup> . Because weighthalf weightwhole, the sum of the estimates for the two halves of a road section is not the same as the estimate for the whole. Under the assumption that *i*= the EB estimates are inconsistent when the two halves have different 's. The alternative assumption that *i*=*L* does not have the same blemish. Now weighthalf=[1+(-/2)/(*L*/2)]−1=[1+-(*L*)]−1=weightwhole. Since the two weights are now the same, the sum of the estimates for the two halves is the same as the estimate for the whole.
+
+That in our thought we divided the road section into halves only made the algebra transparent. Division into unequal parts would lead to the same conclusion. It appears that if it is not true that all subsections of a road that is homogeneous in represented traits have the same expected number of accidents per unit length, then use of the assumption that *i*= would lead to inconsistent EB estimates.
+
+# **6. Summary and discussion**
+
+In NB regression it is common to assume that the overdispersion parameter is the same for all entities. Experience with statistical modelling of accidents on road sections indicates that this assumption may have undesirable consequences. When road sections in the data base differ in length, two problems arise. First, if model parameters are estimated by maximum likelihood, the relative influence of long road sections is much diminished whereas very short road sections exert an unduly large influence. Second, when the expected accident frequency of a road section is estimated by the EB method and the entire road section does not have the same , the sum of estimates for subsections does not add up to the estimate for the road section as a whole.
+
+An alternative is to consider overdispersion for road section *i* to be given by *Li* . This has been motivated by the argument that each road section can be viewed as consisting of many subsections that differ in traits that are not represented in the model. Therefore, that these subsections have different 's drawn independently from the same Gamma distribution. The use of this assumption has been shown to remedy both problems.
+
+It can be argued that reality is somewhere between the unsatisfactory assumption that *i*=*L*<sup>0</sup> and the logically-pleasing-but-not-exactly-true assumption that *i*=*Li* 1 . It is therefore tempting to consider *i*=*Li* with in the range (0, 1). Two problems arise. First, any 1 will give rise to the problem that when the of a road section is not constant, the sum of EB estimates for parts will not be the EB estimate for the whole. Second, examination of residuals may give an illusory confirmation of incorrect assumptions about . The problem is that a value of close to 0 will give little weight to data points with large -. This, will make for large squared residuals at these data points. Large squared residuals for these data points will seem to confirm the assumption that is close to 0. Had a close to 1 been assumed, the data points where is large would have been more influential. The resulting parameter estimates, would make these residuals smaller, and thereby seem to confirm the assumption of  being close to 1. Whether the usual maximum likelihood estimation can overcome this built-in tendency needs to be explored.
+
+The recommendation here is not to regard the overdispersion parameter as a constant common to all entities, at least not without justification that is based on the data. At this time, the relationship *i*= *Li* seems to be the more attractive alternative. In this relationship *Li* is a 'logical variable'. It represents the logical requirement that if a road section is made up of two non-overlapping sub-sections the sum of the model predictions for the two subsections should be the same as the model prediction for the road section as a whole. This can be true only if the model prediction is proportional to section length. At times, section length is used in models not only as a logical variable but also as a statistical covariate. Since the length of a road section is determined mainly by the distance between major intersections, it is entirely possible that section length is correlated with speed, driveway density etc, and therefore legitimately serves as a proxy for several causal variables. Logical variables and statistical covariates must be kept separate. Thus, e.g. should parameter estimation lead to the result that accident frequency is proportional to (section length)0.79, this is best written as proportionality to *L*×*L*<sup>−</sup>0.21 in which *L* is the logical variable and *L* the covariate which serves as proxy for causal variables associated with section length.
+
+Would the generalization suggested by Cameron and Trivedi (1998) to use *i*=*i* with set to either 0 or 1 be a remedy? If is set to be 0 then we have the standard NB model and both noted deficiencies apply. If is chosen to be larger than 0 but lesser than 1, then weightpartweightwhole and the potential for estimation inconsistency discussed in Section 5 remains. When is set to be 1, then, in Eq. (2), weight*i*=(1+1/)<sup>−</sup><sup>1</sup> . This would remove the estimation inconsistency. Unfortunately, use of a weight that does not diminish as *i* increase would deprive EB estimation of its essence. The attraction of EB estimation is that it correctly accounts for differences in the precision of two separate sources of information. When *<sup>i</sup>* is large, most of the information about the expected accident frequency for the specific site should come from the accident count for that site, not the model prediction. Therefore weight*<sup>i</sup>* should be small. Conversely, when *<sup>i</sup>* is small the accident count will be small and therefore much of the information should come from the model prediction. In this case weight*<sup>i</sup>* should be large. It follows that the suggested generalization does not solve the problem.
+
+Heydecker and Wu (1999) suggest that itself be modelled as a function of the various covariates and a new set of parameters. This avenue can be pursued provided that the structure *i*=*Li*×*f*(covariates*i*) is preserved. However, since we suspect that in reality the 's for subsections of the same road are neither uncorrelated nor perfectly correlated, the fundamental remedy is in research that might provide details about the variation of 's among spatially proximate and spatially distant road sections.
+
+To make our point clearly, the numerical example on which Section 2 was based made use of one very short and one very long road section. The question arises whether in real data sets which consist of a mixture of road section lengths, the assumption made about the overdispersion parameter would have a noticeable effect on the results modelling. A related practical question is whether models that have been estimated in the past on the assumption of a single common overdispersion parameter should revisited. These questions can be answered fully only by modelling several real data sets using both assumptions about the overdispersion parameter. It is a task for future research. However, one issues can be settled now. The value of the overdispersion parameter under the differing assumptions is sure to be different. Therefore, if any use is to be made of the overdispersion parameter, the models need to be re-estimated.
+
+Another question is whether the two alternative assumptions about the dispersion parameter will have a noticeable effect on the EB estimation of the expected accidents. This question can be answered affirmatively. The EB estimate depends on the model prediction (-) and the 'weight' (Eq. (2)). Even if the model prediction was affected only mildly by the alternative assumptions, the 'weight' under the two assumptions will, as a rule, be substantially different. This may open another line of empirical inquiry. After fitting a model to data using both assumption, one can then compare the performance of the two alternative EB estimators.
+
+### **References**
+
+- Bonneson, J.A., McCoy, P.T., 1993. Estimation of safety at two-way stop-controlled intersections on rural highways. Transportation Research Record 1401, 83–89.
+- Cameron, A.C., Trivedi, P.K., 1998. Regression Analysis of Count Data. Cambridge University Press, Cambridge.
+- Guo, G., 1996. Negative multinomial regression models for clustered event counts. Sociological Methodology 26, 113–132.
+- Hauer, E., 1997. Observational Before–After Studies in Road Safety. Pergamon, Oxford, UK.
+- Hauer, E., Ng, J.C.N., Lovell, J., 1989. Estimation of safety at signalized intersections. Transportation Research Record 1185, 48–61.
+- Hausman, J., Hall, B.H., Griliches, Z., 1984. Econometric models for count data with an application to the patents-R&D relationship. Econometrica 52 (4), 909–938.
+- Heydecker, B.G, Wu, J., January/1999. Identification of sites for road accident remedial work by Bayesian statistical methods: an example of uncertain inference. The Fifth International Conference on the Application of Artificial Intelligence to Civil and Structural Engineering, Oxford.
+- Maher, M., Summersgill, I., 1996. A comprehensive methodology for the fitting of predictive accident models. Accident Analysis and Prevention 28 (6), 281–296.
+- Maycock, G, Hall, R.D., 1984. Accidents at four-arm roundabouts. Laboratory Report LR 1120, Transport Research Laboratory, Crowthorne, Berkshire, UK.
+- Miaou, S-P., 1996. Measuring the Goodness of Fit of Accident Prediction Models. FHWA- RD-96-040. FHWA, McLean, VA.
+- Miaou, S.P., Lum, H., 1993. Modelling vehicle accidents and highway geometric design relationships. Accident Analysis and Prevention 25 (6), 689–709.
+- Vogt, A., Bared, J., 1998. Accident models for two-lane rural segments and intersections. Transportation Research Record 1635, 18–29.

Zotero/001_artiklid/Road Safety Indicators/hauerpredictionroadsafety2010.md

@@ -0,0 +1,588 @@
+---
+category: literaturenote
+citekey: hauerpredictionroadsafety2010
+title: On prediction in road safety
+authors: "Hauer, Ezra"
+year: 2010
+date: 2010-00-00 2010
+doi: 10.1016/j.ssci.2010.03.003
+publication: Safety Science
+url: "https://doi.org/10.1016/j.ssci.2010.03.003"
+zotero_key: MA7QJD6E
+zotero_storage: AJN8JVZ4
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Hauer, Ezra"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+Contents lists available at [ScienceDirect](http://www.sciencedirect.com/science/journal/09257535)
+
+# Safety Science
+
+![](_page_0_Picture_4.jpeg)
+
+![](_page_0_Picture_5.jpeg)
+
+# On prediction in road safety
+
+# Ezra Hauer \*
+
+35 Merton Street, Toronto, Ontario, Canada M4S 3G4
+
+#### article info
+
+Article history: Received 12 September 2009 Received in revised form 2 March 2010 Accepted 3 March 2010
+
+Keywords: Prediction Road safety Safety targets Evaluation
+
+#### abstract
+
+Prediction is about potential outcomes: what will happen if and what would have happened if. The first question arises when safety targets are set, the second when the effect of an intervention on safety is to be evaluated. There are many ways to predict. For the same data different prediction methods produce different predictions. What targets are set and what estimates of intervention effect are produced will depend on what method of prediction is chosen. Therefore one has to determine what method tends to predict best. To do so empirically one asks what method would have predicted best had it been applied in the past and then one assumes, inductively, that the same would apply in the future. Quantitative measures of prediction quality are suggested and it is shown how these measures of prediction quality allow one to determine which of two prediction methods should be preferred.
+
+The suggested approach was applied to two data sets: The time series of motor vehicle accident fatalities in Province A and in Province B. On the basis of this analysis one may draw tentative conclusions for these jurisdictions and the methods tested; one can say what method seems preferable, what is the average size of bias than needs to be corrected and how accurate is the prediction likely to be. Broader conclusions will emerge once many additional methods of prediction are applied to data from many other jurisdictions and pertaining to a variety of circumstances.
+
+-2010 Elsevier Ltd. All rights reserved.
+
+#### 1. Introduction
+
+Safety predictions are needed for two main purposes:
+
+- 1. To set policy or program targets. For this purpose one needs to predict what would be the future safety of the unit1 if the policy or programs will not implemented. Safety targets are then set on this basis.
+- 2. To estimate what was the effect a policy, program, or treatment on the safety of the unit. For this purpose too one needs to predict what would have been the safety of the unit had the policy, program or treatment not been implemented.
+
+The difference between the two circumstances is one of timing: for target setting the prediction is produced before the program or policy is implemented whereas for effect estimation the prediction is produced after the intervention was implemented.2
+
+There are many commonly used methods of prediction. Some methods rely on the extrapolation of past trends; some make use comparison groups, another family of methods attempts to model the causal factors that govern the evolution of the safety of a unit. In this paper the question is not how to best describe the structure underlying a time series of data. The question is which method predicts best. This is why the emphasis is not on theory-motivated time series analysis methods and the characterization of their statistical properties. The emphasis is inductive and empirical; a method is attractive for future use if, had it been used in the past it would have predicted better than other methods. From this perspective the theoretical underpinnings of a method and the statistical properties of its fit do not matter.
+
+The quality of a method for safety prediction depends on the nature of the unit for which the prediction is prepared. That is, that one method of prediction may be best when the unit is an intersection or road section while a different method may be best when the unit is a province, state, or country. It is also possible that one method is best for making predictions about the next few years while another when predictions into the more distant future is of interest. In this paper the focus is on predictions when the unit is a province, state or country. In this context predictions for both the near and the more distant future are of interest. Near future predictions are needed to evaluate the effect of interventions in the short term; longer term predictions are needed to tell what might be the safety of some jurisdiction if past practices
+
+<sup>\*</sup> Tel.: +1 416 483 4452.
+
+E-mail address: [Ezra.Hauer@Utoronto.ca](mailto:Ezra.Hauer@Utoronto.ca)
+
+<sup>1</sup> Policies, programs, and treatments apply to units. A unit can be a state, a region, a city, a road section, drivers of an age cohort, trucks of some type, etc.
+
+<sup>2</sup> This has implications about the kind of information available at the time of the prediction is made. For target setting predictions one can only forecast what are likely to be the traits of the unit during the period to which the prediction applies. In contrast, since for effect evaluation the period of prediction is in the past, some such traits can be known.
+
+<span id="page-1-0"></span>were to continue and no novel programmes or new policies implemented.
+
+In this paper the approach to answering the question of which method predicts best is empirical. Available historical data will be used to ask how a method of prediction would have performed had it been used in the past. For clarity it is best to define the differences between Prediction, Forecast, and Estimate. Predictions are about a state of the world that did not and will not exist; they pertain to 'potential outcomes'.3 Thus, prediction is always associated with a conditional clause such as 'what would be if' or 'what would have been if'. Should it happen that, in the event, the policy, program, or intervention will not be implemented then, what was a prediction, turns into a forecast. Unlike a prediction, a forecast is about a state of the world that will exist and therefore the quality of a forecast can be judged by juxtaposing it to what eventually materialized. In contrast to forecasts which pertain to events that will materialize in the future, estimates are about events in the past. Thus, e.g., we can use the history of accident occurrence to estimate the expected number of accidents.
+
+The quality of estimates is the subject of much statistical theory and may be regarded well known. Similarly, there exists a very large body of knowledge about forecasting and forecast quality. This lore is embodied in many books and learned journals.4 The quality of 'potential outcome predictions'5 is less well explored.
+
+## 2. Previous work
+
+The issue of prediction in road safety was examined explicitly in [Hauer et al. \(1991\)](#page-11-0). Using a time series of annual fatal and injury accidents in each of 10 Canadian provinces, predictions were formed by four methods: (a) using 1 year's count to predict for the next, (b) using the average for 2 years to predict the for the next, (c) using the least-squares fit to three counts to predict the next and, (d) using comparison group, i.e. multiplying the count in Province A in year y by the ratio of counts Province B for years y + 1 and y to predict the expected accident count in Province A in year y + 1. It turned out that, in spite of using more data than method 'a', methods 'b' and 'c' gave worse predictions. Method 'd' gave good predictions if the comparison group had many accidents. We found that ''one can predict better not using a comparison group than using one that is too small" (p. 599). In spite of common belief in the importance of similarity between the 'treatment' and the 'comparison group', it turned out that similarity played only a minor role. The issue of predicting by method 'd' was analysed in [Hauer \(1991\).](#page-11-0)
+
+A more thorough examination of prediction in road safety is in the doctoral thesis of [Quaye \(1992\).](#page-11-0) Much of what follows draws on his work. Quaye's object was to propose means to measure the performance of prediction methods and, on this basis, to ascertain whether one method is better than another. Quaye made a distinction between prediction methods that extrapolate a time series
+
+![](_page_1_Figure_11.jpeg)
+
+Fig. 1. Count of road accident fatalities in Province A.
+
+of accident counts and between econometric methods that make use of information about some causal variables. For extrapolation methods his data were the fatality counts of Canadian provinces; for the econometric models he used also data about population, vehicle registration, and vehicle-kilometres of travel (Vkmt). Some of his results were published in [Quaye and Hauer \(1993\)](#page-11-0) and presented in [Quaye and Hauer \(1994\)](#page-11-0). In these we found, e.g., that using twelve different methods to predict the number of fatal accidents in Province A it seemed that extrapolation of a quadratic ordinary least-squares fit produced better predictions that the averaging of past 'y' accident counts and was also better than various econometric models that used data about population, registered vehicles and vehicle-kilometres of travel. In 1994 the trail seems to have grown cold. No further research into the quality of road safety predictions as produced by various prediction methods was found.
+
+#### 3. Same data, different predictions
+
+The purpose of this section is to show that methods of prediction are many and that these produce a multitude of diverse predictions. In the face of this multitude and diversity one asks: ''Which method of prediction should one believe more?" To explain the issues arising from this question in a tangible setting data about the number of persons killed in crashes in Province A (Fig. 1) will be used.
+
+This mountain-like shape depicts the evolution of a process over time.6 What forces shaped this process can be said only in general terms: they are the changes in population, the processes of urbanization and motorization, advances in medicine, improved roads, safer vehicles, changes attitude to risk and in other societal norms, etc.
+
+Suppose that we are at the end of 1985 and wish to predict what would be the number of fatalities for the next 10 years if the same forces continued to exert their influence and no substantially new programs and policies were implemented.<sup>7</sup> The predictions for 1986–1995 are to be based on data for the years to the left of the solid line; what is to the right of the solid line is at this point in time unknown.
+
+Prediction methods are many. As the purpose here is illustration, only relatively simple methods of prediction will be used. Thus, e.g., one could extrapolate the two most recent observed counts, or fit an ordinary least squares regression (OLS) a line to the n (=3, 4, 5, ...) most recent observed counts, or fit a polynomial
+
+<sup>3</sup> The 'potential outcome' terminology favoured by Rubin (see, e.g., [Rubin, 2005\)](#page-11-0) is used instead of the equivalent expression: 'counterfactual'.
+
+<sup>4</sup> One list [\(http://www.forecastingprinciples.com/books1.html\)](http://www.forecastingprinciples.com/books1.html) contains entries for 75 books on forecasting, The University of Toronto libraries presently subscribe to eight journal that have 'forecasting' in their title. Most such books and journals are oriented towards business and economics.
+
+<sup>5</sup> In the context of safety evaluation the word 'prediction' is a poor choice. Evaluations are undertaken after an intervention was implemented, and therefore 'prediction', as used in this context, refers to a time and state of affairs which is already in the past at the time when the evaluation commences. This is awkward because in common usage prediction refers to events in the future. It might have been better to use 'postdiction' or 'retrodiction' instead. However, both terms are already in use and have acquired specialized meanings. Besides, the word prediction has been used in the sense defined here earlier ([Hauer, 1997](#page-11-0)) and coining a new term might lead to confusion. In the rest of this paper the word 'prediction' will be used instead of the longhand expression: 'potential outcome safety prediction'.
+
+<sup>6</sup> The process evolved in a similar way in nearly all developed countries.
+
+<sup>7</sup> This is similar to what Broughton et al. (2000) aimed for when preparing the framework for setting the 2010 Target in the UK. Their goal was to predict ''what would be expected if there were no further DESS measures and only the core road safety activities were undertaken (at the 1998 level of effect) during the period to 2010; this was done by extrapolating trends from the 1983–1998 period." ([Brough](#page-11-0)[ton, 2006](#page-11-0)). The acronym DESS stands for Drink/driving, Engineering, Secondary Safety improvements to vehicles.
+
+<span id="page-2-0"></span>Table 1 A variety of predictions.
+
+| Method and variant     | 1986  | 1987  | 1988  | 1989  | 1990  | 1991  | 1992  | 1993  | 1994   | 1995   |
+|------------------------|-------|-------|-------|-------|-------|-------|-------|-------|--------|--------|
+| OLS linear, n = 2      | 207.0 | 200.0 | 193.0 | 186.0 | 179.0 | 172.0 | 165.0 | 158.0 | 151.0  | 144.0  |
+| OLS linear, n = 3      | 202.3 | 191.8 | 181.3 | 170.8 | 160.3 | 149.8 | 139.3 | 128.8 | 118.3  | 107.8  |
+| OLS linear, n = 5      | 199.9 | 188.4 | 176.9 | 165.4 | 153.9 | 142.4 | 130.9 | 119.4 | 107.9  | 96.4   |
+| OLS linear, n = 10     | 215.0 | 207.7 | 200.5 | 193.2 | 185.9 | 178.6 | 171.4 | 164.1 | 156.8  | 149.5  |
+| OLS Quadratic, n = 3   | 214.0 | 221.0 | 235.0 | 256.0 | 284.0 | 319.0 | 361.0 | 410.0 | 466.0  | 529.0  |
+| OLS Quadratic, n = 5   | 210.4 | 209.4 | 211.4 | 216.4 | 224.4 | 235.4 | 249.4 | 266.4 | 286.4  | 309.4  |
+| OLS Quadratic, n = 10  | 181.9 | 156.6 | 128.3 | 96.9  | 62.6  | 25.2  | -15.1 | -58.5 | -104.8 | -154.2 |
+| Holt smoothing,a Ab    | 207.0 | 199.6 | 192.2 | 184.8 | 177.5 | 170.1 | 162.7 | 155.4 | 148.0  | 140.6  |
+| Holt smoothing, Bc     | 204.4 | 200.7 | 190.5 | 203.7 | 194.8 | 188.5 | 175.2 | 163.3 | 151.2  | 131.3  |
+| Hoerl function, n = 5  | 201.2 | 188.4 | 175.3 | 162.2 | 149.3 | 136.8 | 124.8 | 113.4 | 102.6  | 92.6   |
+| Hoerl function, n = 10 | 210.9 | 202.3 | 193.4 | 184.5 | 175.6 | 166.7 | 158.0 | 149.4 | 141.0  | 132.9  |
+| Hoerl function, n = 20 | 218.4 | 212.5 | 206.6 | 200.7 | 194.7 | 188.7 | 182.7 | 176.7 | 170.8  | 164.9  |
+| Oppe–Koornstra         | 256.7 | 254.7 | 252.5 | 250.1 | 247.4 | 244.6 | 241.6 | 238.4 | 235.2  | 231.8  |
+| What materialized      | 245   | 236   | 200   | 192   | 154   | 170   | 143   | 153   | 151    | 157    |
+
+<sup>a</sup> Let yi be the accident count at time i (i = 1, 2, ... , n) and si the 'smoothed' value. Holt smoothing consists of the recursive relationships si = ayi + (1 a)(si-<sup>1</sup> + ui-1) and ui = c(si si-1) + (1 c)ui-
+
+to these points, or use double exponential smoothing, or fit a Hoerl function or, following [Oppe and Koornstra \(1990\) and Oppe \(1991\),](#page-11-0) extrapolate risk and exposure and predict using their product.8 , 9 The resulting predictions are in Table 1.
+
+The juxtaposition of these predictions and the accident counts that materialized during 1986–1995 (the black squares) is in Figs. 2–5.
+
+One should resist the temptation to form conclusions on the basis of an illustration based on few simple methods of prediction that rely on data from only one jurisdiction (Province A), and use predictions for only one point in time (1985). The few rows of Table 1 are just a smattering of what could be listed. Instead of OLS one could use maximum likelihood, instead of linear or quadratic polynomial regressions one could choose from many function families, in addition to the few n's listed many others could be used, instead of Holt smoothing one could use one of the many autoregressive or state space models for time series data etc. It is obvious that as prediction methods are many and each method has many variants, predictions can be indeed very many. Furthermore, as in Table 1, the many predictions usually vary widely. Naturally one asks which method and variant are best in what circumstance. 'Circumstance' may refer to type of jurisdiction, the shape of the time series profile, whether the prediction is for but few years or for further into the future and so on. Determining which approach to prediction is best requires substantive empirical inquiry involving a juxtaposition of what would have been predicted and what has materialized for several jurisdictions and points in time. Before embarking on this task for two jurisdictions, a brief a discussion of two topics is in order: the issue of extrapolation versus causal modeling in prediction, and the question of how the quality of predictions can to be measured.
+
+![](_page_2_Figure_12.jpeg)
+
+Fig. 2. Prediction by straight lines fitted to the last n data points.
+
+![](_page_2_Figure_14.jpeg)
+
+Fig. 3. Predictions by extrapolating a quadratic polynomial fitted to the last n data points.
+
+#### 4. Approaches to prediction: extrapolation versus causation
+
+All the predictions in Table 1 rely on extrapolation. Even the Oppe–Koornstra method relies on the extrapolation of a time series of risks and another extrapolation of a time series of exposures. Extrapolation rests on the assumption that time-trends of the past will continue into the future. It is commonly believed that one would predict better if the causes of the evolution of a process
+
+<sup>1</sup> in which there are two smoothing constants (<sup>a</sup> and <sup>c</sup>) and starting values (s<sup>0</sup> and <sup>u</sup>0). <sup>b</sup> In variant <sup>A</sup> the smoothing and starting values were selected so as to minimize the sum of one-step-ahead absolute forecast errors. The prediction in Table 1 for years Y > 1985 is s<sup>1985</sup> + (Y - 1985) u1985. For the Province A data, 214.3 + (Year - 1985) (-7.37).
+
+<sup>c</sup> In variant B, the a, c, s<sup>0</sup> and u<sup>0</sup> for n-steps-ahead predictions are estimated so as to minimize the sum of n-step-ahead absolute prediction errors.
+
+<sup>8</sup> The Oppe–Koornstra approach entails two equations. The first traces the evolution of risk over time (Rt = exp(at + b) in which Rt is the risk at time t, with a and b as parameters to be estimated from data). The second traces the evolution of exposure over time (Vt = Vm/{1 + exp[-(at + b)]} in which Vt represents the vehiclekilometres of travel at time t, Vm is the maximum (saturation) Vkmt with a and b parameters estimate from data). The number of accidents is the product Rt Vt.
+
+<sup>9</sup> One could also use econometric models making use of variables such a population, income, alcohol, and consumption. Ranging from simple multivariable regressions to complex structural models. However, models of this kind will have difficulty replicating the fatality peak evident in [Fig. 1](#page-1-0) and will require the making predictions for all variables in the model.
+
+<span id="page-3-0"></span>![](_page_3_Figure_2.jpeg)
+
+Fig. 4. Predictions by extrapolating a Hoerl function fitted to the last n data points.
+
+![](_page_3_Figure_4.jpeg)
+
+Fig. 5. Two variants of Holt smoothing and the Oppe–Koornstra prediction.
+
+were understood and the magnitudes of these causes were known or could be predicted. Fig. 6 is an attempt to clarify.
+
+The cause of the change in the sequence of accident counts is not the passage of time; accidents depend causally on various factors. Thus, e.g., the number of traffic fatalities in a province depends on the number of drivers, amount of pedestrian traffic, quality of health care, norms of behaviour, on what roads and in what environment travel takes place, etc. These factors themselves change with time as indicated by the top grey arrow.
+
+Prediction by extrapolation assumes that causal factors will change with time is such a way that the future is a smooth continuation of past trends. This is represented by the curved dashed arrow on the right which bypasses the box containing causal factors. However, if one could know how expected accidents change under the influence of change in causal factors and if the value of these causal factors in the period for which the prediction is needed is could be known, estimated or predicted, then one could do away with the assumption that the future is an unspecified continuation of the past. The hope is that replacing the belief in an unspecified 'continuity of trends' by information about causal factors and knowledge about cause–effect relationship might improve prediction. This is represented by the looping solid arrow on the left.
+
+![](_page_3_Figure_9.jpeg)
+
+Fig. 6. Causal versus extrapolation of time-trend prediction.
+
+If data about the past values of all important causal factors were available, if the functional relationship between expected accident and causal factors was sufficiently understood, and if regression modeling was up to the task of distinguishing between association and causation, then it might be possible to describe how the expected number of accidents would change if the values of some causal factors changed. At present, none of these conditions fully holds. Furthermore, to predict what would be the expected number of accidents at some future time one would also have to be able to predict the magnitudes of all the causal factors for that future time, further complicating the task of prediction by causal models. For these reasons it is not clear whether predictions based on the use of data in statistical regression (structural) models will usually be better than those based on extrapolating time-trends.10
+
+In the present context of prediction for large jurisdictions there is an additional reason for shunning causal models. As in [Fig. 1,](#page-1-0) in most large jurisdictions there is a period during which fatalities increased, reached a peak and followed by a period of decline and stabilization. All the while causal variables such a population, exposure etc. increased. For a satisfactory explanation of the peak in the time series of fatalities one has to include in the model a cause– effect representation of how risk (accidents per unit of exposure) changes as a function of causal variables (such as changes in public mores, investment in infrastructure, medicine, and urbanization) At present, knowledge of this kind is insufficient. Even if it was known how risk depends on, say, public mores or the extent of urbanization, it still would be necessary to predict what these will be in the future. Unless the process of the evolution of public mores or of urbanization could be described as a function of some variables other than time, extrapolation would still be needed. The hope that the evolution of fatalities can be explained by cause and effect, without resorting to extrapolation of time-trends leads to infinite regress. For these reasons, in the present inquiry, only time-based extrapolation approaches will be examined.
+
+#### 5. Prediction quality
+
+In Section [3](#page-1-0) the story was that we are at the end of 1985 and need to predict what would be the number of fatalities in Province A for 1986–1995 if no substantially new policy or program were to be implemented in those 10 years. If we predict by, say, linear extrapolation of an OLS fit with n = 5, then the predictions are those in row 4 of [Table 1](#page-2-0) and repeated below in row P of [Table](#page-5-0) [2](#page-5-0). The count of fatalities that materialized in that period is in row X. The differences X - P between the two are in row D. In row d is the relative difference, the difference as proportion of the count that materialized. Thus e.g., if d (X - P)/X = 0.18, then the predicted value is 18% below the accident count.
+
+Were one to predict similarly at the end of 1984, one would obtain [Table 3.](#page-5-0)
+
+As the first year of data is for 1952 and when n = 5, the first such table could be produced at the end of 1956. Since the last year for which we have fatality counts is 2007, and each table is for 10 'steps ahead' years, the last table that can be produced is for 1997. Thus we can have 42 such tables and 42 rows of relative differences for 1, 2, ... , 10 steps-ahead predictions. Selected rows from this composite table are in [Table 4.](#page-5-0)
+
+<sup>10</sup> In the community of forecasters there is vigorous debate about the quality of forecasts obtainable by the relatively simple methods of time-series extrapolation and forecasts produced by more complex regression models that make use of important causal variables and often are often based on some theory which takes the form of simultaneous equations. The weight of the evidence at this time suggests that complex regression models do not necessarily produce better forecasts than simpler univariate time-trend extrapolations. As [Makridakis and Hibon \(2000\)](#page-11-0) say: ''Statistically sophisticated or complex methods do not necessarily provide more accurate forecasts than simpler ones." p. 452.
+
+<span id="page-4-0"></span>The interpretation of this table is as follows. Suppose that using the count of fatalities in [Fig. 1](#page-1-0) one was to predict what will be the annual number of fatalities in the next 10 years by extrapolating the ordinary-least-squares fit to the last 5 years of fatality counts. A one-step-ahead prediction (for 1957) when the last year of data is 1956 would prove to have d = -0.11 while a seven step ahead prediction (for 1991) when the last year of data is 1984 would prove to have d = 0.17. It should be clear that the entries in the table are not statistically independent. Thus, e.g., for s = 1 the d in 1956 0.11) and the d in 1957 (0.03) share the same four of five data points to estimate their prediction equations. The last three rows of the table contain summary statistics. Judging by d, the predictions (the P's) for the years 1956–1997 will be on the average about 1% higher11 than the count of fatalities (the X's)when s = 1, 2 and 3 while for s = 9 or 10 the P's would be, on the average, 6% higher than X's. Here the bias (d) increases with the number of steps ahead. The standard error of the d's around d is in the before last row of [Table 4.](#page-5-0) The further into the future one predicts the larger is the standard error of predictions. Thus, e.g., when predicting (by OLS with n = 5) 6 years into the future one should expect the average prediction to be 2% too high and its standard error to be ±39%. As is evident, the standard error of prediction increases with s.
+
+In sum, the quality of a method of predictions can be judged by two attributes. First, one would like predictions to be unbiased in the sense that when the prediction is P then the expected value of the corresponding X's is also P. Second, one would like predictions to be 'close' to the expected value of X. The notion of being 'close' can be variously defined. One common measure of closeness is the standard error (SE); another is the mean absolute error (MAE). While many measures of prediction error are in use12, in what follows the measures of prediction quality will be the bias, as measured by the average of d's (the d or d-bar), and the standard error of the d's denoted by SEdbar. The attributes of being unbiased and that of closeness are not independent. Methods that produce predictions with significant bias will tend to also have large SE and MAE. Therefore in many cases one can combine them into one measure of performance, the standard error around 0. This measure, to be denoted by SE0 is defined by SE0 ¼ ðSE2 dbar <sup>þ</sup> bias2 . Having created the yardstick for measuring prediction quality we can now turn to the task of comparing methods and their variants in terms of the quality of predictions that they produce.
+
+#### 6. Comparing variants and methods
+
+The central question is which of two prediction methods and variants would have predicted better. Thus, e.g., had we predicted by the linear extrapolation of an OLS fit, would we have predicted better with n = 5 or with n = 10? Having specified what will measure the quality of predictions an answer can now be attempted. The raw material is in [Table 5](#page-5-0) the top half of which was taken from the summary rows of [Table 4](#page-5-0) and the bottom half of which was similarly generated except that the extrapolation was of an OLS fit to the last ten fatality counts (n = 10).
+
+The three comparisons are in the Figs. 7–9.
+
+The negative bias is larger with n = 10 than with n = 5 for all s. On the other hand, with the exception of s = 1, when n = 10 the pre-
+
+![](_page_4_Figure_10.jpeg)
+
+Fig. 7. Bias as a function of s for n = 5 and n = 10.
+
+![](_page_4_Figure_12.jpeg)
+
+Fig. 8. Standard error around d as a function of s for n = 5 and n = 10.
+
+![](_page_4_Figure_14.jpeg)
+
+Fig. 9. Standard error around 0 as a function of s for n = 5 and n = 10.
+
+dictions have a smaller standard error (around d and around 0) than when n = 5. One may conclude that when s = 1 it would have been better to use n = 5 while n = 10 would have been preferred for s > 1. The predictions would have been better, on the average,13 had they been reduced by d. Thus, e.g., if s = 5 then the predictions obtained by linear extrapolation should have been reduced by 5% (as per [Table 5.](#page-5-0))
+
+The method that allowed the comparison of n = 5 and n = 10 can be used to compare any two n's. How the three measures of prediction quality vary with s and n is shown in the three figures below.
+
+The bias [\(Fig. 10\)](#page-6-0) increases with s and n. The standard error around d is smallest between n = 9 and 10. The standard error around 0 is also smallest between n = 9 and 10. It follows that if
+
+<sup>11</sup> When P > X then D (and also d) are negative. A negative d (or d) indicates the predictions (P) are to high. When a linear prediction equation is used P will tend to be too high when the underlying process is a downward bending curve (the second derivative is negative).
+
+<sup>12</sup> Common measures of forecast accuracy are also the MAPE (Mean Absolute Percentage Error), MdAPE (Median Absolute Percentage Error), sMAPE (Symmetric Mean Absolute Percentage Error), sMdAPE (Symmetric Median Absolute Percentage Error), MdRAE (Median Relative Absolute Error), GMRAE (Geometric Mean Relative Absolute Error), and the MASE (Mean Absolute Scaled Error). From [Hyndman and](#page-11-0) [Koehler \(2006\).](#page-11-0)
+
+<sup>13</sup> d is the right correction if it is not clear whether, at the time when the prediction is made, that the underlying process is non-linear. If at the time the prediction is made it be clear that the underlying process is upward or downward bending then the use of a linear predictor does not make much sense.
+
+<span id="page-5-0"></span>Table 2 Predictions for the 10 years following 1985.
+
+|   | Years               | 1986  | 1987  | 1988  | 1989  | 1990  | 1991  | 1992  | 1993  | 1994  | 1995 |
+|---|---------------------|-------|-------|-------|-------|-------|-------|-------|-------|-------|------|
+| s | Steps ahead         | 1     | 2     | 3     | 4     | 5     | 6     | 7     | 8     | 9     | 10   |
+| P | OLS linear, n = 5   | 199.9 | 188.4 | 176.9 | 165.4 | 153.9 | 142.4 | 130.9 | 119.4 | 107.9 | 96.4 |
+| X | Count of fatalities | 245   | 236   | 200   | 192   | 154   | 170   | 143   | 153   | 151   | 157  |
+| D | Differences         | 45.1  | 47.6  | 23.1  | 26.6  | 0.1   | 27.6  | 12.1  | 33.6  | 43.1  | 60.6 |
+| d | Rel. diff. D/X      | 0.18  | 0.20  | 0.12  | 0.14  | 0.00  | 0.16  | 0.08  | 0.22  | 0.29  | 0.39 |
+
+Table 3 Predictions for the 10 years following 1984.
+
+|   | Years               | 1985  | 1986  | 1987  | 1988  | 1989  | 1990  | 1991  | 1992  | 1993  | 1994  |
+|---|---------------------|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|
+| s | Steps ahead         | 1     | 2     | 3     | 4     | 5     | 6     | 7     | 8     | 9     | 10    |
+| P | OLS linear, n = 5   | 210.1 | 198.6 | 187.1 | 175.6 | 164.1 | 152.6 | 141.1 | 129.6 | 118.1 | 106.6 |
+| X | Count of fatalities | 214   | 245   | 236   | 200   | 192   | 154   | 170   | 143   | 153   | 151   |
+| D | Differences         | 3.9   | 46.4  | 48.9  | 24.4  | 27.9  | 1.4   | 28.9  | 13.4  | 34.9  | 44.4  |
+| 2 | Rel. diff. D/X      | 0.02  | 0.19  | 0.21  | 0.12  | 0.15  | 0.01  | 0.17  | 0.09  | 0.23  | 0.29  |
+
+Table 4 Relative differences [d (X -P)/X] by 'year' and 'steps ahead' with OLS, linear, n = 5.
+
+| Last year of data          | Steps ahead |          |          |          |          |          |          |          |          |          |
+|----------------------------|-------------|----------|----------|----------|----------|----------|----------|----------|----------|----------|
+|                            | 1           | 2        | 3        | 4        | 5        | 6        | 7        | 8        | 9        | 10       |
+| 1956                       | 0.11<br>    | 0.05     | 0.03<br> | 0.05<br> | 0.01     | 0.01     | 0.09     | 0.02     | 0.18     | 0.17     |
+| 1957                       | 0.03        | <br>0.08 | <br>0.12 | <br>0.07 | <br>0.09 | <br>0.01 | <br>0.10 | 0.07     | 0.05     | <br>0.09 |
+| 1958                       | <br>0.02    | <br>0.02 | 0.04     | 0.04     | 0.12     | 0.06     | 0.22     | 0.21     | 0.11     | <br>0.09 |
+|                            |             |          |          |          |          |          |          |          |          |          |
+| 1980                       | <br>0.09    | <br>0.20 | <br>0.24 | <br>0.33 | <br>0.39 | <br>0.22 | <br>0.28 | <br>0.53 | <br>0.61 | <br>1.02 |
+| 1981                       | 0.10<br>    | 0.11<br> | 0.17<br> | 0.19<br> | 0.02<br> | 0.05<br> | 0.22<br> | 0.25<br> | 0.54<br> | 0.37<br> |
+| 1982                       | 0.03        | 0.03     | 0.06     | 0.24     | 0.27     | 0.21     | 0.25     | 0.15     | 0.31     | 0.28     |
+| 1983                       | 0.01        | 0.04     | 0.22     | 0.25     | 0.18     | 0.22     | 0.11     | 0.27     | 0.23     | 0.37     |
+| 1984                       | 0.02        | 0.19     | 0.21     | 0.12     | 0.15     | 0.01     | 0.17     | 0.09     | 0.23     | 0.29     |
+| 1985                       | 0.18        | 0.20     | 0.12     | 0.14     | 0.00     | 0.16     | 0.08     | 0.22     | 0.29     | 0.39     |
+|                            |             |          |          |          |          |          |          |          |          |          |
+| 1995                       | 0.12<br>    | 0.11     | 0.01     | 0.23     | 0.06     | 0.15     | 0.00     | 0.08     | 0.07<br> | 0.09     |
+| 1996                       | 0.14        | 0.05     | 0.25     | 0.09     | 0.18     | 0.02     | 0.10     | 0.04<br> | 0.12     | 0.06     |
+| 1997                       | <br>0.04    | 0.17     | <br>0.03 | 0.06     | <br>0.13 | <br>0.06 | <br>0.25 | <br>0.07 | <br>0.16 | <br>0.11 |
+| <br>d = average d (bias)   | <br>0.01    | <br>0.01 | <br>0.01 | <br>0.02 | <br>0.02 | <br>0.02 | <br>0.03 | <br>0.04 | <br>0.06 | <br>0.06 |
+| Standard error around<br>d | 0.13        | 0.20     | 0.27     | 0.31     | 0.35     | 0.39     | 0.42     | 0.48     | 0.55     | 0.61     |
+| Standard error around 0    | 0.13        | 0.20     | 0.27     | 0.31     | 0.35     | 0.39     | 0.42     | 0.48     | 0.55     | 0.62     |
+
+Table 5 Statistics for n = 5 and n = 10.
+
+| n  |                                                                                   | Steps ahead              |                          |                          |                          |                          |                          |                          |                          |                          |                          |
+|----|-----------------------------------------------------------------------------------|--------------------------|--------------------------|--------------------------|--------------------------|--------------------------|--------------------------|--------------------------|--------------------------|--------------------------|--------------------------|
+|    |                                                                                   | 1                        | 2                        | 3                        | 4                        | 5                        | 6                        | 7                        | 8                        | 9                        | 10                       |
+| 5  | <br>d = average d (bias)<br>Standard error around<br>d<br>Standard error around 0 | 0.01<br><br>0.13<br>0.13 | 0.01<br><br>0.20<br>0.20 | 0.01<br><br>0.27<br>0.27 | 0.02<br><br>0.31<br>0.31 | 0.02<br><br>0.35<br>0.35 | 0.02<br><br>0.39<br>0.39 | 0.03<br><br>0.42<br>0.42 | 0.04<br><br>0.48<br>0.48 | 0.06<br><br>0.55<br>0.55 | 0.06<br><br>0.61<br>0.62 |
+| 10 | <br>d = average d (bias)<br>Standard error around<br>d<br>Standard error around 0 | <br>0.02<br>0.15<br>0.15 | <br>0.02<br>0.19<br>0.19 | <br>0.03<br>0.23<br>0.23 | <br>0.03<br>0.26<br>0.26 | <br>0.05<br>0.28<br>0.28 | <br>0.06<br>0.30<br>0.31 | <br>0.07<br>0.33<br>0.34 | <br>0.08<br>0.37<br>0.38 | <br>0.09<br>0.42<br>0.43 | <br>0.10<br>0.49<br>0.50 |
+
+for Province A one was to predict using the extrapolation of a linear OLS fit, it would be best to do so using an n of about10. (In general there is no reason to expect that the same n is best for all s. In the present case, however, the trough14 of the function is not a function of s.) The performance of such predictions for n = 10 and s = 1–10 was shown in the lower part of Table 5.
+
+The merit of the suggested approach is now evident. First, it can answer questions of practical interest.<sup>15</sup> Second, it provides an estimate of the average bias inherent in the prediction method and thereby a way for removing it. Third it produces straightforward
+
+<sup>14</sup> While the process underlying [Fig. 1](#page-1-0) is not a straight line it can be locally approximated by one. For the approximation to be 'local', n has to be small. However, for the slope of the regression line to be reliably estimated n must not be too small. The trough in [Figs. 11 and 12](#page-6-0) is the result of these two opposite considerations.
+
+<sup>15</sup> Here the question was how many data points should be used in fitting the straight line used for producing predictions.
+
+<span id="page-6-0"></span>![](_page_6_Figure_2.jpeg)
+
+Fig. 10. Linear prediction: bias as a function of s and n.
+
+and realistic estimates of the standard error of the predictions for every 's' (such as those in the two last rows of [Table 5\)](#page-5-0).
+
+Having established what quality of prediction is attainable in Province A by the extrapolation of a linear OLS fit, the next question is whether a different method of prediction would do better. As noted in Section [2,](#page-1-0) Quaye found that the extrapolation of a quadratic OLS fit produced good predictions. Therefore we will attempt to determine whether in Province A the quadratic OLS fit usually out-performs the linear OLS fit (in spite of the impression from [Fig. 3\)](#page-2-0).
+
+Fig. 13 for the quadratic prediction corresponds to Fig. 10 for the linear one. The main difference is that here the bias is mostly positive (P is too low) whereas with the linear prediction the bias is mostly negative (P was too high). This difference is due to the shape of the 'mountain' in [Fig. 1](#page-1-0). With both methods the bias tends to increase with s and n.
+
+Fig. 14 for the quadratic prediction corresponds to Fig. 11 for the linear one. For ease of comparison the vertical scale has been truncated so as to show the same portion of the function in both figures. It is now clear that for Province A the standard error of the quadratic predictions is always larger than that for the linear one.
+
+Fig. 15 corresponds to Fig. 12 and their comparison leads to the same conclusion: for fatalities in Province A predicting by the extrapolation of a straight line OLS fit would have predicted better than using a quadratic polynomial. Once again the suggested approach for measuring the quality of predictions helps to answer the question of interest straightforwardly.
+
+In this section we established the procedure for determining which variant of a method to use under what circumstances and which of two methods would have usually given better predictions. It should be noted that what has been concluded so far pertains to a specific data set, the time series of fatalities in Province A.
+
+![](_page_6_Figure_10.jpeg)
+
+Fig. 11. Linear prediction: standard error around d as a function of s and n.
+
+![](_page_6_Figure_12.jpeg)
+
+Fig. 12. Linear prediction: standard error around 0 as a function of s and n.
+
+![](_page_6_Figure_14.jpeg)
+
+Fig. 13. Quadratic prediction: bias as a function of s and n.
+
+![](_page_6_Figure_16.jpeg)
+
+Fig. 14. Quadratic prediction: standard error around d as a function of s and n.
+
+![](_page_6_Figure_18.jpeg)
+
+Fig. 15. Quadratic prediction: standard error around 0 as a function of s and n.
+
+Table 6 Measures of performance for Holt smoothing A.
+
+|                            | Steps ahead |          |          |          |          |          |          |          |          |          |
+|----------------------------|-------------|----------|----------|----------|----------|----------|----------|----------|----------|----------|
+|                            | 1           | 2        | 3        | 4        | 5        | 6        | 7        | 8        | 9        | 10       |
+| <br>d = average d (bias)   | <br>0.01    | <br>0.02 | <br>0.02 | <br>0.03 | <br>0.04 | <br>0.05 | <br>0.06 | <br>0.07 | <br>0.09 | <br>0.10 |
+| Standard error around<br>d | 0.12        | 0.17     | 0.23     | 0.26     | 0.28     | 0.30     | 0.33     | 0.36     | 0.42     | 0.46     |
+| Standard error around 0    | 0.13        | 0.17     | 0.23     | 0.26     | 0.29     | 0.31     | 0.33     | 0.36     | 0.43     | 0.47     |
+
+Table 7 Measures of performance for Holt smoothing B.
+
+|                            | Steps ahead |          |          |          |          |          |          |          |          |          |
+|----------------------------|-------------|----------|----------|----------|----------|----------|----------|----------|----------|----------|
+|                            | 1           | 2        | 3        | 4        | 5        | 6        | 7        | 8        | 9        | 10       |
+| <br>d = average d (bias)   | 0.01<br>    | 0.04<br> | 0.02<br> | 0.05<br> | 0.03<br> | 0.05<br> | 0.05<br> | 0.10<br> | 0.07<br> | 0.11<br> |
+| Standard error around<br>d | 0.12        | 0.17     | 0.21     | 0.28     | 0.29     | 0.34     | 0.38     | 0.47     | 0.52     | 0.66     |
+| Standard error around 0    | 0.13        | 0.17     | 0.21     | 0.28     | 0.30     | 0.34     | 0.38     | 0.48     | 0.53     | 0.67     |
+
+Whether similar conclusions hold for other jurisdictions can be established only by performing similar analyses for data from many other jurisdictions.
+
+The following two sections are best regarded as more complete illustration of the approach to determining which method tends to predict best. The illustration is still limited to the simple methods first introduced in [Table 1](#page-2-0) and there is no intent to claim that other prediction methods would not perform better. In the next section we return to the Province A data to determine whether the other prediction methods introduced earlier in Section [3](#page-1-0) can do better than the linear prediction which reigns at this point.
+
+## 7. Province A: examining other methods
+
+Section [6](#page-4-0) served to show how the yardstick for measuring prediction quality introduced in Section [5](#page-3-0) can be used to determine which of two variants or prediction methods tends to predict better for the data of Province A. Several other simple prediction methods and their variants were introduced for illustration in [Table 1.](#page-2-0) The present section will examine how these would have performed had they been used.
+
+# 7.1. Holt smoothing A<sup>16</sup>
+
+For each of the 43 years 1955–1997 the predictions (P(s), s = 1, 2, ... , 10) were computed. From these the relative differences d(s) were obtained. The summary statistics d, SE around d and SE around 0 are in Table 6.
+
+These measures of performance are very similar to those obtained by linear regression with n = 10 as shown in [Table 5.](#page-5-0) It follows that, in Province A, the Holt smoothing (A) is as good or slightly better as the extrapolation of a linear OLS fit with 10 data points. Both outperform the extrapolation of a quadratic fit.
+
+# 7.2. Holt smoothing B<sup>17</sup>
+
+The results for s = 1–10 for Holt smoothing B are in Table 7. Except at s = 3 they are not as good as those for Holt smoothing A.
+
+## 7.3. Fitting a Hoerl function
+
+The function aeb(t-(t d) <sup>c</sup> was fitted to n = 5, 6, ... , 20 data points by minimizing the sum of absolute deviations. How the d depends on n and s is shown in Fig. 16.
+
+![](_page_7_Figure_19.jpeg)
+
+Fig. 16. Hoerl prediction: bias as a function of s and n.
+
+The shape of the bias function lacks regularity. It depends heavily on how well the chosen function fits the data and may be influenced by how well the Excel 'SOLVER' determines the minimum. Bias is minimal when n = 7, 8 or 9 and when n = 18, 19. How the two standard errors depend on s and n is shown in Figs. 17 and 18.
+
+Considering the size of bias and of standard error, the best predictions for all s are when n = 19. The measures of performance at n = 19 are in [Table 8](#page-8-0). They are significantly better than those of the linear prediction and the Holt smoothing A.
+
+The measures of performance of the Hoerl function when n = 10 is in [Table 9.](#page-8-0) These too are better than those of Holt smoothing A. Thus, for Province A, even when only 10 points were available for fitting the Hoerl function, the predictions were, on the average, better than by the other methods examined so far.
+
+![](_page_7_Figure_24.jpeg)
+
+Fig. 17. Hoerl prediction: standard error around d as a function of s and n.
+
+<sup>16</sup> For description see Footnotes 9 and 10.
+
+<sup>17</sup> See Footnote 12.
+
+<span id="page-8-0"></span>![](_page_8_Figure_2.jpeg)
+
+Fig. 18. Hoerl prediction: standard error around 0 as a function of s and n.
+
+## 7.4. The Oppe–Koornstra model
+
+The Oppe–Koornstra model consists of two equations. The first equation traces the evolution of risk over time [Rt = exp(at + b) in which Rt is the risk at time t, with a and b as parameters to be estimated from data]. The second equations traces the evolution of exposure over time [Vt = Vm/{1 + exp - [(a(t c) + b]} in which Vt represents the vehicle-kilometres of travel (Vkmt) at time t, Vm is the maximum (saturation) Vkmt and a, b and c are parameters estimated from data]. The number of accidents is the product Rt Vt.
+
+The Vkmt information for Province A is available from the period 1960–2007. For each year between 1966 and 1998 the parameters a, b, a, b, c and Vm were estimated using all the data from 1960 till that year. Using these parameters Rt and Vt were predicted for s = 1, 2, ... , 10 steps ahead and the relative differences [d(s)] calculated. The usual measures of performance are in Table 10. As is obvious the biases (the d and all the d's that make it up) are negative and large. Had the Oppe–Koornstra model been applied to Province A it would have predicted more fatalities than what actually occurred. However, even if the bias is disregarded, the standard error around d is larger than when prediction is based on fitting a Hoerl function.
+
+#### 7.5. Summary for Province A
+
+The performance of several methods and their variants for predicting the number of fatalities in Province A up to ten steps ahead was examined. It seems that of the many options compared, fitting a Hoerl function with n = 10 and n = 19 would have performed best. This conclusion has to be qualified in several ways. First, there are many other prediction methods and variants that could be tried. Thus, e.g., with Holt smoothing one could take the last n u's (slopes), extrapolate them in some suitable way, and use the sum of extrapolated u's in the prediction. Or, e.g., one could use the Oppe–Koornstra model so that the fitted Risk and Exposure functions are based on the most recent n data points, rather than all data points. Second, the method that, had it been used, would have performed best in Province A, may not be best in another jurisdiction. To draw more general conclusions, similar analyses need to be performed on data from many other jurisdictions.
+
+Table 8 Measures of performance for Hoerl function {a exp[b(t d)](t d) } when n = 19.
+
+|                            | Steps ahead |          |          |          |          |          |          |          |          |          |
+|----------------------------|-------------|----------|----------|----------|----------|----------|----------|----------|----------|----------|
+|                            | 1           | 2        | 3        | 4        | 5        | 6        | 7        | 8        | 9        | 10       |
+| <br>d = average d (bias)   | 0.03<br>    | 0.02<br> | 0.02<br> | 0.02<br> | 0.02<br> | 0.02<br> | 0.03<br> | 0.03<br> | 0.03<br> | 0.03<br> |
+| Standard error around<br>d | 0.13        | 0.14     | 0.16     | 0.18     | 0.19     | 0.22     | 0.24     | 0.27     | 0.29     | 0.32     |
+| Standard error around 0    | 0.13        | 0.15     | 0.16     | 0.18     | 0.19     | 0.22     | 0.24     | 0.27     | 0.30     | 0.33     |
+
+Table 9 Measures of performance for Hoerl function {a exp[b(t d)](t d) } when n = 10.
+
+|                                                       | Steps ahead  |              |              |              |              |              |              |              |              |              |
+|-------------------------------------------------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|
+|                                                       | 1            | 2            | 3            | 4            | 5            | 6            | 7            | 8            | 9            | 10           |
+| <br>d = average d (bias)                              | 0.00         | 0.00         | 0.00         | 0.00         | 0.01<br>     | 0.02<br>     | 0.04<br>     | 0.05<br>     | 0.07<br>     | 0.08<br>     |
+| Standard error around<br>d<br>Standard error around 0 | 0.15<br>0.15 | 0.19<br>0.19 | 0.19<br>0.19 | 0.22<br>0.22 | 0.24<br>0.25 | 0.27<br>0.27 | 0.28<br>0.29 | 0.31<br>0.31 | 0.35<br>0.35 | 0.40<br>0.41 |
+
+Table 10 Measures of performance for the Oppe–Koonstra model.
+
+|                            |          | Steps ahead |          |          |          |          |          |          |          |          |  |  |  |
+|----------------------------|----------|-------------|----------|----------|----------|----------|----------|----------|----------|----------|--|--|--|
+|                            | 1        | 2           | 3        | 4        | 5        | 6        | 7        | 8        | 9        | 10       |  |  |  |
+| <br>d = average d (bias)   | <br>0.30 | <br>0.34    | <br>0.39 | <br>0.43 | <br>0.47 | <br>0.51 | <br>0.56 | <br>0.59 | <br>0.64 | <br>0.69 |  |  |  |
+| Standard error around<br>d | 0.28     | 0.30        | 0.31     | 0.31     | 0.30     | 0.31     | 0.33     | 0.35     | 0.36     | 0.37     |  |  |  |
+| Standard error around 0    | 0.42     | 0.46        | 0.50     | 0.53     | 0.56     | 0.59     | 0.65     | 0.69     | 0.74     | 0.78     |  |  |  |
+
+Table 11 Predictions for 1986–1995.
+
+| Year       | 1986  | 1987  | 1988  | 1989  | 1990  | 1991  | 1992  | 1993  | 1994  | 1995  |
+|------------|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|
+| Prediction | 216.6 | 210.0 | 203.4 | 196.6 | 189.8 | 183.0 | 176.2 | 169.4 | 162.7 | 156.1 |
+
+<span id="page-9-0"></span>![](_page_9_Figure_2.jpeg)
+
+Fig. 19. Time series of fatalities, population, etc. in Province B.
+
+![](_page_9_Figure_4.jpeg)
+
+Fig. 20. Amount of travel in Province B.
+
+To round out this section recall the story line around which the narration revolved. We imagined that we are at the end of 1985 and wish to predict for the years 1986–1995. It now seems that the Hoerl function with n = 19 would be sensible choice of method and variant. The parameter estimates based on the 1967–1985 data are a = 0.3438, b = -0.0982, c = 2.822 and d = 1944.7. Using these parameters the predictions are in [Table 11.](#page-8-0)
+
+Based on [Table 8](#page-8-0) the average bias is -0.02 to -0.03. It would therefore seem appropriate to reduce the prediction by 2–3%. With this the unbiased prediction, e.g., for 1990 (s = 5) is 189.8 0.98 = 186.0. Thus, at the end of 1985 one should predict that if the prevailing trends will continue and no substantially new policies and measures are implemented, in 1990 there will be in Province A about 186 fatalities. Also based on [Table 8](#page-8-0) the standard error is ±0.19 186 = ±35.3 fatalities. Using the usual rule of thumb of ±2 standard deviations, the number of fatalities is almost certain to be in the 116–256 range.
+
+# 8. Province B
+
+The main time series information for Province B is in Figs. 19 and 20. While fatalities reached a peak and later declined, population, licensed drivers, and registered vehicles continued to increase over time. Till 1999 the Vkmt information (in Fig. 20) was based on fuel sales data and after 2000 it is based on the Annual Canadian Vehicle Survey by Statistics Canada. Because of the uncertainties surrounding this data the number of Licensed Drivers is used as a proxy for Exposure.
+
+#### 8.1. Linear prediction
+
+The performance of the linear OLS predictor on the Province B data is best for n between 8 and 13 as shown in Table 12.
+
+#### 8.2. Quadratic prediction
+
+For this method the quality of predictions improves with n. The measures of performance for n from 15 to 19 are in [Table 13.](#page-10-0) For s = 1 and 2 the quadratic predictor has a slight edge over the linear one but is significantly worse for larger s.
+
+#### 8.3. Holt smoothing A
+
+It appears that the extrapolation of a linear fit to n between 8 and 13 is about as good as the Holt Smoothing 'A'.
+
+Table 12 Measures of performance for the linear OLS predictor.
+
+| n           | Steps ahead |          |          |          |          |          |          |          |          |          |
+|-------------|-------------|----------|----------|----------|----------|----------|----------|----------|----------|----------|
+|             | 1           | 2        | 3        | 4        | 5        | 6        | 7        | 8        | 9        | 10       |
+| <br>d, bias |             |          |          |          |          |          |          |          |          |          |
+| 8           | <br>0.01    | <br>0.02 | <br>0.03 | <br>0.03 | <br>0.03 | <br>0.03 | <br>0.03 | <br>0.03 | <br>0.04 | <br>0.04 |
+| 9           | 0.01<br>    | 0.02<br> | 0.03<br> | 0.03<br> | 0.03<br> | 0.02<br> | 0.03<br> | 0.03<br> | 0.04<br> | 0.04<br> |
+| 10          | <br>0.02    | <br>0.02 | <br>0.03 | <br>0.02 | <br>0.02 | <br>0.02 | <br>0.03 | <br>0.03 | <br>0.04 | <br>0.04 |
+| 11          | <br>0.02    | <br>0.02 | <br>0.02 | <br>0.02 | <br>0.02 | <br>0.02 | <br>0.03 | <br>0.03 | <br>0.04 | <br>0.05 |
+| 12          | <br>0.02    | <br>0.02 | <br>0.01 | <br>0.01 | <br>0.02 | <br>0.02 | <br>0.03 | <br>0.04 | <br>0.04 | <br>0.05 |
+| 13          | 0.01<br>    | 0.01<br> | 0.01<br> | 0.01<br> | 0.02<br> | 0.02<br> | 0.03<br> | 0.04<br> | 0.05<br> | 0.06<br> |
+| SE around   | d           |          |          |          |          |          |          |          |          |          |
+| 8           | 0.13        | 0.17     | 0.22     | 0.24     | 0.26     | 0.28     | 0.31     | 0.35     | 0.39     | 0.42     |
+| 9           | 0.13        | 0.18     | 0.21     | 0.23     | 0.25     | 0.26     | 0.30     | 0.33     | 0.37     | 0.40     |
+| 10          | 0.14        | 0.17     | 0.20     | 0.22     | 0.23     | 0.26     | 0.29     | 0.32     | 0.36     | 0.39     |
+| 11          | 0.14        | 0.17     | 0.19     | 0.20     | 0.23     | 0.26     | 0.29     | 0.32     | 0.35     | 0.38     |
+| 12          | 0.13        | 0.16     | 0.18     | 0.20     | 0.23     | 0.25     | 0.29     | 0.32     | 0.35     | 0.38     |
+| 13          | 0.13        | 0.15     | 0.17     | 0.20     | 0.23     | 0.26     | 0.29     | 0.32     | 0.35     | 0.39     |
+| SE around 0 |             |          |          |          |          |          |          |          |          |          |
+| 8           | 0.13        | 0.18     | 0.22     | 0.24     | 0.26     | 0.28     | 0.31     | 0.35     | 0.39     | 0.42     |
+| 9           | 0.14        | 0.18     | 0.21     | 0.23     | 0.25     | 0.27     | 0.30     | 0.34     | 0.37     | 0.40     |
+| 10          | 0.14        | 0.18     | 0.21     | 0.22     | 0.23     | 0.26     | 0.30     | 0.33     | 0.36     | 0.39     |
+| 11          | 0.14        | 0.17     | 0.19     | 0.20     | 0.23     | 0.26     | 0.29     | 0.32     | 0.36     | 0.39     |
+| 12          | 0.14        | 0.16     | 0.18     | 0.20     | 0.23     | 0.25     | 0.29     | 0.32     | 0.36     | 0.39     |
+| 13          | 0.13        | 0.15     | 0.18     | 0.20     | 0.23     | 0.26     | 0.29     | 0.32     | 0.36     | 0.39     |
+
+<span id="page-10-0"></span>Table 13 Measures of performance for the quadratic OLS predictor.
+
+| n           | Steps ahead |          |          |          |          |          |          |          |          |          |
+|-------------|-------------|----------|----------|----------|----------|----------|----------|----------|----------|----------|
+|             | 1           | 2        | 3        | 4        | 5        | 6        | 7        | 8        | 9        | 10       |
+| <br>d, bias |             |          |          |          |          |          |          |          |          |          |
+| 15          | 0.01        | 0.01     | 0.01     | 0.01     | 0.01     | 0.01     | 0.00     | 0.00     | 0.01<br> | 0.01<br> |
+| 16          | 0.00        | 0.00     | 0.00     | 0.00     | 0.01<br> | 0.01<br> | 0.01<br> | 0.02<br> | 0.03<br> | 0.03<br> |
+| 17          | 0.00        | <br>0.01 | <br>0.01 | <br>0.01 | <br>0.02 | <br>0.02 | <br>0.03 | <br>0.04 | <br>0.05 | <br>0.05 |
+| 18          | <br>0.01    | <br>0.01 | <br>0.02 | <br>0.02 | <br>0.03 | <br>0.03 | <br>0.04 | <br>0.05 | <br>0.05 | <br>0.06 |
+| 19          | 0.01<br>    | 0.01<br> | 0.02<br> | 0.02<br> | 0.03<br> | 0.03<br> | 0.04<br> | 0.05<br> | 0.05<br> | 0.06<br> |
+| 20          | <br>0.01    | <br>0.01 | <br>0.02 | <br>0.03 | <br>0.03 | <br>0.03 | <br>0.04 | <br>0.04 | <br>0.05 | <br>0.05 |
+| SE around   | d           |          |          |          |          |          |          |          |          |          |
+| 15          | 0.12        | 0.17     | 0.22     | 0.28     | 0.35     | 0.41     | 0.50     | 0.59     | 0.68     | 0.78     |
+| 16          | 0.12        | 0.16     | 0.22     | 0.27     | 0.33     | 0.40     | 0.48     | 0.56     | 0.65     | 0.75     |
+| 17          | 0.12        | 0.17     | 0.22     | 0.27     | 0.33     | 0.39     | 0.47     | 0.55     | 0.64     | 0.73     |
+| 18          | 0.12        | 0.17     | 0.22     | 0.27     | 0.32     | 0.38     | 0.46     | 0.54     | 0.63     | 0.72     |
+| 19          | 0.13        | 0.17     | 0.22     | 0.26     | 0.32     | 0.38     | 0.45     | 0.53     | 0.62     | 0.71     |
+| 20          | 0.13        | 0.17     | 0.21     | 0.26     | 0.32     | 0.38     | 0.45     | 0.52     | 0.61     | 0.70     |
+| SE around 0 |             |          |          |          |          |          |          |          |          |          |
+| 15          | 0.12        | 0.17     | 0.22     | 0.28     | 0.35     | 0.41     | 0.50     | 0.59     | 0.68     | 0.78     |
+| 16          | 0.12        | 0.16     | 0.22     | 0.27     | 0.33     | 0.40     | 0.48     | 0.56     | 0.66     | 0.75     |
+| 17          | 0.12        | 0.17     | 0.22     | 0.27     | 0.33     | 0.39     | 0.47     | 0.55     | 0.64     | 0.74     |
+| 18          | 0.12        | 0.17     | 0.22     | 0.27     | 0.32     | 0.38     | 0.46     | 0.54     | 0.63     | 0.72     |
+| 19          | 0.13        | 0.17     | 0.22     | 0.26     | 0.32     | 0.38     | 0.46     | 0.53     | 0.62     | 0.71     |
+| 20          | 0.13        | 0.17     | 0.22     | 0.26     | 0.32     | 0.38     | 0.45     | 0.52     | 0.61     | 0.70     |
+
+Table 14 Measures of performance for Holt smoothing A.
+
+|                            | Steps ahead |          |      |      |      |      |      |          |          |          |
+|----------------------------|-------------|----------|------|------|------|------|------|----------|----------|----------|
+|                            | 1           | 2        | 3    | 4    | 5    | 6    | 7    | 8        | 9        | 10       |
+| <br>d = average d (bias)   | 0.00        | 0.01<br> | 0.00 | 0.01 | 0.01 | 0.01 | 0.00 | 0.01<br> | 0.03<br> | 0.04<br> |
+| Standard error around<br>d | 0.11        | 0.15     | 0.20 | 0.21 | 0.21 | 0.23 | 0.28 | 0.31     | 0.36     | 0.40     |
+| Standard error around 0    | 0.11        | 0.15     | 0.20 | 0.21 | 0.22 | 0.23 | 0.28 | 0.31     | 0.36     | 0.40     |
+
+Table 15 Measures of performance for the Hoerl function.
+
+|                            | Steps ahead |      |      |      |      |      |      |      |      |      |
+|----------------------------|-------------|------|------|------|------|------|------|------|------|------|
+|                            | 1           | 2    | 3    | 4    | 5    | 6    | 7    | 8    | 9    | 10   |
+| <br>d = average d (bias)   | 0.01        | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.07 | 0.08 | 0.09 |
+| Standard error around<br>d | 0.16        | 0.18 | 0.19 | 0.21 | 0.24 | 0.26 | 0.29 | 0.31 | 0.34 | 0.36 |
+| Standard error around 0    | 0.16        | 0.18 | 0.19 | 0.22 | 0.24 | 0.27 | 0.29 | 0.32 | 0.35 | 0.37 |
+
+# 8.4. The Hoerl function
+
+For the Province B time series of fatalities the Hoerl Function performs best when n is about 12. The measures of performance at this point are in Table 15.
+
+### 8.5. The Oppe–Koornstra method
+
+To apply this method one has to compute risk which, in the Oppe–Koornstra formulation is the ratio of fatalities per vehiclekilometres of travel. Inasmuch as for Province B the information about Vkmt is sketchy as shown in [Fig. 20](#page-9-0), the number of licensed drivers will be the proxy for exposure.18 Even though the predictions of risk and exposure are quite good, their product does not yield satisfactory predictions as is obvious from comparing the measures of performance in [Table 16](#page-11-0) to these in [Table 12](#page-9-0) or Table 14.
+
+For Province B the count of fatalities begins in 1931. To produce [Table 16](#page-11-0) the parameters of the risk and exposure function were estimated using data from 1931 till the year before the prediction for s = 1. This is in line with the spirit of the Oppe–Koornstra thinking which regards the regularities in the evolution of risk and exposure as representing some deeper theoretical principles. It is possible the predictions would be better if the parameter estimates were based on the n most recent data points rather than on all available data.
+
+#### 8.6. Summary for Province B
+
+The performance of several methods and their variants for predicting the number of fatalities in Province B for up to ten steps (years) ahead was examined. It seems that the linear predictor with n around 10, Holt smoothing A, and the Hoerl function predictor perform similarly. The quadratic predictor was better than the
+
+<sup>18</sup> Vkmt and the number of licensed drivers are usually closely correlated. It is possible that the number of kilometres of travel per licensed driver changes gradually over time. However, in the period 2000–2007 for which reliable Vkmt data are available there is no evidence of any such trend.
+
+<span id="page-11-0"></span>Table 16 Measures of performance for the Oppe–Koornstra method.
+
+|                            | Steps ahead |          |          |          |          |          |          |          |          |          |
+|----------------------------|-------------|----------|----------|----------|----------|----------|----------|----------|----------|----------|
+|                            | 1           | 2        | 3        | 4        | 5        | 6        | 7        | 8        | 9        | 10       |
+| <br>d = average d (bias)   | <br>0.09    | <br>0.10 | <br>0.12 | <br>0.13 | <br>0.15 | <br>0.17 | <br>0.19 | <br>0.20 | <br>0.21 | <br>0.22 |
+| Standard error around<br>d | 0.31        | 0.33     | 0.34     | 0.36     | 0.37     | 0.38     | 0.39     | 0.41     | 0.42     | 0.44     |
+| Standard error around 0    | 0.32        | 0.34     | 0.36     | 0.38     | 0.40     | 0.42     | 0.44     | 0.46     | 0.47     | 0.49     |
+
+linear one for one or two steps ahead and significantly worse for predicting further into the future. The Oppe–Koornstra method has a relatively large average bias and standard error. It is possible that its performance would improve if the number of data points used in the curve fitting was limited.
+
+## 9. Summary and discussion
+
+Prediction is about potential outcomes: what will happen if and about what would have happened if. As there are many ways to predict one has to determine what method tends to predict best. To do so empirically one asks what method would have predicted best had it been applied in the past and assumes, inductively, that the same will be true in the future.
+
+To say which of two methods is better, one has to have a yardstick for prediction quality. It is suggested that d (the average bias) and the standard error of d be used for this purpose. The d tells by what proportion, on the average, the count of accidents would exceed the prediction if the method was applied year after year. The standard error (around d or around 0) measures the variability of the d's in the usual manner. It is shown how these measures of prediction quality allow one to determine which of two methods should be preferred in what circumstance.
+
+The approach was applied to two data sets: The time series of motor vehicle accident fatalities in Province A and in Province B. On the basis of this analysis one may draw tentative conclusions for these jurisdictions and the methods tested; one can say what method seems preferable, what is the average size of bias that needs to be corrected, and how accurate is the prediction likely to be.
+
+Broader conclusions will emerge once many additional methods of prediction are applied to data from many other jurisdictions and pertaining to other circumstances. As there are many jurisdictions that have long time series of road safety related data, research along these lines is both feasible and attractive. The hope is that the approach outlined here will prove attractive to others and that the cumulative results of research on prediction will make for better safety evaluation and improved setting of safety targets.
+
+#### Acknowledgements
+
+The help of Transport Canada, Road Safety is gratefully acknowledged. Kwei Quaye of the Saskatchewan Government Insurance, Leo Tasca and Chris Janusz of the Ontario Ministry of Transport provided important data.
+
+#### References
+
+Broughton, J., Allsop, R.E., Lynam, D.A., McMahon, C.M., 2000. The numerical context for setting national casualty reduction targets. TRL Report 382, TRL Limited, Wokingham.
+
+Broughton, J., 2006. Monitoring progress towards the GB casualty reduction target. In: Proceedings, European Transport Conference, Strasbourg, France.
+
+Hauer, E., Ng, J.N.C., Papaioannou, P., 1991. Prediction in road safety studies: an empirical inquiry. Accident Analysis and Prevention 23 (6), 595–607.
+
+Hauer, E., 1991. Comparison groups in road safety studies: an analysis. Accident Analysis and Prevention 23 (6), 609–622.
+
+Hauer, E., 1997. Observational Before–After Studies in Road Safety. Pergamon.
+
+Hyndman, R.J., Koehler, A.B., 2006. Another look at measures of forecast accuracy. International Journal of Forecasting 22 (4), 679–688.
+
+Makridakis, S., Hibon, M., 2000. The M3-competition: results, conclusions, and implications. International Journal of Forecasting 16, 451–476.
+
+Oppe, S., Koornstra, M.J., 1990. A mathematical theory for related long term developments of road traffic and safety. In: Koshi, M. (Ed.), Transportation and Traffic Theory. Elsevier, New York, pp. 113–132.
+
+Oppe, S., 1991. Development of traffic and traffic safety: global trends and incidental fluctuations. Accident Analysis and Prevention 23 (5), 413–422.
+
+Quaye, K.E., 1992. Forecasting models in road safety studies. PhD Dissertation, Department of Civil Engineering, University of Toronto.
+
+Quaye, K.E., Hauer, E., 1993. The use of forecasting models in the evaluation of safety interventions: a theoretical inquiry. In: Daganzo, C.F. (Ed.), Transportation and Traffic Theory. Elsevier Science Publishers, pp. 313–332.
+
+Quaye, K.E., Hauer, E., 1994. Assessing forecasting methods used in before after studies. Paper presented at the 72nd Annual Meeting of the Transportation Research Board, Washington.
+
+Rubin, D.B., 2005. Causal inference using potential outcomes: design, modeling, decisions. Journal of the American Statistical Association 100 (469), 322–331.

Zotero/001_artiklid/Road Safety Indicators/helseffectroundaboutdesignfeatures2007.md

@@ -0,0 +1,309 @@
+---
+category: literaturenote
+citekey: helseffectroundaboutdesignfeatures2007
+title: The effect of roundabout design features on cyclist accident rate
+authors: "Hels, Tove; Orozova-Bekkevold, Ivanka"
+year: 2007
+date: 2007-00-00 2007
+doi: 10.1016/j.aap.2006.07.008
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2006.07.008"
+zotero_key: 2RFJFRDB
+zotero_storage: N4984AUP
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Hels, Tove"
+status: converted
+---
+![](_page_0_Picture_0.jpeg)
+
+Accident Analysis and Prevention 39 (2007) 300–307
+
+![](_page_0_Picture_2.jpeg)
+
+# The effect of roundabout design features on cyclist accident rate
+
+Tove Hels ∗, Ivanka Orozova-Bekkevold
+
+*Danish Transport Research Institute, Knuth-Wintherfeldts All´e, Building 116 West, DK-2800 Lyngby, Denmark* Received 27 January 2006; received in revised form 27 July 2006; accepted 30 July 2006
+
+### **Abstract**
+
+Roundabouts are known to result in fewer traffic accidents than traditional intersections. However, this is to a lesser degree true for bicycles than for vehicles. In this paper, we aimed at establishing statistical relationships through Poisson regression and logistic regression analyses between yearly rate of cyclist accidents on one hand and roundabout geometry, age and traffic volume (vehicles and cyclists) on the other. We related all roundabout cyclist accidents recorded by the hospital emergency department of the town of Odense, Denmark, through the years 1999–2003 (*N*= 171) to various geometric features, age and traffic volume of all roundabouts on the Danish island of Funen (*N*= 88). Cyclist and vehicle volumes turned out to be significant predictors in most of our models—the higher the volumes, the more accidents. Moreover, potential vehicle speed was a significant predictor, and so was age of the roundabout—older roundabouts related to more accidents and higher accident probability. Excluding 48 single cyclist accidents strengthened the relationship between accidents on one hand and vehicle and cyclist volume and potential vehicle speed on the other. This stresses the significance of speed and traffic volume for traffic accidents with more than one partner involved. The 48 single cyclist accidents were significantly related to the traffic volume of cyclists only. Due to our limited number of observations, the models should be regarded as indicative.
+
+© 2006 Elsevier Ltd. All rights reserved.
+
+*Keywords:* Roundabout; Cyclist accident; Geometry; Speed
+
+#### **1. Introduction**
+
+Roundabouts are popular. During the latest 20 years, more and more intersections have been converted to roundabouts, both in Europe (e.g. Britain: [Brown, 1995, D](#page-6-0)enmark: [Jørgensen and](#page-7-0) [Jørgensen, 1994,](#page-7-0) Finland: Ras¨ [anen and Summala, 2000](#page-7-0) ¨ , Sweden: [Brude and Larsson, 2000](#page-6-0) ¨ , Switzerland: [Huber, 1995,](#page-7-0) The Netherlands: [Schoon and van Minnen, 1994,](#page-7-0) France: Guichet pers. com.) and in the US, although the development in the US has come later than in Europe [\(Perseaud et al., 2001; Flannery,](#page-7-0) [2001\).](#page-7-0) The advantages of roundabouts as compared to intersections concern two aspects: safety and mobility. Safety: roundabouts produce slowed and smooth traffic flows with few stops and no left turns in front of left coming traffic in countries where traffic drives on the right. Mobility: roundabouts have a larger capacity than give way intersections and signalised junctions due to the fact that left turns are omitted ([Elvik and Vaa, 2004\).](#page-7-0) [Elvik and Vaa \(2004\)](#page-7-0) have reviewed 34 studies from Northern Europe, Australia and the US evaluating the effect on accidents
+
+The aim of this study was to identify reasons for the relatively higher prevalence of cyclist accidents in Danish roundabouts compared to intersections [\(Herslund and Jørgensen, 2003\).](#page-7-0) Our primary aim was to find relations between the number of cyclist accidents in roundabouts and the geometric features of a roundabout, its age and traffic volume (vehicles as well as cyclists).
+
+of converting an intersection into a roundabout. They find a mean reduction in injury accidents of 10 to 40%. Generally, the reduction in serious injury accidents is larger than that in less serious injury accidents, while property damage only accidents typically increase in number, although this effect is highly uncertain. Roundabouts, however, do not have the same positive effect on the number of cyclist accidents ([Schoon and van](#page-7-0) [Minnen, 1994; Jørgensen and Jørgensen, 1994\).](#page-7-0) There may be more than one explanation to this. Firstly, a roundabout represents a more challenging orientation task for all road users than an ordinary junction because of the circular design. This may increase the probability of orientation failure and the risk of accidents. Secondly, passing through a roundabout is physically more demanding due to the circular deflection of the road. This may increase the number and proportion of one-cyclist accidents.
+
+<sup>∗</sup> Corresponding author. Tel.: +45 45 25 65 00; fax: +45 45 93 65 33. *E-mail addresses:* [ths@dtf.dk](mailto:ths@dtf.dk) (T. Hels), [iob@dtf.dk](mailto:iob@dtf.dk) (I. Orozova-Bekkevold).
+
+| Variables recorded for the 88 roundabouts the number of bicycle accidents is the dependent variable, the rest are the independent variables |                     |                              |              |    |
+|---------------------------------------------------------------------------------------------------------------------------------------------|---------------------|------------------------------|--------------|----|
+| Variable (NAME)                                                                                                                             | Type of variable    | Range                        | Mean/median  | N  |
+| Number of legs (LEGS)                                                                                                                       | Numeric, discrete   | 2–6                          | Median: 4    |    |
+| Diameter of central island (ISLAND)                                                                                                         | Numeric, continuous | 5–53 m                       | Mean: 19.0 m | 85 |
+| Circulatory roadway width (CIRC)                                                                                                            | Numeric, continuous | 4–15 m                       | Mean: 6.4 m  | 86 |
+| Apron width (APRON)                                                                                                                         | Numeric, continuous | 0–6 m                        | Mean: 1.6 m  | 84 |
+| Drive curve (DRIVCURV)                                                                                                                      | Numeric, continuous | 9–40 m                       | Mean: 21 m   | 62 |
+| Cyclist facility (CYCFAC)                                                                                                                   | Binary              | 0 (no), 1 (yes)              | –            | 81 |
+| Traffic volume of cyclists (CYCLISTS)                                                                                                       | Semi-quantitative   | 1 (≤1000/day), 2 (>1000/day) | –            | 60 |
+| Traffic volume of vehicles (VEHIC)                                                                                                          | Numeric, continuous | 1100–24,500/day              | Mean: 7200   | 69 |
+| Year of construction (CONSTRYEAR)                                                                                                           | Numeric, discrete   | 1981–2004                    | Median: 1996 | 80 |
+
+Location (urban or rural zone) (ZONE) Qualitative Urban, rural, urban/rural – 88 Number of bicycle accidents in each roundabout Numeric, discrete 0–29 Median: 1 152
+
+<span id="page-1-0"></span>Table 1 Variables recorded for the 88 roundabouts the number of bicycle accidents is the dependent variable, the rest are the independent variables
+
+Secondly, we wanted to clarify the types and prevalence of various cyclist accidents. Thirdly, our aim was to quantify the degree of underreporting of cyclist accidents (cyclist accidents recorded by the police compared to the cyclist accidents recorded by the hospital emergency department).
+
+#### *1.1. Hypotheses*
+
+The main hypotheses of this study were:
+
+- The number of accidents with cyclists in a roundabout increases with increasing of the traffic volume of cyclists and vehicles.
+- Roundabout geometry, which allows higher speed of the vehicles leads to higher number of cyclist accidents.
+- Existence of a cyclist facility makes the roundabout safer for cyclists if the traffic volume of vehicles is high.
+
+## **2. Data**
+
+The data set consisted of data on cyclist accidents and data on geometric features, age and position of the roundabouts, besides the daily number of vehicles and cyclists entering the roundabouts (measured per 24 h).
+
+The cyclist roundabout accidents were recorded in two different ways: by the police and by the emergency department of the hospital in Odense, Funen, in Denmark. Accidents involving cyclists between the years 1999 and 2003 on the Danish island of Funen were used.
+
+For all roundabouts at the island of Funen (*N*= 88) the following features were recorded (abbreviated variable name in parentheses, cf. Table 1):
+
+The number of legs was recorded from digital maps and from construction maps of the roundabouts. The diameter of the central island, the circulatory roadway width and the apron width were all measured from construction maps of the roundabouts. If there was a cyclist facility, this was not considered part of the circulatory roadway. The drive curve was measured from construction maps according to [CROW, 1998](#page-6-0) (cf. Fig. 1):
+
+$$D = \frac{(0.25 \times L)^2 + (0.50 \times (U+2))^2}{U+2},$$
+ (1)
+
+where *D* (in m) is the drive curve (R on Fig. 1), and *L* and *U* can be seen from Fig. 1. Drive curve was measured for symmetrical roundabouts only (*N*= 62). In these roundabouts, drive curves were the same for all directions. The drive curve variable is a proxy for potential vehicle speed through roundabout in that it is a measure of how much deflection is needed over a given distance to pass through the roundabout. The drive curve was considered a continuous variable.
+
+Information on the existence of a cyclist facility was provided by the road authorities, i.e. the municipalities (*N*= 46), the county of Funen (*N*= 38) or in the four cases of large roundabouts, the Danish Road Directorate. Any kind of cyclist facility was considered to have equal potential impact on cyclist accidents. Traffic volume of cyclists was assessed by the road authorities as the total number of cyclists riding into the roundabout per 24 h. The number was placed into an ordinal scale: 1: 0–250, 2: 250–500, 3: 500–1000, 4: 1000–2000 and 5: above 2000. The traffic volume of vehicles was counted by the road authorities as the total number of ingoing vehicles per 24 h in the roundabout. The traffic volume was counted in different years; in order to compare the numbers directly, they were all standardised to year 2001-numbers. This standardisation was done by calculating the development in the total number of kilometres driven by motor vehicles in Denmark. In 2002 the
+
+![](_page_1_Picture_18.jpeg)
+
+Fig. 1. A sketch of a roundabout showing the measures necessary for the calculation of the drive curve. After [CROW \(1998\).](#page-6-0)
+
+<span id="page-2-0"></span>number of kilometres driven equalled 1.22 times the number of kilometres driven in 1992 (source: Danish Road Directorate: [http://www.vd.dk](http://www.vd.dk/)), corresponding to a yearly increase by 1.02. This factor was then used—either to multiply or to divide traffic volume numbers from years other than 2001 to standardise them to 2001-numbers. The construction years of the roundabouts were provided by the road authorities and was considered a discrete variable with values ranging from 1981 to 2004 (cf. [Table 1\).](#page-1-0) The location of the roundabouts (urban surroundings, country surroundings or a combination) was found on digital GIS-maps and was considered a class variable.
+
+#### **3. Data analysis**
+
+The goal of the analysis was to explain as much variation as possible in the number of cyclist accidents in the roundabouts by the independent variables mentioned above. We have used a
+
+Table 2 Number of roundabouts and cyclist accidents in the various categories
+
+| Variable               | Category               | Number of<br>roundabouts | Number of emergency<br>department recorded accidents |  |
+|------------------------|------------------------|--------------------------|------------------------------------------------------|--|
+| LEGS (no.)             | 2–3                    | 24                       | 18                                                   |  |
+|                        | 4                      | 59                       | 108                                                  |  |
+|                        | 5–6                    | 5                        | 26                                                   |  |
+| Total                  |                        | 88                       | 152                                                  |  |
+|                        |                        |                          |                                                      |  |
+| ISLAND (m)             | 0–10                   | 10                       | 5                                                    |  |
+|                        | 10–20                  | 34                       | 96                                                   |  |
+|                        | 20–30                  | 26                       | 21                                                   |  |
+|                        | 30–40                  | 12                       | 21                                                   |  |
+|                        | 40–50                  | 2                        | 9                                                    |  |
+|                        | 50–60                  | 1                        | 0                                                    |  |
+| Total                  |                        | 85                       | 152                                                  |  |
+| CIRC (m)               | 4–8                    | 76                       | 116                                                  |  |
+|                        | 8–12                   | 7                        | 35                                                   |  |
+|                        | 12–16                  | 3                        | 0                                                    |  |
+| Total                  |                        | 86                       | 151a                                                 |  |
+| APRON (m)              | 0–2                    | 46                       | 111                                                  |  |
+|                        | 2–4                    | 36                       | 40                                                   |  |
+|                        | 4–6                    | 2                        | 0                                                    |  |
+| Total                  |                        | 84                       | 151a                                                 |  |
+|                        |                        |                          |                                                      |  |
+| DRIVCURV (m)           | 9–20                   | 28                       | 30                                                   |  |
+|                        | 20–30                  | 28                       | 53                                                   |  |
+| Total                  | 30–40                  | 6<br>62                  | 49<br>132a                                           |  |
+|                        |                        |                          |                                                      |  |
+| CYCFAC                 | 0 (no)                 | 19                       | 18                                                   |  |
+|                        | 1 (yes)                | 62                       | 129                                                  |  |
+| Total                  |                        | 81                       | 147a                                                 |  |
+| CYCLISTS (no. per day) | <1000                  | 46                       | 28                                                   |  |
+|                        | ≥1000                  | 14                       | 87                                                   |  |
+| Total                  |                        | 60                       | 115a                                                 |  |
+| VEHIC (no. per day)    | 0–5000                 | 23                       | 19                                                   |  |
+|                        | 5000–10,000            | 34                       | 45                                                   |  |
+|                        | 10,000–15,000          | 7                        | 19                                                   |  |
+|                        | 15,000–20,000          | 4                        | 50                                                   |  |
+|                        | 20,000–25,000          | 1                        | 0                                                    |  |
+| Total                  |                        | 69                       | 133a                                                 |  |
+|                        |                        |                          |                                                      |  |
+| CONSTRYEAR             | 1981–1995<br>1996–2004 | 38<br>42                 | 108<br>37                                            |  |
+| Total                  |                        | 80                       | 145a                                                 |  |
+|                        |                        |                          |                                                      |  |
+| ZONE                   | Urban                  | 60                       | 132                                                  |  |
+|                        | Rural                  | 10                       | 2                                                    |  |
+|                        | Urban/rural            | 18                       | 18                                                   |  |
+| Total                  |                        | 88                       | 152                                                  |  |
+
+Categorisation of the continuous variables is arbitrary and only serves the purpose of giving an overview of data.
+
+<sup>a</sup> There are 152 cyclist accidents registered by the emergency department for which the location is known. Some of the totals are less than 152; this is because the rest of the accidents took place in roundabouts where the variable in question was not recorded.
+
+<span id="page-3-0"></span>Poisson model with the formal expression:
+
+$$\log(y) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \ldots + \beta_n x_n + \varepsilon, \tag{2}$$
+
+where *y* is the dependent variable (number of accidents per year after the construction of the roundabout), *x*1, ..., *xn* denote the independent variables mentioned above, and β0, ..., β*<sup>n</sup>* are coefficients scaling the variables. Binary variables (e.g. existence of a cyclist facility) are given the value of one when a cyclist facility is present and 0 otherwise. Ordinally scaled variables are given a separate coefficient for each level of the variable. One level is set to be the reference level with a coefficient equal to 0. ε is the variation in the dependent variable not accounted for by the model.
+
+However, because of the limited number of roundabouts, building a complete model and subsequently eliminate insignificant variables was not possible. Therefore, we adopted a different strategy: We checked each independent variable separately for significance and eventually put the significant variables together into a model. Proc GENMOD (SAS 9.1) was used to quantify the coefficients and the significance of the independent variables (*F*-tests). In order to save degrees of freedom, the categorization of the variables LEGS, CYCLISTS and CONSTRYEAR was slightly changed (cf. [Table 2\):](#page-2-0)
+
+*LEGS.* Three categories were used: category 1: two to three legs, category 2: four legs and category 3: five to six legs.
+
+*CYCLISTS.* Instead of five categories, two were used: above and below 1000 cyclists per 24 h.
+
+*CONSTRYEAR.* Two categories were established: 1: 1981– 1995 and 2: 1996–2004.
+
+Secondly, we built a logistic regression model where the dependent variable instead of being the number of cyclist accidents as above, was the probability of a cyclist accident taking place given certain values of independent variables. The same categories for variables LEGS, CYCLISTS and CONSTRYEAR as mentioned above, were used in the logistic regression analyses.
+
+A logistic relation is described by the general formula:
+
+$$P(y) = \frac{\exp(y)}{1 + \exp(y)},\tag{3}$$
+
+where *P*(*y*) denotes the probability of an event, and log(*y*) is a linear expression of *x*, given in Eq. (2) [\(Hosmer and Lemeshow,](#page-7-0) [1989\).](#page-7-0) The model building strategy was the same as for the Poisson model given above. Proc LOGISTIC (SAS 9.1) was used to quantify the coefficients and the significance of the independent variables (goodness of fit-tests).
+
+Table 3 Details of Poisson models including one variable at a time
+
+| Variable    | Poisson regression univariate analyses |      |       |        |                                                               | Logistic regression univariate analyses |                              |                                                               |                                                       |
+|-------------|----------------------------------------|------|-------|--------|---------------------------------------------------------------|-----------------------------------------|------------------------------|---------------------------------------------------------------|-------------------------------------------------------|
+|             | N                                      | d.f. | F     | P      | Parameter estimate                                            | χ2                                      | P                            | Parameter estimate                                            | Odds ratio estimate                                   |
+| LEGS        | 75                                     | 2    | 2.89  | 0.06   | Int.: 0.26<br>Cat. 1: −2.01<br>Cat. 2: −1.07<br>Cat. 3: ref.  | 0.36                                    | 0.83                         | Int: 4.5 × 10−16<br>3: −0.20<br>4: 0.12<br>5: ref.            | Legs 3 vs. 5: 0.82<br>Legs 4 vs. 5: 1.13              |
+| DIAMETER    | 74                                     | 1    | 0.21  | 0.65   | Int.: −1.08<br>0.71<br>0.40<br>Diameter: 0.01                 |                                         | Int: 0.46<br>Diameter: −0.02 | Island: 0.98                                                  |                                                       |
+| CIRC        | 75                                     | 1    | 2.41  | 0.13   | Int.: −2.04<br>0.28<br>0.60<br>Circ: 0.18                     |                                         | Int: 0.49<br>Circ: −0.07     | Circ: 0.93                                                    |                                                       |
+| APRON       | 73                                     | 1    | 6.52  | 0.01   | Int: −0.28<br>2.75<br>Apron: −0.42                            |                                         | 0.10                         | Int: 0.59<br>Apron: −0.30                                     | Apron: 0.74                                           |
+| DRIVCURV    | 55                                     | 1    | 32.95 | 0.0001 | Int: −3.78<br>Drivcurv: 0.13                                  | 3.97                                    | 0.05                         | Int: −1.95<br>Drivcurv: 0.10                                  | Drivcurv: 1.11                                        |
+| CYCFAC      | 70                                     | 2    | 1.81  | 0.18   | Int: −0.69<br>0: −0.96<br>1: ref.                             | 2.00                                    | 0.16                         | Int: 0.29<br>0: −0.88<br>1: ref.                              | Cycfac 0 vs. 1: 0.42                                  |
+| CYCLISTS    | 53                                     | 2    | 33.89 | 0.0001 | Int: 1.81<br>≤1000: ref.<br>>1000: 2.17                       | 6.02                                    | 0.01                         | Int: −0.45<br><1000: ref.<br>>1000: 2.06                      | Cyclists > 1000 vs. ≤ 1000: 7.81                      |
+| VEHIC       | 63                                     | 1    | 19.71 | 0.0001 | Int: −2.04<br>Vehic: 0.0001                                   | 2.21                                    | 0.14                         | Int: −0.60<br>Vehic: 9.4 × 10−5                               | Vehic: 1.00                                           |
+| CON-STRYEAR | 75                                     | 1    | 2.70  | 0.10   | Int: −1.33<br>1: 0.76<br>2: ref.                              | 4.70                                    | 0.03                         | Int: −0.50<br>1: 1.04<br>2: Ref.                              | Constryear 1 vs. 2: 2.82                              |
+| ZONE        | 75                                     | 2    | 2.67  | 0.08   | Int: −3.81<br>Urban: 3.17<br>Urban/rural: 2.63<br>Rural: ref. | 4.64                                    | 0.10                         | Int: −2.08<br>Urban: 2.37<br>Urban/rural: 2.20<br>Rural: ref. | Urban vs. rural: 10.67<br>Urban/rural vs. rural: 9.00 |
+
+Table 4 Details of the multiple Poisson model including all significant variables from the single-variable models
+
+| Variable  | d.f. | F    | P     | Coefficient estimate | P      |
+|-----------|------|------|-------|----------------------|--------|
+| INTERCEPT | 1    | –    | –     | −3.23                | 0.0001 |
+| CYCLISTS  | 2    | 3.86 | 0.028 | ≤1000: ref.          | ref.   |
+|           |      |      |       | >1000: 1.28          | 0.0045 |
+| VEHIC     | 1    | 3.68 | 0.062 | 0.00006              | 0.057  |
+| DRIVCURV  | 1    | 6.50 | 0.014 | 0.06                 | 0.010  |
+
+*N*= 50. Poisson regression analyses at left, logistic regressions at right.
+
+Since we have data only on accidents in the period 1999–2003, roundabouts constructed in 2003 and 2004 (*N*= 5) were excluded from the regression analyses.
+
+Both Poisson and logistic regression models were built with and without single cyclist accidents. Finally, logistic regression models were built with single cyclist accidents alone.
+
+## **4. Results**
+
+All roundabouts at the island of Funen were included in the analysis. Details on the variables describing them are summarised in [Tables 1 and 2.](#page-1-0)
+
+A total of 171 accidents involving cyclists were recorded by the emergency department of the hospital of Odense through the period 1999–2003, 152 of which were unambiguously assigned to a given roundabout. Only 42 of these 171 cyclist accidents were recorded by the police. Of the 171 accidents recorded by the emergency department, 69 (40%) were one-cyclist accidents; of these, 48 were unambiguously assigned to a given roundabout. Almost half of the accidents (i.e. 81, equal to 47%) were vehiclefailed-to-give-way-accidents; in 57 accidents the vehicle hit a circulating cyclist on its way into the roundabout; in 24 accidents the vehicle was about to leave the roundabout when it hit the cyclist. The rest of the emergency department cyclist accidents (21 equal to 13%) were of various kinds – e.g. cyclists hitting each other – or insufficiently described.
+
+The pattern of the 42 accidents recorded by the police in the same roundabouts in the years 1999–2003 was similar to the one above except that the one-cyclist accidents were fewer, namely no more than two accidents (equal to 5% of the recorded). Vehicle-failed-to-give-way-accidents had a higher proportion: 34 were recorded, equal to 81%. Of these, in 22 accidents the vehicle hit a circulating cyclist on its way into the roundabout; in 12 accidents the vehicle hit the cyclist on its way out of the roundabout. Six of the accidents (14%) contained too little information to be categorized.
+
+In the univariate Poisson model drive curve, apron width and traffic volume of cyclists and of vehicles significantly explained the variation in the number of cyclist accidents per year in the roundabouts ([Table 3\).](#page-3-0) More cyclist accidents per year were connected to roundabouts with a narrow apron, a large drive curve and with high volumes of cyclists and vehicles. The existence of a cyclist facility did not significantly explain the variation in the number of cyclist accidents, and neither did the number of legs. There was a trend for roundabouts with more legs to have more cyclist accidents. Just about the same variables were significant in the univariate logistic regression analysis. The probability of an accident was significantly positively related to the number of cyclists and to the age of the roundabout: the older the roundabout, the higher the probability of a cyclist accident. Like in the Poisson models, there was a higher accident probability connected to roundabouts with a small apron (non-significant) and a large drive curve (significant).
+
+In the multiple models the explanatory variables from the univariate analyses were put together into the same model to correct for each other. In Tables 4 and 5 are reported the coefficients and odds ratios only for the significant variables. In doing this, it turned out that apron width was not a significant predictor of the number of accidents in the linear model, whereas cyclist and vehicle volume and drive curve still were. In the multiple logistic model, the drive curve was significant, the number of cyclists was only an insignificant trend whereas construction year was close to being significant, the older roundabouts related to the higher accident probability.
+
+Excluding 48 single cyclist accidents from the analyses did not qualitatively change the outcome of the univariate or the multiple Poisson models. However, it did somewhat change the results of the logistic regression analysis. The significant variables (i.e. the number of cyclists and the age of roundabout) became more significant (*P* = 0.001 and *P* = 0.004, respectively). Moreover, apron, drive curve and traffic volume of vehicles changed from insignificant to significant (*P*-values 0.03, 0.03 and 0.03, respectively). In the multiple logistic model, traffic
+
+Table 5 Details of the multiple logistic model including all variables that were significant in the single-variable models
+
+| Variable   | d.f. | χ2   | P    | Coefficient estimate | Odds ratio estimate | 95% confidence limits |
+|------------|------|------|------|----------------------|---------------------|-----------------------|
+| INTERCEPT  | 1    | 5.91 | 0.02 | −4.80                | –                   | –                     |
+| CONSTRYEAR | 1    | 3.94 | 0.05 | 1.60                 | 1 vs. 2: 4.95       | 1.02–23.99            |
+| DRIVCURV   | 1    | 5.24 | 0.02 | 0.22                 | 1.24                | 1.03–1.50             |
+
+Table 6 Details of the multiple logistic model excluding single cyclist accidents
+
+| Variable   | d.f. | χ2    | P     | Coefficient estimate | Odds ratio estimate | 95% confidence limits |
+|------------|------|-------|-------|----------------------|---------------------|-----------------------|
+| INTERCEPT  | 1    | 10.23 | 0.001 | −2.25                | –                   | –                     |
+| CONSTRYEAR | 1    | 8.36  | 0.004 | 1.68                 | 1 vs. 2: 5.38       | 1.72–16.84            |
+| VEHIC      | 1    | 4.33  | 0.04  | 0.00015              | 1.00                | 1.00–1.00             |
+
+*N*= 63. For simplicity, variables which showed insignificant effect were omitted from the table.
+
+volume of vehicles stayed in the model instead of drive curve (Table 6).
+
+Analysing the 48 single cyclist accidents in the 88 roundabouts separately revealed very few significant relations to the recorded independent variables. Thus, only drive curve and traffic volume of cyclists turned out to be significantly related to the probability of single cyclist accidents in a roundabout, for both variables in a positive way (*N*= 55, *P* = 0.03 and *N*= 53, *P* = 0.004, respectively). The significance of the drive curve disappeared in a multiple logistic model containing these two variables.
+
+## **5. Discussion**
+
+The 88 roundabouts included in this analysis differ considerably, both in terms of geometry and traffic volumes. All of them are built in the speed-reducing way with legs more or less perpendicularly radiating from the circulatory roadway as opposed to the dynamic design that gives the roundabout a higher capacity but is not recommended where there are many cyclists ([Jørgensen and Jørgensen, 2002\).](#page-7-0)
+
+Single-cyclist accidents were the type of accident most seriously underreported by the police, i.e. only 3% of this type of accident gets reported by the police. This was expected due to the nature of these accidents, since in single-cyclist accidents there are no conflicting interests for the involved partner regarding insurance matters, and the cyclist – if seriously injured – probably goes straight to the emergency department without contacting the police. It is highly likely that the number of single-cyclist accidents is vastly underreported by the emergency department as well, since most cyclists – if not seriously injured – probably go home after such an accident and then, if needed, later consults his or her general practitioner.
+
+About 75% of the cyclist accidents recorted by the hospital were not found in the police reports. In a comparable Danish study ([Bach, 2004\)](#page-6-0) the author found that cyclist accidents were by far the most underreported group of accidents since the percentage of reported cyclist accidents is only 5% of the actual number.
+
+A large part of the accidents were vehicle-failed-to-give-wayaccidents. This is in accordance with other findings [\(Alphand](#page-6-0) [et al., 1991; Jørgensen and Jørgensen, 2002\).](#page-6-0) Some or all of these may be what is also described as looked-but-failed-to-seeaccidents ([Hills, 1980\).](#page-7-0) In this type of accident the cyclist usually reports the driver to have looked at him but then moves on as if the cyclist was not there. [Herslund and Jørgensen \(2003\)](#page-7-0) have examined this type of accident closely and concluded tentatively that the drivers' visual search strategy and/or mental processing somehow fails, so the driver sees but does not perceive the cyclist.
+
+As a rule of thumb when building causal regression models, the number of observations must be about ten times the number of independent variables, including the number of levels of the semi quantitative variables. This is to ensure a fair number of observations in all categories. With a limited amount of data (i.e. 171 accidents distributed over 88 roundabouts) building models with a method like backwards elimination would clearly violate this rule. Therefore, we started out constructing univariate models to get an indication of the significance of the various variables. Later the explanatory variables were put together into multiple models to quantify the effects of the variables when correcting for other significant variables. The models developed here must be seen only as indicative due to the limited number of observations.
+
+When working with discrete data such as counts, the most used distributions are the Poisson and the binomial distributions. The negative binomial distribution also provides a good model for discrete data, and it can be derived from the Poisson distribution by providing a prior distribution to the mean. We decided to use a Poisson instead of a negative binomial distribution since Poisson provides a good model for our data and does not require a prior distribution (which means additional assumptions) for the mean.
+
+Many of the same explanatory variables turned out to be significantly related to the number of accidents and to the predicted accident probability (Poisson and logistic regression, respectively). It seems intuitively obvious that traffic volume is an important predictor of the number of accidents. We found significantly positive relationships between the number of cyclists and the number of accidents per year after construction of the roundabout as well as accident probability—a relation that was strengthened by excluding single cyclist accidents. By relating the original five class ordinal scale for cyclist traffic volume to the number of cyclist accidents and cyclist accident probability, it became evident that there was a significant increase in both the number of accidents and the accident probability at 1000 cyclists/day. Other authors have found positive relationships between the number of cyclists and the number of accidents as well ([Schoon and van Minnen, 1994;](#page-7-0) [Brude and Larsson, 1997, 2000](#page-7-0) ¨ ). In most other investigations of cyclist safety in roundabouts, traffic volume of cyclists is not included. In cases where it is included, the relation turns out to be like the one we found: the more cyclists, the more cyclist accidents. [Ekman \(1996\)](#page-6-0) suggested that even if the number of accidents increases with increasing number of cyclists, accident risk per cyclist is higher for low cyclist volumes. Unfortunately, <span id="page-6-0"></span>our data does not allow calculating accident risk per cyclist because the number of cyclists is recorded on a semi quantitative scale.
+
+According to the literature the relation between vehicle volume and number of cyclist accidents is clearly positive ([Layfield](#page-7-0) [and Maycock, 1986; Schoon and van Minnen, 1994; Brude and](#page-7-0) ¨ [Larsson, 2000; Perseaud et al., 2001; Elvik, 2003; Alphand et](#page-7-0) [al., 1991\).](#page-7-0) This was confirmed by our models. In the logistic regression models the relation between accident probability and vehicle volume was strengthened considerably, not surprisingly, by excluding single cyclist accidents. In doing so, focus was put on accidents involving cyclists and vehicles, which is probably why vehicle volume turned out to be more significant. Refined models usually fit a levelling off of number of accidents per circulating vehicle for large vehicle volumes. We, however, regard our models indicative due to the limited amount of data.
+
+Speed of the circulating vehicles has an intuitively obvious impact on accident risk; the speed also has a very high impact on the severity of the potential injury. We used a measure of deflection related to the length of the road deflected as a proxy for potential vehicle speed through the roundabout (i.e. drive curve). This measure turned out to be positively related to the number of cyclist accidents. A positive relation between vehicle speed and the number of cyclist accidents is also found by Brude ¨ and Larsson (2000) who measured speed of entering, circulating and exiting vehicles directly.
+
+Excluding single cyclist accidents also strengthened the positive relation between the traffic volume of cyclists and the probability of cyclist accidents. Overall, the probability of cyclist accidents turned out to be high in older roundabouts with high vehicle volumes. These are roundabouts often situated in the town centres where the cyclist volume is high, too. The location of the roundabouts is probably more important than its age, but since the location was not recorded accurately enough, age became a significant predictor of accident probability.
+
+Of the geometric features recorded, surprisingly few had any relation whatsoever to the number of cyclist accidents or cyclist accident probability. The effect of a cyclist facility on cyclist accidents was investigated by [Schoon and van Minnen \(1994\).](#page-7-0) In their much cited paper they found no effect on the number of accidents of a cyclist facility in roundabouts with vehicle volumes below 8000 vehicles/day. Above 8000 vehicles/day, however, they recommended construction of cycle paths completely separated from the circulatory roadway. [Jørgensen and](#page-7-0) [Jørgensen \(2002\)](#page-7-0) did not find a difference in number of cyclist accidents with different types of cyclist facilities (no facility, cycle lane, white markings on the asphalt) or cycle path (not separated from the circulatory roadway). We did not find a relation between the existence of a cyclist facility (0/1-variable) and the number of cyclist accidents or cyclist accident probability, but it might be due to the limited amount of observations, and it does not necessarily mean that there is no relation. In Denmark roundabouts without cyclist facilities tend to be rare, and in our data set we have only 19 of them. Moreover, in our data set we have different kinds of cyclist accidents lumped together due to the relatively small number of observations. It may be that the number of, e.g. vehicle-failed-to-give-way-accidents is related to the existence of a cyclist facility whereas other types of accidents are not.
+
+Excluding the single cyclist accidents strengthened the significance of the relation between drive curve, apron width and cyclist accident probability considerably. This stresses the connection between potential speed of vehicles and accident probability in accidents with more than one partner involved.
+
+According to our results, the probability of single cyclist accidents is apparently related to the number of cyclists entering the roundabout, only. As it turns out, the geometric features recorded in these particular roundabouts do not pose a challenge to bicycle riders. A look at the verbal descriptions of the single cyclist accidents reveals that single cyclist accidents often happen in situations where cyclists collide with the kerb or permanent installations like lamp posts, where cyclists are driving under influence and fall over, where the road is slippery due to winter weather conditions and/or where the cyclist fails at manoeuvring the bicycle properly, e.g. by braking too hard.
+
+One of the limitations of our study is clearly the relatively small number of observations, all of which were collected on the island of Funen. Another limitation, which is a general problem all over, is the high percentage of underreporting of cyclist accidents. Clearly, a larger database is needed, both concerning roundabouts and number of accidents, in order to construct a reliable statistical model. Until then, our results may serve as preliminary ones.
+
+## **Acknowledgements**
+
+We thank Lars Binderup-Larsen, consultant orthopaedic surgeon, for a great help in extracting relevant data from the databases of the Accident Analysis Group of the Odense University Hospital. Moreover, we thank all employees of the road authorities who provided us with invaluable information: The Danish Road Directorate, The County and the Communities of Funen. This work was supported by a grant from the Danish Ministry of Transport and Energy.
+
+# **References**
+
+Alphand, F., Noelle, U., Guichet, B., 1991. Roundabouts and road safety. State of the art in France. In: Brilon, W. (Ed.), Intersections Without Traffic Signals II. Springer-Verlag, pp. 126–140.
+
+Bach, U., 2004. Færdselsuheld registreret pa Sydvestjysk Sygehus [Traffic acci- ˚ dents recorded at the hospital in South West Jutland]. Carl Bro., 30 (in Danish).
+
+Brown, M., 1995. The design of roundabouts. HMSO, 270.
+
+Brude, U., Larsson, J., 1997. The safety of cyclists at roundabouts—a compari- ¨ son between Swedish, Danish and Dutch results. Nordic Road Transp. Res. 1, 23–25.
+
+Brude, U., Larsson, J., 2000. What roundabout design provides the highest pos- ¨ sible safety? Nordic Road Transp. Res. 2, 17–21.
+
+CROW, 1998. Eenheid in rotondes [in Dutch]. Publicatie 126, 112.
+
+Ekman, L., 1996. On the treatment of flow in traffic safety analysis. A nonparametric approach applied on vulnerable road users. Lunds Tekniska Hogskola, Bull. 136, 99. ¨
+
+Elvik, R., 1847. Effects on road safety of converting intersections to roundabouts. Transport. Res. Rec., 1–10.
+
+- <span id="page-7-0"></span>Elvik, R., Vaa, T., 2004. The Handbook of Road Safety Measures. Elsevier, 1078 pp.
+- Flannery, A., 1751. Geometric design and safety aspects of roundabouts. Transport. Res. Rec., 76–82.
+- Herslund, M.-B., Jørgensen, N.O., 2003. Looked-but-failed-to-see-errors in traffic. Accid. Anal. Prevent. 35, 885–891.
+- Hills, B.L., 1980. Vision, visibility and perception in driving. Perception 9, 183–216.
+- Hosmer, D.W., Lemeshow, S., 1989. Applied Logistic Regression. Wiley, 307 pp.
+- Huber, C.A., 1995. Sicherheit von Kreiselanlagen [in German]. Zeitschrift fur¨ Verkehrssicherheit 41 (2), 83–85.
+- Jørgensen, E., Jørgensen, N.O., 1994. Trafiksikkerhed i 82 danske rundkørsler—anlagt efter 1985 [Traffic safety in 82 Danish roundabouts], Institut for Veje, Trafik og Byplan, DTU, 49 pp. (in Danish).
+- Jørgensen, E., Jørgensen, N.O., 2002. Trafiksikkerhed i rundkørsler i Danmark. En analyse af uheld i danske rundkørsler i arene 1991–1996 [Traffic safety in ˚ roundabouts in Denmark. An analysis of accidents in Danish roundabouts in 1991–1996], Vejdirektoratet, rapport 235, 50 pp. + 16 siders notat (in Danish).
+- Layfield, R.E., Maycock, G., 1986. Pedal-cyclists at roundabouts. Traff. Eng. Contr. Juni, 343–349.
+- Perseaud, B.N., Retting, R.A., Garder, P.E., Lord, D., 1751. Safety effect of roundabout conversions in the United States: empirical bayes observational before-after study. Transport. Res. Rec., 1–9.
+- Ras¨ anen, M., og Summala, H., 2000. Car drivers' adjustments to cyclists at ¨ roundabouts. Transport. Hum. Factors 2 (1), 1–17.
+- Schoon, C., van Minnen, J., 1994. The safety of roundabouts in The Netherlands. Traffic Eng. Contr. 35 (3), 142–148.

Zotero/001_artiklid/Road Safety Indicators/hiseliusestimatingrelationshipbetweenaccident2004.md

@@ -0,0 +1,268 @@
+---
+category: literaturenote
+citekey: hiseliusestimatingrelationshipbetweenaccident2004
+title: Estimating the relationship between accident frequency and homogeneous and inhomogeneous traffic flows
+authors: "Hiselius, Lena Winslott"
+year: 2004
+date: 2004-00-00 2004
+doi: 10.1016/j.aap.2003.11.002
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2003.11.002"
+zotero_key: WHT97JG4
+zotero_storage: VA46EBG7
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Hiselius, Lena Winslott"
+status: converted
+---
+![](_page_0_Picture_0.jpeg)
+
+Accident Analysis and Prevention 36 (2004) 985–992
+
+![](_page_0_Picture_2.jpeg)
+
+# Estimating the relationship between accident frequency and homogeneous and inhomogeneous traffic flows
+
+# Lena Winslott Hiselius∗
+
+*Department of Economics or Department of Technology and Society, Lund University, P.O. Box 118, Lund 22100, Sweden* Received 5 January 2002; received in revised form 19 November 2003; accepted 24 November 2003
+
+#### **Abstract**
+
+This paper estimates the relationship between accident frequency and the traffic flow empirically treating the hourly traffic flow in two different ways, as consisting of homogenous vehicles and as consisting of cars and lorries. Rural roads in Sweden are studied using Poisson and Negative Binominal regression models. It is found that important information is lost if no consideration is taken to differences between vehicle types when estimating the marginal effect of the traffic flow. The accident rate decreases when the traffic flow is treated as if homogeneous. However, when cars are studied separately the result suggests that the accident rate is constant or increases. The result with respect to lorries is reversed, indicating a decreasing number of accidents as the number of lorries increases. © 2004 Elsevier Ltd. All rights reserved.
+
+*Keywords:* Accident prediction models; Road accidents; Hourly traffic flow; Poisson; Negative Binomial
+
+# **1. Introduction**
+
+Studies that analyse the relationship between accident frequency and traffic flows show a great variation in their results. An early report from [Vickery \(1968\)](#page-7-0) suggests that the marginal accident rate is 1.5 times the average accident rate. On the other hand, [Vitaliano and Held \(1991\)](#page-7-0) cannot detect any significant increase in the accident rate as the traffic flow increases and, according to [Hauer and Bamfo \(1997\), a](#page-7-0)nd a majority of the results reviewed in [Ardekani et al. \(1997\),](#page-7-0) the accident rate even decreases with an increasing number of vehicles. Furthermore, only a few empirical studies analyse the effect of the traffic flow separated for different road user groups. One example is the study by [Jovanis and](#page-7-0) [Chang \(1986\), w](#page-7-0)here they found that the accident frequency increased with the number of car and lorry miles travelled. However, it is difficult to distinguish the marginal effect of another car or lorry as the number of miles travelled varies in their study.
+
+There is, thus, an interest in studying the effect of the traffic flow more thoroughly, and the aim of this article is to estimate the relationship between the accident frequency and the traffic flow treated as both homogeneous, i.e. consisting of homogeneous vehicles, and inhomogeneous, i.e.
+
+*E-mail address:* lena.hiselius@tft.lth.se (L.W. Hiselius).
+
+consisting of cars and lorries. The data is restricted to accidents on sections in rural areas of Sweden.
+
+### **2. Model**
+
+The model seeks to describe the relationship between the number of vehicles per hour and the accident frequency. It is developed under the assumption that a limited and homogeneous road system is studied and only the traffic flow affects the number of accidents. In the first case the flow of vehicles per hour, *q*, is assumed to be homogeneous, and with all vehicles influencing the occurrence of an accident the same way. In the second case the traffic flow is defined as the number of cars and lorries per hour, *q*<sup>c</sup> and *q*l. The accident frequency, *Z*, is calculated as the expected number of accidents per unit of time and kilometer in order to apply the model to time periods and road systems of different length.
+
+### **3. Data**
+
+Data is collected from 83 road sections in rural areas of Sweden, where the number of passing vehicles is continuously counted by the Swedish National Road Administration. The assumption is made that the counted traffic flow at a stationary place is valid along the section. Information is collected on police reported accidents with personal
+
+<sup>∗</sup> Fax: +46-46-12-32-72.
+
+Table 1 Number of accidents in the data set
+
+| Road<br>type |       | Number of accidents | Total number of |                   |  |
+|--------------|-------|---------------------|-----------------|-------------------|--|
+|              | Total | Single-vehicle      | Multi-vehicle   | daytime accidents |  |
+| I            | 59    | 29                  | 30              | 49                |  |
+| II           | 83    | 52                  | 31              | 51                |  |
+| III          | 179   | 107                 | 72              | 111               |  |
+| IV           | 186   | 121                 | 65              | 129               |  |
+
+casualties that occurred on the studied road sections from 1989 to the middle of 1995. Given the time and date of the accidents, the hourly traffic flow that prevailed at the time of each accident is obtained. Furthermore, in order to calculate traffic flow frequencies, information on the number of hours that each traffic flow was observed during 1990 is also collected on the assumption that this is a representative year.
+
+Only accidents occurring on sections without intersections are included. Accidents involving animals are excluded together with accidents that may be considered specific for each road section. Since it is not possible to obtain information on driving speed, the analysis will be made bearing in mind that the estimated effect of the traffic flow may also be an effect of speed adjustment. Moreover, there are several other factors likely to influence the occurrence of accidents, e.g. weather, road conditions, type of vehicle and drivers' characteristics. In order to take some of these factors into account, daylight accidents are studied separately.
+
+Four road types are distinguished. Road type I with speed limit 70 km/h and road width 6–9.7 m, road type II with speed limit 90 km/h and road width 6–7.9 m, road type III with speed limit 90 or 110 km/h and road width 8–13 m, without separated road lanes, and road type IV, motorways, with speed limit 90 or 110 km/h. The numbers of accidents that occurred on the studied sections are presented in Table 1.
+
+For the road types I, II and III the traffic flow is counted as the number of vehicles per hour driving in both directions. Thus, there is no information on the traffic flow per road lane. This is a problem since depending on the distribution of the traffic flow, different types of accidents may occur. The same situation occurs when studying road type IV. Even though the traffic flow is counted for each direction for this road type, there is no information on the traffic in each lane.
+
+### **4. Accident frequency**
+
+In order to analyse the accident frequency, *Z*, the distribution model of the number of accidents, *Y*, is first discussed. Several authors use a Poisson distribution to approximate the number of accidents occurring during a certain time interval, e.g. [Hall \(1986\)](#page-7-0) and [Nicholson and Wong \(1993\).](#page-7-0) The assumption is based on the fact that although the total number of accidents is reasonably stable over a period, each accident is unpredictable. A reasonable assumption is furthermore that accidents are independent events. However, it may be argued that individual collisions in a multi-collision accident are not mutually independent events. This problem is avoided here, since a traffic accident in this study is defined without consideration being given to the number of vehicles involved. Let *Y*<sup>i</sup> denote the number of accidents occurring on a specific road site during a given time period. If we assume that the number of accidents follows a Poisson distribution with expected value, and thus variance, equal to λi, the probability that the number of accidents during this period will be equal to *y*i, may be written as
+
+$$P(Y_i = y_i) = \frac{\lambda_i^{y_i} e^{-\lambda_i}}{y_i!}$$
+  $y_i = 0, 1, 2, 3, ...$ 
+
+There is, however, no reason to believe that λi is same at all sites, and there will usually be a variation of the λi around a mean, λ. The variation of the *Y*<sup>i</sup> (the number of accidents occurring at different sites) will thus comprise both the variation of *y*<sup>i</sup> around λi and the variation of λi around λ. The variance will therefore tend to be larger than the mean, i.e. we can observe so called over dispersion with respect to the Poisson distribution. The variation in λi is often assumed to be Gamma distributed. Assuming λi to have a Gamma distributed parameter κ, it can be shown that the number of accidents, *Y*, follows a Negative Binomial distribution, e.g. in [Kulmala \(1995\).](#page-7-0) The probability density function for *Y* can now be written as
+
+$$P(Y = y) = \frac{\Gamma(\kappa + y)}{\Gamma(\kappa)y!} \left(\frac{\kappa}{\kappa + \lambda}\right)^{\kappa} \left(\frac{\lambda}{\kappa + \lambda}\right)^{y}$$
+$$y = 0, 1, 2, 3, \dots$$
+
+and variance
+
+$$\sigma^2 = \lambda + \frac{\lambda^2}{\kappa}$$
+
+The Negative Binomial distribution is thus an extension of the Poisson distribution, where the gamma parameter, κ, determines the shape of the distribution. As κ → ∞, the rate of over dispersion declines and the distribution approaches the Poisson.
+
+So far the distribution model of the number of accidents has been discussed. However, when analysing accident frequency, there is also need for a regression model that can describe this phenomenon. An exponential function is a common formulation, since this function ensures that the expected number of accidents is a positive number. Furthermore, measuring the explanatory variables on the logarithmic scale, Gj = ln(Xj), the accident frequency may be written as a multiplicative function for which the value of the exponents can be estimated directly
+
+$$E(Y) = \exp\left[\sum \beta_j \ln(X_j)\right] = \Pi X_j^{\beta_j}$$
+
+The result of a regression model is, however, often dependent on the choice of model function, [Hauer and Bamfo](#page-7-0) [\(1997\). U](#page-7-0)sing an exponential regression model, as described
+
+<span id="page-2-0"></span>Table 2 Statistics on traffic flow data
+
+| Road type | Number of traffic flow intervals |                               | Average traffic flow        |                               |         | Maximum traffic flow        |                               |         |  |
+|-----------|----------------------------------|-------------------------------|-----------------------------|-------------------------------|---------|-----------------------------|-------------------------------|---------|--|
+|           | Homogeneous<br>traffic flow      | Inhomogeneous<br>traffic flow | Homogeneous<br>traffic flow | Inhomogeneous<br>traffic flow |         | Homogeneous<br>traffic flow | Inhomogeneous<br>traffic flow |         |  |
+|           |                                  |                               |                             | Cars                          | Lorries |                             | Cars                          | Lorries |  |
+| I         | 98                               | 408                           | 153                         | 139                           | 12      | 2712                        | 2328                          | 312     |  |
+| II        | 38                               | 159                           | 94                          | 81                            | 8       | 987                         | 938                           | 132     |  |
+| III       | 131                              | 595                           | 250                         | 232                           | 24      | 1687                        | 1645                          | 267     |  |
+| IV        | 121                              | 458                           | 550                         | 498                           | 55      | 3088                        | 2937                          | 326     |  |
+
+Table 3 Statistics on total number of accidents (Tot), single-vehicle (Sing) and multi-vehicle accidents (Mult)
+
+| Road type |                             |      | Number of accidents per traffic flow interval, average | Number of accidents per traffic flow interval, variance |      |                             |     |      |                               |     |      |      |
+|-----------|-----------------------------|------|--------------------------------------------------------|---------------------------------------------------------|------|-----------------------------|-----|------|-------------------------------|-----|------|------|
+|           | Homogeneous traffic<br>flow |      |                                                        | Inhomogeneous<br>traffic flow                           |      | Homogeneous traffic<br>flow |     |      | Inhomogeneous traffic<br>flow |     |      |      |
+|           | Tot                         | Sing | Mult                                                   | Tot                                                     | Sing | Mult                        | Tot | Sing | Mult                          | Tot | Sing | Mult |
+| I         | 0.6                         | 0.3  | 0.3                                                    | 0.1                                                     | 0.1  | 0.1                         | 1.4 | 0.7  | 0.4                           | 0.4 | 0.2  | 0.1  |
+| II        | 2.2                         | 1.4  | 0.8                                                    | 0.5                                                     | 0.3  | 0.2                         | 4.4 | 3.8  | 0.7                           | 1.7 | 0.9  | 0.4  |
+| III       | 1.4                         | 0.8  | 0.5                                                    | 0.3                                                     | 0.2  | 0.1                         | 4.5 | 2.5  | 0.9                           | 0.6 | 0.3  | 0.2  |
+| IV        | 1.5                         | 1.0  | 0.5                                                    | 0.4                                                     | 0.3  | 0.1                         | 3.8 | 2.2  | 0.7                           | 1.0 | 0.5  | 0.2  |
+
+previously, restricts the model to the description of monotonic functions, i.e. the functions may be either increasing or decreasing, not both. The use of this model function is supported by studying the data visually and by analysing the residuals.
+
+### *4.1. Variables and regression models*
+
+In this study, the relationship between the accident frequency and the traffic flow is estimated empirically. In order to apply the result to time periods and road systems of different length, the accident frequency, *Z*, is defined as the expected number of accidents, *E*(*Y*), per hour and kilometre. A parameter, *H*, is also defined, see [Appendix A.](#page-7-0) This parameter describes the exposure, i.e. the number of hours and kilometres that each road type has been studied.
+
+The accident frequency cannot be estimated as a function of the traffic flow directly, since the variable describing the accident frequency is not Poisson or Negative Binomial distributed. The regression model is therefore rewritten as E(Yt) = Z<sup>t</sup> ×H<sup>t</sup> so that the accident frequency may be estimated separately as a function of the traffic flow. The complete regression model for the case of homogeneous traffic flow is estimated as
+
+$$E(Y_1) = \exp[\beta_1 + \beta_2 \ln(q_1) + \beta_3 \ln(H_1)] = [e^{\beta_1} \times (q_1)^{\beta_2}]H_1$$
+ with the restriction,  $\beta_3 = 1$ . When treating the traffic flow as consisting of cars and lorries, the following regression model is estimated
+
+$$\begin{split} E(Y_t) &= \exp[\beta_1 + \beta_2 \ln(q_c)_t + \beta_3 \ln(q_l)_t + \beta_4 \ln(H)_t] \\ &= \left| e^{\beta_1} \times (q_c)_t^{\beta_2} \times (q_l)_t^{\beta_3} \right| H_t \end{split}$$
+
+with the restriction β<sup>4</sup> = 1.
+
+The regression models are estimated applying both the Poisson and the Negative Binomial distribution. The statistical program LIMDEP version 7 is used. In order to evaluate the goodness-of-fit, four indicators are studied; the scaled deviance statistic, the over dispersion parameter, κ, estimated in the Negative Binomial regression model and two test functions for over dispersion.
+
+# **5. Results**
+
+The regression analysis of this study is based on aggregates of hours with similar traffic flows, see [Appendix A.](#page-7-0) Traffic flow statistics per road type are presented in Table 2 together with the number of traffic flow intervals used. In Table 3, statistics on the aggregated number of accidents per traffic flow interval is presented for various accident types1. The average number of accidents per traffic flow interval is generally low when inhomogeneous traffic flows or different accidents types are studied.
+
+The results from the regression analysis generally suggest a good fit for both the Poisson and the Negative Binominal models. Distributional assumptions do not seem to affect the results since there are only small differences. The test statistics that are used for over dispersion occasionally contradict the size of the dispersion parameter. Contradictions tend to arise when the scaled deviance is almost the same for the Poisson and the Negative Binominal regression models, suggesting a good fit for both. Since the estimated coefficients are practically the same, only the results of the
+
+<sup>1</sup> Statistics for daytime traffic flows and accidents are left out since these statistics corresponds largely to the presented statistics.
+
+<span id="page-3-0"></span>
+
+<span id="page-4-0"></span>Table 6Expected number of accidents occurring all day and in daylight, inhomogeneous traffic flowa
+
+|                                                                                                                                  | Ex<br>ted<br>mb<br>of<br>pec<br>nu<br>er                                                                                  | ide<br>nts<br>acc                                                                                                         |                                                                                                                           |                                                                                                                                     | Ex<br>ted<br>mb<br>of<br>ide<br>in<br>day<br>lig<br>ht<br>nts<br>pec<br>nu<br>er<br>acc                                   |                                                                                                                           |                                                                                                                           |                                                                                                                            |  |
+|----------------------------------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------------------------------------|--|
+|                                                                                                                                  | Ro<br>ad<br>e I<br>typ                                                                                                    | Ro<br>ad<br>e I<br>I<br>typ                                                                                               | Ro<br>ad<br>e I<br>II<br>typ                                                                                              | Ro<br>ad<br>e I<br>V<br>typ                                                                                                         | Ro<br>ad<br>e I<br>typ                                                                                                    | Ro<br>ad<br>e I<br>I<br>typ                                                                                               | Ro<br>ad<br>e I<br>II<br>typ                                                                                              | Ro<br>ad<br>e I<br>V<br>typ                                                                                                |  |
+| Co<br>nst<br>ant<br>ln(<br>)<br>qc<br>ln(<br>)<br>ql<br>ln(<br>H)                                                                | b<br>−3<br>1.5<br>4 (<br>−1<br>6.4<br>2)<br>b<br>3.6<br>2 (<br>7.1<br>0)<br>b<br>−2<br>.66<br>(−<br>5.4<br>0)<br>1.0<br>0 | b<br>−2<br>3.4<br>8 (<br>−3<br>3.0<br>3)<br>b<br>1.4<br>3 (<br>7.2<br>8)<br>b<br>−0<br>.94<br>(−<br>5.6<br>9)<br>1.0<br>0 | b<br>−2<br>1.1<br>6 (<br>−5<br>0.5<br>6)<br>b<br>1.0<br>4 (<br>9.9<br>6)<br>b<br>−0<br>.77<br>(−<br>9.0<br>7)<br>1.0<br>0 | b<br>−2<br>2.8<br>4 (<br>−3<br>5.3<br>8)<br>b<br>1.9<br>0 (<br>13.<br>04<br>)<br>b<br>−1<br>.84<br>(−<br>14.<br>44<br>)<br>1.0<br>0 | b<br>−3<br>1.1<br>4 (<br>−1<br>5.6<br>5)<br>b<br>3.6<br>0 (<br>6.9<br>5)<br>b<br>−2<br>.66<br>(−<br>5.4<br>1)<br>1.0<br>0 | b<br>−2<br>4.1<br>8 (<br>−2<br>3.8<br>4)<br>b<br>1.6<br>8 (<br>6.1<br>8)<br>b<br>−1<br>.26<br>(−<br>5.4<br>0)<br>1.0<br>0 | b<br>−2<br>1.4<br>2 (<br>−2<br>9.9<br>7)<br>b<br>1.1<br>6 (<br>7.3<br>6)<br>b<br>−0<br>.92<br>(−<br>7.8<br>5)<br>1.0<br>0 | b<br>−1<br>9.5<br>2 (<br>−2<br>1.6<br>0)<br>b<br>1.3<br>0 (<br>7.0<br>2)<br>b<br>−1<br>.61<br>(−<br>11.<br>39)<br>1.0<br>0 |  |
+| d.f<br>Sc<br>ale<br>d d<br>evi<br>atio<br>n<br>LM<br>-st<br>at<br>t-S<br>isti<br>1<br>tat<br>cs<br>t-S<br>isti<br>2<br>tat<br>cs | 40<br>4<br>56<br>.31<br>61<br>c<br>.97<br>0.6<br>0<br>3.6<br>9c                                                           | 15<br>5<br>35<br>.44<br>17.<br>52c<br>−1<br>.21<br>0.0<br>3                                                               | 59<br>1<br>16<br>0.2<br>3<br>12.<br>55c<br>1.4<br>6<br>0.9<br>4                                                           | 45<br>4<br>15<br>3.9<br>7<br>14.<br>40<br>c<br>−0<br>.39<br>0.6<br>3                                                                | 39<br>6<br>55<br>.49<br>54<br>c<br>.67<br>0.7<br>4<br>4.2<br>8c                                                           | 15<br>4<br>26<br>.54<br>7.4<br>6<br>−2<br>.11<br>c<br>−3<br>.26<br>c                                                      | 57<br>1<br>11<br>3.8<br>8<br>5.5<br>4<br>−2<br>.11<br>c<br>−2<br>.43<br>c                                                 | 45<br>4<br>17<br>8.1<br>9<br>19<br>.70<br>c<br>1.2<br>5<br>1.7<br>9                                                        |  |
+
+Values in parenthesis are *<sup>t</sup>*-statistics.
+
+Table 7Expected number of single and multi-vehicle accidents, inhomogeneous traffic flowa
+
+|                                      | Ex<br>ted<br>mb<br>of<br>pec<br>nu<br>er | sin<br>le-<br>veh<br>icl<br>cci<br>den<br>ts<br>g<br>e a |                                          |                                          | Ex<br>ted<br>mb<br>of<br>lti-<br>veh<br>icl<br>cci<br>den<br>ts<br>pec<br>nu<br>er<br>mu<br>e a |                                          |                                          |                                          |  |
+|--------------------------------------|------------------------------------------|----------------------------------------------------------|------------------------------------------|------------------------------------------|-------------------------------------------------------------------------------------------------|------------------------------------------|------------------------------------------|------------------------------------------|--|
+|                                      | Ro<br>ad<br>e I<br>typ                   | Ro<br>ad<br>e I<br>I<br>typ                              | Ro<br>ad<br>e I<br>II<br>typ             | Ro<br>ad<br>e I<br>V<br>typ              | Ro<br>ad<br>e I<br>typ                                                                          | Ro<br>ad<br>e I<br>I<br>typ              | Ro<br>ad<br>e I<br>II<br>typ             | Ro<br>ad<br>e I<br>V<br>typ              |  |
+| Co<br>nst<br>ant                     | b<br>−2<br>9.0<br>0 (<br>−1<br>2.5<br>7) | b<br>−2<br>2.8<br>5 (<br>−2<br>7.2<br>3)                 | b<br>−2<br>0.3<br>6 (<br>−4<br>2.9<br>9) | b<br>−2<br>2.0<br>8 (<br>−3<br>0.0<br>4) | b<br>−3<br>7.4<br>1 (<br>−1<br>0.8<br>0)                                                        | b<br>−2<br>6.5<br>8 (<br>−2<br>0.1<br>8) | b<br>−2<br>4.9<br>9 (<br>−2<br>9.0<br>2) | b<br>−2<br>6.8<br>9 (<br>−2<br>0.4<br>7) |  |
+| ln(<br>)<br>qc                       | b<br>2.9<br>0 (<br>4.6<br>5)             | b<br>1.1<br>4 (<br>4.8<br>2)                             | b<br>0.7<br>6 (<br>6.1<br>0)             | b<br>1.7<br>0 (<br>9.9<br>2)             | b<br>4.6<br>9 (<br>5.2<br>3)                                                                    | b<br>1.9<br>6 (<br>5.5<br>7)             | b<br>1.6<br>1 (<br>8.3<br>8)             | b<br>2.4<br>0 (<br>8.6<br>4)             |  |
+| ln(<br>)<br>ql                       | b<br>−2<br>.29<br>(−<br>3.8<br>3)        | b<br>−0<br>.85<br>(−<br>4.3<br>4)                        | b<br>−0<br>.73<br>(−<br>7.0<br>9)        | b<br>−1<br>.82<br>(−<br>11.<br>91<br>)   | b<br>−3<br>.15<br>(−<br>3.6<br>7)                                                               | b<br>−1<br>.11<br>(−<br>3.6<br>8)        | b<br>−0<br>.83<br>(−<br>5.6<br>9)        | b<br>−1<br>.90<br>(−<br>8.2<br>3)        |  |
+| ln(<br>H)                            | 1.0<br>0                                 | 1.0<br>0                                                 | 1.0<br>0                                 | 1.0<br>0                                 | 1.0<br>0                                                                                        | 1.0<br>0                                 | 1.0<br>0                                 | 1.0<br>0                                 |  |
+| d.f                                  | 40<br>4                                  | 15<br>5                                                  | 59<br>1                                  | 45<br>4                                  | 40<br>4                                                                                         | 15<br>5                                  | 59<br>1                                  | 45<br>4                                  |  |
+| Sc<br>ale<br>d d<br>evi<br>atio<br>n | 59<br>.34                                | 47<br>.30                                                | 22<br>8.5<br>5                           | 17<br>4.2<br>7                           | 78<br>.34                                                                                       | 49<br>.39                                | 19<br>2.3<br>0                           | 19<br>0.3<br>1                           |  |
+| LM<br>-st<br>at                      | c<br>13<br>.54                           | 95c<br>13.                                               | 8.9<br>3                                 | c<br>11<br>.15                           | 4c<br>9.9                                                                                       | 7.1<br>9                                 | c<br>13<br>.67                           | c<br>56<br>.60                           |  |
+| t-S<br>isti<br>1<br>tat<br>cs        | 0.0<br>2                                 | −0<br>.03                                                | 0.0<br>3                                 | −0<br>.04                                | −0<br>.01                                                                                       | −0<br>.05                                | 0.0<br>4                                 | 0.0<br>2                                 |  |
+| t-S<br>isti<br>2<br>tat<br>cs        | 0c<br>0.3                                | 0.0<br>2                                                 | 0c<br>0.2                                | −0<br>.02                                | 0.1<br>2                                                                                        | c<br>−0<br>.22                           | −0<br>.09                                | 0.0<br>8                                 |  |
+
+Values in parenthesis are *<sup>t</sup>*-statistics.
+
+Significantly different from zero at the 5% level.
+
+The null-hypothesis of over dispersion is significantly rejected at the 5% level.
+
+Significantly different from zero at the 5% level.
+
+The null-hypothesis of over dispersion is significantly rejected at the 5% level.
+
+<span id="page-5-0"></span>![](_page_5_Figure_2.jpeg)
+
+Fig. 1–4. Number of accidents per hour and kilometre (×1E7) at averge number of cars per hour +/−20%. *Remark*: The number of dots does not correspond to the number of accidents since more than one accident have occurred at some traffic flows. The total number of accidents in the figure of road type I–IV is 8, 18, 15 and 37.
+
+![](_page_5_Figure_4.jpeg)
+
+Fig. 5–8. Number of accidents per hour and kilometre (×1E7) at averge number of lorries per hour +/−20%. *Remark*: The total number of accidents in the figure of road type I–IV is 16, 10, 21 and 14.
+
+Poisson regression model will be presented. A complete report of the results is to be found in [Winslott \(1998\).](#page-7-0)
+
+### *5.1. Homogeneous traffic flow*
+
+The results of the regression analysis are presented in [Table 4](#page-3-0) for accidents occurring all day and in daylight only. All estimated parameter values are significantly different from zero. Furthermore, for all road types, except for road type I, the estimated exponent is significantly different from 1. Thus, we may reject the hypothesis that the expected accident frequency increases in proportion with the traffic flow for these road types, i.e. that the accident rate is constant. Instead, an additional vehicle lowers the accident rate and increases the traffic safety.
+
+For the road types I, II and III the estimated parameters are lower than when studying accidents that have occurred throughout the day. The difference is, however, not significant. The result for road type IV suggests that the model of the expected number of accidents occurring in daylight is not a correct choice. The estimated value of the exponent should be interpreted with caution.<sup>2</sup>
+
+The results for single and multi-vehicle accidents are presented in [Table 5.](#page-3-0) The results differ in some respects from the case when all accident types are studied together. The estimated values of the exponent of the traffic flow are significantly larger for multi-vehicle accidents than single-vehicle accidents. For multi-vehicle accidents, we cannot reject the hypothesis that the expected number of multi-vehicle accidents increases in proportion with the traffic flow. However,
+
+<sup>2</sup> The chosen regression model may only describe a monotonic function, and when the graph of the number of daytime accidents versus the number of vehicles per hour is studied for road type IV, there is no sign of a monotonic relationship, either increasing or decreasing.
+
+for single-vehicle accidents, the estimated exponents of the traffic flow are significantly less than one for all road types.
+
+### *5.2. Inhomogeneous traffic flow*
+
+[Table 6](#page-4-0) shows the results for all accidents and accidents occurring in daylight only when separating the traffic flow for cars and lorries. The estimated coefficients are significantly different from zero for all road types. For road type I and IV the exponents are significantly different from 1, indicating that the expected number of accidents increases more than proportionally with the number of cars per hour, i.e. that the accident rate increases. However, for the road types II and III the null-hypothesis cannot be rejected. The estimated exponent for the flow of lorries is generally negative and different from zero. This suggests that an increasing number of lorries per hour will lower the expected number of accidents independently of the flow of cars. When studying accidents in daylight, the results are not significantly different.
+
+The estimates when studying single and multi-vehicle accidents are presented in [Table 7.](#page-4-0) The value of the exponent for the number of cars per hour is generally different from 1 when both single and multi-vehicle accidents are studied. Accordingly, the expected number of accidents increases more than proportionally for both single and multi-vehicle accidents. However, the exponent for cars is significantly lower for single-vehicle accidents than for multi-vehicle accidents. The exponent for the number of lorries per hour is also negative here. The value of the exponent is generally, however not significantly, lower when multi-vehicle accidents are studied compared to when all accidents are studied together.
+
+The difference between the studied road types is also tested. The results for the road types I and IV are significantly higher and lower than the results for the road type II and III, respectively. This corresponds to the results when all accidents are studied together.
+
+The result using an inhomogeneous traffic flow differs greatly to that of homogeneous traffic flow. Since the estimates raise certain questions the data is also studied visually. [Figs. 1–4](#page-5-0) present the number of accidents per hour and kilometre in relation to the flow of lorries per hour. [Figs. 5–8](#page-5-0) present the number of accidents per hour and kilometre in relation to the flow of cars per hour. The figures are drawn for the average number of cars ([Figs. 1–4\) a](#page-5-0)nd lorries ([Figs. 5–8\)](#page-5-0) per hour for each road type±20%.<sup>3</sup> Generally, there are few accidents in the data set and when restricting the analysis to certain traffic flows, the number of accidents is reduced even further. In spite of a limited number of observations, the plots can be argued to indicate the type of relation between the accident frequency and the flow of cars and lorries.
+
+Data presented in [Figs. 1–4](#page-5-0) indicates that the number of accidents is constant or decreasing with an increasing number of lorries on the road, whereas in [Figs. 5–8](#page-5-0) the number of accidents is increasing with an increasing number of cars. Consequently, the visual analysis is not in conflict with the estimates received in the regression analysis.
+
+### **6. Discussion**
+
+The estimated relationship between the expected number of accidents per hour and kilometre, and the traffic flow differs considerably depending on whether different types of traffic modes are considered or not, i.e. if a homogeneous or a inhomogeneous traffic flow analysis is carried out. Differences appear both in relation to accidents occurring throughout the day, accidents occurring in daylight, singleand multi-vehicle accidents. In the homogeneous traffic flow analysis regarding accidents that have occurred throughout the day, we reject the null-hypothesis that the expected number of accidents increases proportionally with the traffic flow for the road types II, III and IV. The value of the exponent is less than 1. This result is in line with a number of studies reviewed in [Satterthwaite \(1981\),](#page-7-0) [Ardekani et al. \(1997\)](#page-7-0) and [Hauer and Bamfo \(1997\).](#page-7-0)
+
+As cars constitute the main part of the traffic flow, one may expect the outcome, with respect to the number of cars per hour, to be similar to that of the homogeneous traffic flow analysis. However, here the suggestion is that the expected number of accidents increases in proportion or more, to the number of cars per hour. Furthermore, studying the effect of an increasing number of lorries one might expect that the presence of more lorries on the road would increase the number of accidents since the incidence of possibly dangerous overtaking manoeuvres increases. The result of this study indicates, however, that at a given number of cars per hour, the expected number of accidents will decrease with increasing number of lorries per hour. This safety effect of lorries may be regarded as counter intuitive at first glance. It is, however, possible that the number of accidents decreases as an effect of a speed reduction as the speed limit for lorries is lower than for cars. An increasing number of lorries per hour will accordingly slow down the average speed. The result with respect to the flow of lorries may also be a result of people's unease when sharing the road-space with a lorry causing the attention to increase. There is, furthermore, a possibility that the number of lorries per hour is correlated with an unknown factor that lowers the number of accidents. Hours with many lorries may coincide with hours with good road conditions and hours when experienced drivers are driving.
+
+It is obvious that there are shortcomings in analysis indicating that some of the results in this paper can be questioned. For instance, the model for inhomogeneous traffic flow analysis may be considered as a better model in relation to differences between traffic modes. As this model can be assumed to have more explanatory power, no over dispersion is expected in the Poisson regression model. However, in the inhomogeneous traffic flow model, over dispersion becomes
+
+<sup>3</sup> The same pattern is given irrespective of the flow of cars and lorries used.
+
+<span id="page-7-0"></span>evident, while in the homogeneous traffic flow model, when the traffic flow is treated without consideration to different types of traffic modes, there is generally no over dispersion.
+
+In addition, the accident sample is rather small even though data on accidents for 6.5 years is used. According to [Table 3, t](#page-2-0)he average numbers of accidents per traffic flow interval is generally less than 1, indicating low statistical power in especially the analysis of inhomogeneous traffic flows. Random fluctuations may then be a source of error. The maximum hourly flow of lorries is furthermore a fraction of that of cars which may lead to unstable results.<sup>4</sup>
+
+Consequently, the marginal effect of the flow of different vehicle types may have been miscalculated and exaggerated in the regression analysis due to shortcomings in the data set and in the regression models. However, visual analysis of the inhomogeneous traffic flow data, presented in [Figs. 1–8,](#page-5-0) indicates that the flow of lorries affects the accident frequency differently than the flow of cars. The result implies that important information is lost if no consideration is taken to vehicle type. Literature of today shows limited interest in studying traffic safety models distinguishing between different vehicle types, though. This is a shortcoming since the characteristics of our vehicles and their driving pattern differ in a number of respects, undoubtedly affecting the accident frequency in various ways. Other model structures ought to be applied in future research including, for instance, speed as an explanatory variable.
+
+### **Acknowledgements**
+
+Finical support from the Swedish National Road Administration is gratefully acknowledged. I am grateful for insightful suggestions from Risto Kulmala, and Karin Brundell-Freij. Any errors are my sole responsibility.
+
+### **Appendix A**
+
+Traffic flow data from the stationary counters are applied to calculate traffic flow frequencies for each road section, i.e. set of hours of different traffic volumes. An interval level of 25 vehicles per hour is used when treating the traffic flow as homogeneous. When treating the traffic flow as inhomogeneous, an interval level of 25 cars per hour is used together with an interval level of five lorries per hour. The traffic flow frequency for interval *t* is denoted by τt. For each road type the number of kilometres, η, is calculated for each section, i.e. ηr kilometres for road type r = I, II, III and IV. The number of kilometres is then multiplied by the traffic flow frequencies for the studied road section. The analysis is based on four road types and each road type is represented on several road sections. If *s* is the number of sections of road type *r*, the exposure, *H*, is calculated as the total number of hours and kilometres studied.
+
+$$H_{r,t} = \sum_{s=1}^{n} \eta_{r,s} \times \tau_{t,s}$$
+  
+for  $t = 1, \dots, p$  and  $r = I, \dots, IV$ 
+
+The traffic flow variables are calculated as the average number of vehicles, *q*, and the average number of cars and lorries per hour, *q*<sup>c</sup> and *q*l, for each traffic flow interval.
+
+The number of accidents throughout the day and in daylight is calculated by aggregation using the same traffic flow interval as when calculating the traffic flow frequencies. These variables are used as dependent variables. When calculating the exposure for daytime traffic, frequencies of traffic flows observed between 6 a.m. and 6 p.m. are used.
+
+## **References**
+
+Ardekani, S., Hauer, E., Jamei, B., 1997. In: Gartner, N. et al. (Eds.), Traffic Impact Models in Traffic Flow Theory—A State-of-the-Art Report, Oak Ridge National Laboratory.
+
+Hall, R.D.,1986. Accidents at Four-arm Single Carriageway Urban Traffic Signals. Transport and Road Research Laboratory Contractor Report, 65.
+
+Hauer, E., Bamfo, J., 1997. In: Proceedings of the ICTCT conference on Two Tools for Finding what Function Links the Dependent Variable to the Explanatory Variables, Lund.
+
+Jovanis, P.P., Chang, H.L., 1986. Modelling the relationship of accidents to miles travelled. Transportation Res. Rec. 1068, 42–51.
+
+Kulmala, R., 1995. Safety at Rural Three- and Four-arm Junctions, VTT-Publications, 233.
+
+Nicholson, A., Wong, Y.D., 1993. Are accidents Poisson distributed? a statistical test. Accid. Anal. Prev. 1, 91–97.
+
+Satterthwaite, P.S., 1981. A Survey of Research into Relationship Between Traffic Accidents and Traffic Volumes, Transport and Road Research Laboratory Contractor Report, 692.
+
+Vickery, W.S., 1968. Automobile Accidents, Tort Law, Externalities and Insurance. Journal of Law and Contemporary problems 33, 464–484. Vitaliano, D.F., Held, J., 1991. Road Accident External Effects: an
+
+Empirical Assessment. Applied Economics 23, 373–378. Winslott, L., 1998. The external costs of traffic accidents: an empirical analysis of the traffic flow. Department of Economics Working Paper Series, 1.
+
+<sup>4</sup> In this type of data one could also expect a correlation between the flow of cars and lorries, leading to spurious effects in the regression analysis. No correlation can be detected, however. Even though high flows of lorries are more frequently found at high flows of cars there are also hours in the data set with high flows of lorries and low flows of cars.

Zotero/001_artiklid/Road Safety Indicators/huangeffectsdaylightsavingtime2010.md

@@ -0,0 +1,353 @@
+---
+category: literaturenote
+citekey: huangeffectsdaylightsavingtime2010
+title: The effects of daylight saving time on vehicle crashes in Minnesota
+authors: "Huang, Arthur; Levinson, David"
+year: 2010
+date: 2010-00-00 2010
+doi: 10.1016/j.jsr.2010.10.006
+publication: Journal of Safety Research
+url: "https://doi.org/10.1016/j.jsr.2010.10.006"
+zotero_key: BJ9EXQDF
+zotero_storage: 8NGEZ4AN
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Huang, Arthur"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+# Journal of Safety Research
+
+j o u r n a l h om e p a g e : www. e l s ev i e r. c om / l o c a t e / j s r
+
+![](_page_0_Picture_5.jpeg)
+
+# The effects of daylight saving time on vehicle crashes in Minnesota
+
+Arthur Huang a, ⁎, David Levinson <sup>b</sup>
+
+- <sup>a</sup> University of Minnesota, Department of Civil Engineering, 500 Pillsbury Drive SE, Minneapolis, MN 55455 USA
+- <sup>b</sup> RP Braun-CTS Chair of Transportation Engineering, University of Minnesota, Department of Civil Engineering, USA
+
+## article info abstract
+
+Available online 25 November 2010
+
+Keywords: Daylight saving time Vehicle crashes Exposure Traffic volume
+
+Introduction: Daylight saving time (DST), implemented as an energy saving policy, impacts many other aspects of life; one is road safety. Based on vehicle crash data in Minnesota from 2001 to 2007, this paper evaluates long- and short-term effects of DST on daily vehicle crashes. Method: To provide evidence to explain the causes of more/fewer crashes in DST, we examine the impact of DST on crashes in four periods of a day: 3 a.m.-9 a.m., 9 a.m.-3 p.m., 3 p.m.-9 p.m., 9 p.m.-midnight. The effects of risk and exposure to traffic are also separated. Our statistical models not only include weather conditions and dummy variables for days in DST as independent variables, but also consider traffic volumes on major roads in different periods of a day. Our major finding is that the short-term effect of DST on crashes on the morning of the first DST is not statistically significant. Moreover, it is interesting to notice that while DST per se is associated with fewer crashes during dusk, this is in part offset because it is also associated with more traffic on roads (and hence more crashes). Our path analysis shows that overall DST reduces crashes. Impact on industry: Daylight saving time can lead to fewer crashes on roads by providing better visibility for drivers.
+
+© 2010 National Safety Council and Elsevier Ltd. All rights reserved.
+
+## 1. Introduction
+
+Daylight saving time (DST) in the United States dates to World War I as an energy saving measure, and in its current incarnation was officially enacted by Congress in 1974. DST aims to match activities with hours of daylight by adjusting the standard time (ST) forward or backward one hour at different times of year. This keeps sunset and sunrise one hour later in spring, summer, and early fall, so that people have one more evening hour of daylight (and therefore one less morning hour of daylight) in DST [\(Coate & Markowitz, 2003\)](#page-7-0). In addition to its claimed feature of energy conservation, DST may impact many other aspects of daily life, one of which is road safety. Therefore, quantitatively examining the effects of DST on road safety is of interest.
+
+The relationship between DST and road injuries or fatalities has been a topic of extensive research. [Sood and Ghosh \(2007\)](#page-7-0) indicated that there were generally two schools in studying the effects of DST on road safety, which respectively concentrated on long-term effects and shortterm effects. Here short-term effects indicate the impact of DST on vehicle crashes in a few days after time changes, while long-term effects refer to its impact of a longer time period, say, a few months or year round.
+
+URL: <http://nexus.umn.edu> (D. Levinson).
+
+To be specific, one of the schools claimed that DST in the long run decreases vehicle crashes due to better visibility in the evenings, reducing the likelihood of crashing in darkness. [Table 1](#page-1-0) summarizes previous research. These studies argued that if driving in darkness contributes to more traffic crashes, the delayed sunset time can lead to fewer crashes.
+
+The other school contended that time change to DST resulted in more vehicle crashes in the short run. Some representative studies are reviewed in [Table 2](#page-1-0), where many research reported an increase of general/fatal crashes after time change in spring. The main hypothesis was that time change in spring deprived people of one-hour sleep, which, in the short run, could induce drivers' sleepiness or fatigue while driving. Some clinical research performed controlled experiments to examine the relationship between hours of sleep and drivers' response time and vigilance; their main conclusion was that one-hour less sleep can boost the rise of traffic crashes. In addition, some other studies found that time change to DST was associated with an increase of fatal vehicle crashes, and attributed it to possible drivers' alcohol-drinking or late-night driving behavior out of one extra hour during dusk.
+
+In contrast, other research found that time change did not have a statistically significant impact on vehicle crashes. For example, [Lambe](#page-7-0) [and Cummings \(2000\)](#page-7-0), based on the Swedish vehicle crash data from 1984 to 1995, concluded that the shifts to and from DST had no effect on vehicle crashes. [Vincent \(1998\)](#page-7-0) also drew a similar conclusion using the crash data from Canada.
+
+While different data may produce different results, the effects of DST do not seem to have been sufficiently investigated. First of all,
+
+<sup>⁎</sup> Corresponding author.
+
+E-mail addresses: [huang284@umn.edu](mailto:huang284@umn.edu) (A. Huang), [dlevinson@umn.edu](mailto:dlevinson@umn.edu) (D. Levinson).
+
+<span id="page-1-0"></span>Table 1 Selected previous studies supporting positive effects of daylight saving time on road safety.
+
+| Study                                 | Data set                                                           | Method                             | Results                                                                                                                                                                                          |
+|---------------------------------------|--------------------------------------------------------------------|------------------------------------|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
+| Meyerhoff (1978)                      | US fatal vehicle crash<br>data (1973-1974)                         | Fourier analysis                   | Several weeks after time change from ST to DST<br>witnessed a net reduction of about 0.7% in fatal<br>vehicle crashes; little effect of DST was found on<br>fatal crashes in the winter.         |
+| Ferguson (1995)                       | US fatal vehicle crash<br>data (1987-1991)                         | Generalized linear models<br>(GLM) | Year-round DST from 1987 to 1991 would have<br>helped reduce 901 fatal crashes (727 involving<br>pedestrians, 174 involving vehicle occupants).                                                  |
+| Broughton and Sedmand (1989)          | UK accident data (1968-1971)                                       | GLM                                | Year-round DST in 1987 would have helped reduce<br>159 fatal crashes, 654 seriously injured crashes<br>and 1247 overall injured crashes.                                                         |
+| Hillman (1989)                        | Survey data in Britain                                             | GLM                                | Transferring one hour of daylight from the early<br>morning to daytime may save more lives on road,<br>especially for rural areas.                                                               |
+| Ward et al. (1994)                    | Survey data in Northampton, England                                | GLM                                | 237 pedestrians were injured per 100 million road<br>in darkness, compared with 52 pedestrians in daylight.<br>This implied that better light condition at night could<br>lead to fewer crashes. |
+| Adams (2005)                          | UK crash data (1968–1971)                                          | GLM                                | Year-round DST was associated with a 6.9%<br>decrease in serious or fatal road injuries of children.                                                                                             |
+| Coate and Markowitz (2003)            | US fatal vehicle crash data (1998–1999)                            | GLM                                | Year-round DST would reduce pedestrian fatalities in<br>morning peak hours and afternoon peak hours by 13%;<br>motor vehicle occupant fatalities would downsize by 3%.                           |
+| Broughton and Stone (1998)            | UK vehicle crash data (1969 to 1994)                               | GLM                                | Year-round DST would reduce 104 to 138 fatal crashes<br>per year.                                                                                                                                |
+| Broughton, Hazelton, and Stone (1999) | UK vehicle crash data (1969–1973<br>and 1985–1994), USA (19911995) | Statistical models                 | Year-round DST from 1987 to 1991 would have helped<br>reduce 833 fatal crashes involving pedestrians, and 140<br>fatal crashes involving vehicle occupants .                                     |
+| Stevens and Lord (2006)               | Fatal and nonfatal jnjury<br>Crashes in Texas                      | GLM                                | Time change to DST can improve traffic safety due to<br>better light condition.                                                                                                                  |
+
+most studies only looked at crashes for the whole day, which, while useful in comparing crashes in DST and ST on a daily basis, cannot provide insight about the effects of DST on different periods of a day. Therefore, to test the hypotheses about the reasons for more crashes (such as sleepiness or late-night driving) calls for studying crashes in different time periods of a day where the causes may exist. Second, previous research has not addressed the issue of exposure. In addition to a direct influence on crashes, DST may change traffic flow patterns near dawn and dusk, which can further impact crashes.
+
+This paper, based on the crash data in Minnesota (MN) from 2001 to 2007, endeavors to fill this research niche by considering vehicles' exposure to traffic. The research question is whether the policy of DST can impact vehicle crashes in MN. Over the long-term, we hypothesize that the impact of DST on crashes includes two parts: exposure and risk of crashes. First, in terms of exposure, we hypothesize that DST may reduce traffic during dawn yet increase traffic during dusk. Second, DST may lower the risk of crashes during dusk due to one more hour of visibility for drivers. In the short term, DST, per se, may be associated with more crashes on morning of the first day of DST because of drivers' tiredness/sleepiness after losing one hour's sleep.
+
+The rest of the paper is organized as follows. Section 2 describes the background of daylight saving time and the dates of time change in the United States. [Section 3](#page-2-0) shows the results of exploratory analysis on crashes. [Section 4](#page-2-0) introduces the data sets used in this research. In [Section 5,](#page-4-0) statistical models are constructed to assess the effects of DST on crashes in MN; two-stage least squared models are built to look at the effects of DST on traffic as well as on crashes. The results and analysis are depicted in [Section 6](#page-5-0). Lastly, [Section 7](#page-7-0) concludes the paper.
+
+## 2. Background of daylight saving time
+
+The idea of daylight saving time is to transfer an hour of morning daylight to the evening. It is achieved by setting clocks ahead one hour in spring and turning the clocks back one hour in fall. To illustrate, [Fig. 1](#page-2-0) plots the times of sunrise and sunset (with DST adjustment as separate lines) in Minneapolis/St. Paul, MN, USA for 2007. Note that not all places in the United States observe daylight time; in particular, Hawaii and most of Arizona do not use it. In 2006 and earlier years, DST begins at 2 a.m. local time on the first Sunday in April, and ends at
+
+Table 2 Selected previous studies supporting negative effects of daylight saving time on road safety.
+
+| Study                               | Data set                                 | Method                 | Results                                                                                                                                               |
+|-------------------------------------|------------------------------------------|------------------------|-------------------------------------------------------------------------------------------------------------------------------------------------------|
+| Ledger (1994)                       | US vehicle crashes 1998                  | Human capital approach | Sleepiness and sleep disorders are major<br>causes of traffic crashes.                                                                                |
+| Coren (1996a)                       | US fatal crash data (3 years)            | GLM                    | Sleep loss in spring can increase the likelihood<br>of fatal crashes on road.                                                                         |
+| Coren (1996b)                       | Canada traffic crash data (1991 to 1992) | GLM                    | Deprivation of one-hour sleep can boost the risk<br>of traffic crashes.                                                                               |
+| Hicks, Lindseth, and Hawkins (1983) | US fatal vehicle crashes                 | Qualitative analysis   | Crashes increased significantly in the first seven<br>days after the time change both in spring and in fall.                                          |
+| Hicks, Davis, and Hicks (1998)      | Fatal crashes in New Mexico (1989-1992)  | GLM                    | Percentage of alcohol-related fatal crashes increased<br>significantly in the first seven days after the time change.                                 |
+| Varughese and Allen (2001)          | US fatal vehicle crashes (21 years)      | GLM                    | Crashes on Sundays in fall rised significantly after shifting<br>from DST to ST, which may be highly associated with drivers'<br>alcohol consumption. |
+
+<span id="page-2-0"></span>![](_page_2_Figure_2.jpeg)
+
+Fig. 1. Times of sunrise and sunset (with DST adjustment as separate lines) in Minneapolis/St. Paul, MN, USA for 2007 (plotted by Daylight Chart 3.0 ([Fatehi, 2009\)](#page-7-0)).
+
+Table 3 Beginning and ending dates of daylight saving time in the US (Hawaii and Arizona excluded).
+
+| Year | DST Begins | DST ends |  |
+|------|------------|----------|--|
+| 2000 | 4/2        | 10/29    |  |
+| 2001 | 4/1        | 10/28    |  |
+| 2002 | 4/7        | 10/27    |  |
+| 2003 | 4/6        | 10/26    |  |
+| 2004 | 4/4        | 10/31    |  |
+| 2005 | 3/3        | 10/30    |  |
+| 2006 | 3/2        | 10/29    |  |
+| 2007 | 3/11       | 11/4     |  |
+| 2008 | 3/9        | 11/2     |  |
+| 2009 | 3/8        | 11/1     |  |
+
+2 a.m. local time on the last Sunday in October. Starting in 2007, DST begins on the second Sunday in March, and ends on the first Sunday in November. The dates of time change from 2000 to 2009 are shown in Table 3.
+
+Before we perform statistical analysis, it may be important to first obtain an intuitive sense of the trend of crashes as time changes (from ST to DST or from DST to ST).
+
+## 3. Exploratory analysis
+
+The exploratory analysis involves two parts. First, based on historical crash data in Minnesota (2001–2007), this study plots weekly crashes during the 16 weeks crossing the time change in spring (from ST to DST) and in fall from (DST to ST), categorized by four parts of day (hereafter dayparts) (3 a.m.-9 a.m., 9 a.m.-3 p.m., 3 p.m.-9 p.m., 9 p.m.-midnight).1 [Fig. 2](#page-3-0) shows the crashes during the transition period in spring. The first week in DST in all four sections has fewer crashes than its previous week, although afterwards crashes rise again. However, the average weekly crashes in the first eight weeks in DST are lower than the last eight weeks in ST. [Fig. 3](#page-3-0) sketches weekly crashes during the transition period in fall. During 3 p.m. to 9 p.m., the first week in ST suffers from more crashes than the last week in DST, while the opposite holds for 3 a.m. to 9 p.m. For other time periods, the difference is not easy to be discerned. It is interesting to notice that in the 5th week in ST (around the first week in November), there is a surge of crashes for most periods of a day. This is around the time when snow starts to fall in Minnesota, increasing the probability of crashes.
+
+Second, to inspect the short-term effect of DST, we single out the first day (Sunday) of time change and compare it with other Sundays. [Fig. 4](#page-4-0) displays the crashes on Sundays during the transition period in spring. It can be found that crashes decline sharply in the 3rd week before DST, and level off afterwards. For 9 a.m.-3 p.m. the first week in DST has slightly fewer crashes than the last week in ST; yet during other periods more crashes happen in the first week of DST. [Fig. 5](#page-4-0) further shows the Sunday crashes during the transition period in fall. The first Sunday in ST seems to be afflicted with more crashes than its previous week in all four periods; the difference of crashes, nevertheless, are generally less than 5%.
+
+Overall, based on the above analysis, it is hard to pinpoint the effect of time change. For example, as shown in [Figs. 2 and 4](#page-3-0), the first day and first week of DST have fewer crashes than their previous ones, but a longer effect of time change cannot be clearly eyeballed. In addition, we have not yet controlled for other crash-related factors such as weather and season. Further statistical analysis is thus needed.
+
+One frequently asked question is: how much more dangerous, if any, is travel in daylight saving time? This question includes two aspects. First, what natural conditions in DST, different from those in ST, lead to more crashes? Second, what changes in the amount of driving happen? The next two sections will introduce the variables and models used to answer this question.
+
+### 4. Data
+
+To answer the first aspect of the question, the data we use are the crash data of Minnesota, USA from year 2001 and 2007. The crash data set, complied by the Minnesota Office of Traffic Safety (OTS), documents all reported vehicle-related crash data for Minnesota. The complied crash data have one record for each crash and contain overall information about the crash, such as date and time, location, whether property besides vehicles was damaged, and type of roadway.
+
+<sup>1</sup> While our major interests are in the dawn and dusk periods, we also would like to explore if there is any ripple effect for other dayparts.
+
+<span id="page-3-0"></span>![](_page_3_Figure_2.jpeg)
+
+Fig. 2. Weekly vehicle crashes from eight weeks before DST through the first eight weeks in DST in MN (2001-2007), categorized by four periods of day (3 am-9 am, 9 am-3 pm, 3 pm-9 pm, 9 pm-midnight).
+
+![](_page_3_Figure_4.jpeg)
+
+Fig. 3. Weekly vehicle crashes from eight weeks before ST through the first eight weeks in ST in MN (2001-2007), categorized by four periods of day (3 am-9 am, 9 am-3 pm, 3 pm-9 pm, 9 pm-midnight).
+
+The second aspect of the question concerns the issue of exposure to traffic. It would be ideal to use vehicle travel by hour of day, but this information is not available statewide. But we can obtain hourly traffic volume data on some major roads, which are counted by Automatic Traffic Recorder Stations (ATR) located on the state's interstates, trunk highways, county state aid highways, and municipal state aid streets. The ATR data are maintained by the MN Traffic Forecast and Analysis Section of the Office of Transportation data and Analysis.2 In this paper total traffic volumes from all ATR stations in different dayparts are used to indicate changes in the amount of travel statewide.3 The basic statistics of crashes and ATR traffic volumes in different periods of a day from 2001 to 2007 are shown in [Table 4.](#page-4-0)
+
+As noted above, we categorize crashes by four dayparts. The dependent variables are statewide daily vehicle crashes given a period of day from 2001 to 2007. Regarding dependent variables, we are mainly interested in dummy variables for the first Sunday of DST, first Monday of DST, first Sunday of ST, first Monday of ST, and the rest of the days of DST (the base case being the rest of the days in ST). Other independent variables include: traffic volumes (from statewide ATR stations); weather indicators such as rain and snow; dummy variables for year, dummy variables for month; day length <sup>4</sup> ; monthly gas price statewide. It should be noted that dummy variables for month and day length (sunlight hours per day) are included to separate the seasonal effects from DST. To elaborate, monthly dummy variables indicate
+
+<sup>2</sup> The data can be downloaded from [http://www.dot.state.mn.us/traf](http://www.dot.state.mn.us/traffic/data/atr/atr.html)fic/data/atr/atr.
+
+<sup>3</sup> It is assumed that the fluctuation of traffic volumes in different dayparts from the ATR data set is representative of the general trend of traffic flows statewide.
+
+<sup>4</sup> Day length is calculated as the period between the rising and setting of the sun. The statewide sunrise/sunset time is averaged over sunrise/sunset times of all counties in Minnesota. The data set was downloaded from the website of Astronomical Applications Department of the U.S. Naval Observatory: [http://aa.usno.navy.mil/data/](http://aa.usno.navy.mil/data/docs/RS-OneYear.php) [docs/RS-OneYear.php.](http://aa.usno.navy.mil/data/docs/RS-OneYear.php)
+
+<span id="page-4-0"></span>![](_page_4_Figure_2.jpeg)
+
+Fig. 4. Total vehicle crashes on Monday from eight weeks before DST through the first eight weeks in DST in MN (2001-2007), categorized by four periods of day (3 am-9 am, 9 am-3 pm, 3 pm-9 pm, 9 pm-12 midnight).
+
+![](_page_4_Figure_4.jpeg)
+
+Fig. 5. Total vehicle crashes on Monday from eight weeks before ST through the first eight weeks in ST in MN (2001-2007), categorized by four periods of day (3 am-9 am, 9 am-3 pm, 3 pm-9 pm, 9 pm-12 midnight).
+
+seasonal effects on a monthly basis, while day length is used to represent the effect of hours of daylight.
+
+## 5. The models
+
+The dependent variables are crash counts, so the Poisson or negative binomial distribution is often used to describe them ([Long, 1997](#page-7-0)). Deviance and Pearson Chi-square divided by the degrees of freedom are used to detect over-dispersion and under-dispersion. If the value is larger than 1, then variance is larger than the mean, indicating overdispersion, and under-dispersion if the value is less than 1. For count data with over-dispersion, the negative binomial distribution is often used. Our initial analysis reveals that the Minnesota crash data show over-dispersion. We further test the null hypothesis equality of mean and variance imposed by the Poisson distribution against the alternative that the variance exceeds the mean. The method we use is based on [Cameron and Trivedi \(1998\)](#page-7-0): first we run the regression model using negative binomial distribution, and keep the value of LLnb (loglikelihood value); second, the log-likelihood value for fitting the Poisson distribution (LLp) is also documented; third, we calculate the log likelihood ratio statistic of −2(LLp−LLnb). The asymptotic distribution of the log likelihood ratio statistic follows the Chi-square distribution with one degree of freedom. For all our models, we reject the null hypothesis that the mean equals and the variance at the 5% level.
+
+Table 4 Descriptive statistics of crashes and traffic volumes.
+
+|                | Time period   | Mean   | St. dev. | Min    | Max     |
+|----------------|---------------|--------|----------|--------|---------|
+| Crashes        | 3 am-9 am     | 42.78  | 3.30     | 5      | 360     |
+|                | 9 am-3 pm     | 75.12  | 41.06    | 19     | 562     |
+|                | 3 pm-9 pm     | 84.74  | 33.01    | 23     | 483     |
+|                | 9 pm-midnight | 21.89  | 13.51    | 1      | 108     |
+| Traffic volume | 3 am-9 am     | 415596 | 160994   | 76469  | 574082  |
+|                | 9 am-3 pm     | 792097 | 77569    | 363397 | 1015998 |
+|                | 3 pm-9 pm     | 813235 | 97030    | 292490 | 1004692 |
+|                | 9 pm-midnight | 177408 | 32536    | 49633  | 255546  |
+
+<span id="page-5-0"></span>Table 5 Regressing log(traffic) on days in daylight saving time (2001-2007).
+
+| Dependent Var.    |                  | Log-traffic volumes in the following periods |                    |                  |                   |  |  |  |
+|-------------------|------------------|----------------------------------------------|--------------------|------------------|-------------------|--|--|--|
+|                   | 3 am-9 am        | 9 am-3 pm                                    | 3 pm-9 pm          | 9 pm-midnight    | all day           |  |  |  |
+| 1st Sunday(DST)   | 0.004(0.072)     | -0.008(0.023)                                | -0.005(0.030)      | -0.057(0.032)    | -0.009(0.027)     |  |  |  |
+| 1st Monday(DST)   | 0.090(0.072)     | -0.006(0.023)                                | 0.039(0.030)       | 0.066(0.032) *   | 0.033(0.027)      |  |  |  |
+| Rest of days(DST) | 0.017(0.027)     | 0.015(0.009)                                 | 0.045(0.011) **    | 0.049(0.012) *   | 0.03(0.010) **    |  |  |  |
+| 1st Sunday(ST)    | 0.028 (0.072)    | 0.0003(0.023)                                | -0.005(0.030)      | -0.059(0.032) *  | -0.006(0.027)     |  |  |  |
+| 1st Monday(ST)    | 0.114(0.067)     | 0.010(0.021)                                 | 0.026(0.027)       | 0.0324(0.029)    | 0.039(0.025)      |  |  |  |
+| rain              | -0.032(0.013) *  | -0.025(0.005) **                             | -0.029(0.006) **   | -0.049(0.006) ** | -0.021(0.005) **  |  |  |  |
+| snow              | -0.059(0.017) ** | -0.092(0.005) **                             | -0.115(0.007) **   | -0.091(0.007) ** | -0.079(0.006) **  |  |  |  |
+| cloudy            | 0.002(0.008)     | -0.002(0.002)                                | -0.008(0.003) *    | -0.011(0.003) ** | -0.008(0.003) **  |  |  |  |
+| windy             | -0.040 (0.025)   | 0.014 (0.015)                                | 0.0003(0.018)      | 0.0182(0.013)    | -0.001(0.016)     |  |  |  |
+| mon               | 1.246(0.014) **  | 0.068 (0.004) **                             | 0.124(0.006) **    | 0.056 (0.006) ** | 0.023(0.005) **   |  |  |  |
+| tue               | 1.329(0.013) **  | 0.078(0.004) **                              | 0.160(0.005) **    | 0.153(0.006) **  | 0.270(0.005) **   |  |  |  |
+| wed               | 1.333(0.013)**   | 0.096(0.004) **                              | 0.179 (0.005) **   | 0.192 (0.006) ** | 0.287 (0.005) **  |  |  |  |
+| thur              | 1.316 (0.013) ** | 0.115(0.004) **                              | 0.186 (0.005) **   | 0.248 (0.006) ** | 0.299 (0.005) **  |  |  |  |
+| fri               | 1.288 (0.013) ** | 0.213 (0.004) **                             | 0.253 (0.005) **   | 0.408 (0.006) ** | 0.364 (0.005) **  |  |  |  |
+| sat               | 0.498(0.013) **  | 0.131(0.004) **                              | -0.00007(0.006) ** | 0.331(0.006) **  | 0.128(0.005) **   |  |  |  |
+| y2002             | -0.003(0.014)    | 0.008(0.004)                                 | -0.005(0.006)      | 0.001(0.006)     | 0.0002(0.005)     |  |  |  |
+| y2003             | -0.016(0.013)    | -0.0005 (0.004)                              | -0.016(0.005) **   | -0.006 (0.006)   | -0.009(0.005)     |  |  |  |
+| y2004             | 0.036(0.016) **  | 0.035(0.005) **                              | 0.014(0.006) **    | 0.028(0.007) **  | 0.025(0.006) **   |  |  |  |
+| y2005             | 0.067(0.021) **  | 0.058(0.007) **                              | 0.035(0.008) **    | 0.041(0.009) **  | 0.047(0.008) **   |  |  |  |
+| y2006             | 0.067(0.026) **  | 0.059(0.008) **                              | 0.032 (0.011) **   | 0.031(0.0112) ** | 0.045(0.010) **   |  |  |  |
+| y2007             | 0.050(0.029)     | 0.045(0.009) **                              | 0.008(0.012) **    | -0.001(0.013)    | 0.024(0.011) *    |  |  |  |
+| Feb.              | 0.052(0.022) *   | 0.013(0.007)                                 | 0.023(0.009) **    | 0.016(0.009)     | 0.020(0.008) *    |  |  |  |
+| Mar.              | 0.085(0.032 ) ** | 0.023(0.010) **                              | 0.040(0.013) **    | 0.0004(0.014)    | 0.037(0.012) **   |  |  |  |
+| Apr.              | 0.110(0.049) *   | 0.043(0.016) **                              | 0.050(0.020) *     | 0.015(0.021)     | 0.053(0.018) *    |  |  |  |
+| May               | 0.102(0.061)     | 0.049(0.019) **                              | 0.049(0.025) *     | 0.0436(0.026)    | 0.057(0.023) *    |  |  |  |
+| June              | 0.157(0.067)     | 0.083(0.021) **                              | 0.085(0.0273) **   | 0.136(0.029) **  | 0.099(0.025) **   |  |  |  |
+| July              | 0.084(0.064)     | 0.075(0.020) **                              | 0.054(0.026) *     | 0.156(0.028) **  | 0.077(0.024) **   |  |  |  |
+| Aug.              | 0.138(0.054) **  | 0.109(0.017) **                              | 0.086(0.022) **    | 0.141(0.023) **  | 0.104(0.020) **   |  |  |  |
+| Sep               | 0.117(0.0424) ** | 0.078( 0.013) **                             | 0.063(0.0174) **   | 0.028(0.018)     | 0.071(0.016) **   |  |  |  |
+| Oct.              | 0.142(0.033) **  | 0.099(0.010) **                              | 0.077(0.0135) **   | 0.035(0.014) *   | 0.087(0.012) **   |  |  |  |
+| Nov.              | 0.070(0.018 ) ** | 0.093(0.006) **                              | 0.059(0.007) **    | 0.056(0.008) **  | 0.071(0.007) **   |  |  |  |
+| Dec.              | -0.025(0.018)    | 0.094(0.006) **                              | 0.035(0.007) **    | 0.119(0.008) **  | 0.052(0.007) **   |  |  |  |
+| daylight hours    | 0.003(0.010)     | 0.013(0.003) **                              | 0.003(0.004)       | 0.015(0.004) **  | 0.006(0.004)      |  |  |  |
+| gas price (log)   | -0.042(0.042)    | -0.056(0.013) **                             | -0.031(0.0170)     | -0.047(0.018) ** | -0.041(0.0155) ** |  |  |  |
+| R2                | 0.89             | 0.70                                         | 0.68               | 0.83             | 0.79              |  |  |  |
+
+<sup>\*\*</sup> statistically significant at the 1% level.
+
+Note 1: the OLS model is used to run the regressions.
+
+Note 2: number in (parentheses) indicates std. error.
+
+Therefore, the negative binomial distribution is used to characterize the crash counts in this paper.
+
+Among the independent variables, special attention is paid to the traffic volume variable in that we suspect that it may be an endogenous one. Our model fit tests show that the variable of traffic is endogenous in all our models. Based on [Wooldridge \(2003\),](#page-7-0) the 2SLS model can be considered if an independent variable is endogenous and the instrumental variables (IV) of this variable are uncorrelated with crashes. In our search for the IVs, we have found that the dummy variables for day of week are highly associated with traffic yet are not correlated with crashes at the 5% level; moreover, our tests further show that gas price (at the monthly level statewide) is also uncorrelated with crashes. Therefore dummy variables for day of week and the variable of gas price are used as instrumental variables (IV) for traffic.5 Therefore the 2SLS model is adopted in our analysis.
+
+We also look at fatal crashes statewide. Yet the fatal crash data, when grouped by different dayparts, display obvious under-dispersion, especially for 3 a.m.-9 a.m. (most of the days where fatal crashes happen only have one or two observations). Therefore we examine fatal crash data on a daily basis; the traffic variable is calculated as daily average traffic volume from all ATR stations. The model fit test also allows us to use the 2SLS model; the model structure is otherwise the same as other models mentioned above.
+
+## 6. Results and analysis
+
+A 2SLS model includes two functions: the reduced function to predict traffic and the structural function to predict crashes. These are discussed in turn.
+
+## 6.1. Traffic
+
+In the reduced function, traffic volumes are used as dependent variables, with day-of-week dummy variables and gas price being instrumental variables; dummy variables for year, month, and weather are also included. The hypothesis is that all else equal, weekdays are associated with more traffic than Sunday in daytime; higher gas price is associated with lower traffic volume. The results for the reduced functions are listed in Table 5; we differentiate the estimated coefficients at the 1% and 5% level. As can be seen, the coefficients of first Sunday and first Monday in DST are generally not statistically significant (except for the first Monday in DST during 9 a.m.-midnight), meaning that traffic in the first two days of DST, ceteris par ribus, is no different from an ordinary day in ST. The rest of the days in DST, while not statistically significant in the 3 a.m.-9 a.m. or 9 a.m.-3 p.m. dayparts, are positively associated with traffic during 3 p.m.-9 p.m. and 9 p.m.-midnight. These results accord with our hypotheses that one more hour daylight in the afternoon in DST encourages more out-of-home activities, and thus results in more vehicles on the road. The signs of the coefficients of other variables are consistent with our hypotheses too. For example, during the first two dayparts: 3 a.m.-9 a.m. and 9 a.m.-3 p.m., traffic in weekdays
+
+<sup>\*</sup> statistically significant at the 5% level.
+
+<sup>5</sup> In two of the four models, the correlation between gas price and traffic is not statistically significant; but we still include them in the model for the convenience of comparing the results with those of other models.
+
+are higher than weekends; yet the opposite holds for dusk and night (probably due to night-time activities on weekends). Moreover, many of the monthly dummy variables are statistically significant, displaying a seasonal trend; the coefficients of day length, representing the impact of daily sunlight differences on traffic, are positive; yet the variable is only statistically significant for the off-peak 9 a.m.-3 p.m. and 9 p.m.-midnight periods.
+
+## 6.2. Crashes
+
+After obtaining the OLS estimate of traffic volume, we employ it as an independent variable in our structural function with crashes as the dependent variable. Table 6 shows the results of the structural forms given different time periods. The coefficient of first Sunday of DST equals 0.026 (the increase may result from our previous hypothesis that losing one hour's sleep affects drivers' behavior); however, it is not statistically significant. During 3 p.m.-9 p.m., the coefficient of the first Sunday in DST equals -0.29, suggesting approximately 0.28% fewer crashes on the first day of DST than a day in ST. We also find that during mid-day (9 a.m.-3 p.m.) the first day of DST also has a negative and statistically significant coefficient, but the rest of the days in DST are not statistically significant; it is still unclear why the crashes on the mid-day of the first day in DST are affected, but it must be remembered this first day of DST only occurs on 7 days between 2001 and 2007, so may be anomalous. For the rest of the days in DST, they have negative and statistically significant coefficients for most periods of day except for 9 a.m.-3 p.m. In particular, for sunset and dusk period, the decrease of crashes may be attributed to longer visibility for drivers. Furthermore, none of the estimates for the first Sunday of ST and first Monday of ST are statistically significant, which probably illustrates that crashes on the first two days of ST are no different from other days in ST.
+
+Clearly, traffic has a salient impact on crashes. Particularly for 3 p.m.- 9 p.m., a 1% increase of traffic volume, all else equal, is associated with
+
+Table 7 The overall effect of DST on crashes(log form), MN crashes data, 2001-2007 (base case: days in ST).
+
+|                            | 3 am-9 am | 9 am-3 pm | 3 pm-9 pm | 9 pm-midnight | Fatal<br>(all day) |
+|----------------------------|-----------|-----------|-----------|---------------|--------------------|
+| first Sunday<br>in DST     | -         | -0.367    | -0286     | -             | -                  |
+| first Monday<br>in DST     | -         | -         | - 0.246   | 0.107         | -                  |
+| Rest of the<br>days in DST | -0.164    | -         | -0.090    | -0.075        | -0.010             |
+
+2.2% more crashes. Take note that the exposure issue is also influenced by DST. To assess the overall effect of DST on crashes, therefore, we need to consider both the impact of DST per se and exposure to traffic affected by DST (using path analysis). For example, for 3 p.m.-9 a.m., the overall effect of the rest of the days in DST on log-crashes is calculated as: 0.045×2.214−0.190=−0.090. Thus, a day in DST, all else equal, is associated with about 0.09% fewer crashes than a day in ST. This reminds us that the benefit of better visibility during dusk may be diminished (but not eliminated) by the corresponding uptick in traffic.
+
+The final results of the influence of DST on crashes are described in Table 7 (in calculation we only consider the statistically significant coefficients). They demonstrate that a day DST, largely speaking, is associated with fewer crashes than its counterpart in ST when all else is equal. In addition, we do not find a statistically significant rise of crashes on the first day of DST during 3 a.m.-9 a.m.
+
+## 6.3. Fatal crashes
+
+Our results from the fatal crash data show that none of the coefficients for the first Sunday day in DST, first Monday in DST, and the rest of days in DST are statistically significant (after controlling for
+
+Table 6 Regressing vehicle crashes on days in daylight saving time (MN crash data , 2001-2007).
+
+| Dependent Var.      | Log-crashes in the following periods |                   |                    |                  |                 |  |  |
+|---------------------|--------------------------------------|-------------------|--------------------|------------------|-----------------|--|--|
+|                     | 3 am-9 am                            | 9 am-3 pm         | 3 pm-9 pm          | 9 pm-midnight    | Fatal(all day)  |  |  |
+| 1st Sunday(DST)     | 0.026(0.171)                         | -0.367(0.134) **  | -0.286(0.111) **   | -0.032(0.179)    | -0.346(0.410)   |  |  |
+| 1st Monday(DST)     | -0.201 (0.160)                       | -0.115(0.128)     | -0.246(0.106) *    | -0.293(0.178)    | 0.168(0.234)    |  |  |
+| Rest of days(DST)   | -0.164 (0.062) **                    | -0.093 (0.049)    | -0.190(0.039) **   | -0.155(0.060) ** | -0.058(0.113)   |  |  |
+| 1st Sunday(ST)      | -0.168(0.170)                        | -0.142(0.129)     | -0.129(0.1066)     | 0.237(0.164)     | 0.168(0.234)    |  |  |
+| ^<br>1st Monday(ST) | -0.204 (0.147)                       | -0.080(0.116)     | 0.126 (0.092)      | -0.024(0.146)    | -0.167(0.256)   |  |  |
+| traff<br>(log)      | 0.634(0.017) **                      | 1.978(0.104) **   | 2.214 (0.059) **   | 1.634(0.058) **  | -0.323(0.104) * |  |  |
+| rain                | 0.296(0.029 ) **                     | 0.375(0.025) **   | 0.238(0.020) **    | 0.314(0.028) **  | 0.006(0.056)    |  |  |
+| snow                | 0.639(0.037) **                      | 0.926(0.033) **   | 0.752(0.025) **    | 0.814(0.033) **  | 0.082(0.065)    |  |  |
+| cloudy              | -0.033(0.018) **                     | -0.013(0.013)     | 0.0106(0.011)      | 0.040(0.018) *   | -0.002(0.031)   |  |  |
+| windy               | 0.100(0.057) **                      | -0.077(0.085)     | -0.010(0.063)      | 0.038(0.063)     | 0.052(0.089)    |  |  |
+| y2002               | -0.028(0.030)                        | -0.067(0.023) **  | -0.003(0.019)      | -0.069(0.030) *  | 0.088(0.051)    |  |  |
+| y2003               | -0.288(0.030) **                     | -0.313(0.023) **  | -0.056(0.019) **   | 0.291(0.029) **  | 0.056(0.051)    |  |  |
+| y2004               | -0.126(0.030) **                     | -0.248(0.023) **  | -0.033(0.019)      | 0.288(0.029) **  | -0.015(0.052)   |  |  |
+| y2005               | -0.191 (0.030) **                    | -0.310 (0.023) ** | -0.076 (0.0191) ** | 0.256(0.029) **  | -0.018(0.053)   |  |  |
+| y2006               | -0.223(0.030) **                     | -0.385(0.023) **  | -0.149(0.019) **   | 0.197(0.030) **  | -0.022(0.054)   |  |  |
+| y2007               | -0.231(0.030) **                     | -0.344(0.023) **  | -0.128(0.0194) **  | 0.154(0.030) **  | -0.067(0.055)   |  |  |
+| Feb.                | 0.177(0.047) **                      | -0.010(0.037)     | 0.004(0.031)       | -0.0120(0.046)   | -0.161(0.091)   |  |  |
+| Mar.                | -0.001(0.071)                        | -0.116(0.056) *   | -0.181(0.046) **   | -0.100(0.070)    | -0.300(0.133) * |  |  |
+| Apr.                | 0.012(0.109)                         | -0.091(0.085)     | 0.007(0.067)       | 0.057(0.105)     | -0.071(0.192)   |  |  |
+| May                 | 0.208(0.135)                         | 0.156(0.106)      | 0.208(0.087) *     | 0.240(0.131)     | -0.080(0.238)   |  |  |
+| June                | 0.301(0.149) *                       | 0.280(0.116) **   | 0.229(0.095) *     | 0.401(0.144) **  | -0.025(0.261)   |  |  |
+| July                | 0.178(0.142)                         | 0.228(0.111) *    | 0.214(0.091) *     | 0.261(0.138)     | -0.049(0.248)   |  |  |
+| Aug.                | 0.063(0.118)                         | 0.094(0.093)      | 0.081(0.076)       | 0.060(0.115)     | 0.090(0.209)    |  |  |
+| Sep                 | 0.201(0.093)                         | -0.027(0.073)     | 0.119(0.060) **    | 0.067(0.090)     | -0.007(0.166)   |  |  |
+| Oct.                | 0.055 (0.073)                        | -0.153 (0.059) ** | 0.031(0.047)       | 0.084(0.071)     | 0.121(0.130)    |  |  |
+| Nov.                | -0.091(0.039) *                      | -0.273(0.032) **  | 0.019(0.026)       | -0.018(0.039)    | 0.036(0.072)    |  |  |
+| Dec.                | -0.056(0.039)                        | -0.086(0.032) **  | 0.010(0.025)       | -0.124(0.039)    | 0.141(0.069) *  |  |  |
+| daylight hours      | -0.1010(0.023) **                    | -0.090(0.018) **  | -0.051(0.015) **   | -0.064(0.022) ** | 0.039(0.040)    |  |  |
+
+<sup>\*\*</sup> statistically significant at the 1% level.
+
+Note 2: number in (parentheses) indicates std. error.
+
+<sup>\*</sup> statistically significant at the 5% level.
+
+Note 1: the negative binomial (NB) model is used to run the regressions.
+
+<span id="page-7-0"></span>traffic and months). Yet it is interesting to notice that the coefficient of traffic in the structural model equals -0.323 and is statistically significant at the 5% level, meaning that the increase of traffic is associated with a decrease of fatal crashes. This may be because higher traffic volumes limit vehicles' speed, which makes drivers less likely to drive recklessly; a similar theory can be found in John Adams's thoughts in Smeed's Law (Adams, 1987). In the reduced function with traffic as the dependent variable, the coefficient of the rest of the days in DST equals 0.027. The overall effect of DST on vehicle crashes can be calculated as −0.323× 0031= 0.010, indicating that a day in DST, all else equal, is associated with about 0.008% fewer fatal crashes than a day in ST.
+
+## 7. Conclusions
+
+Considering the issue of exposure to traffic, this paper explores long- and short-term effects of DST on vehicle crashes. The contribution of this research includes two aspects. First, we separate the effects of DST on traffic volume and the effect of DST per se on crashes. The two stage least square models are further developed to examine crashes in four periods of day (including dawn and dusk periods). Second, we have looked at crashes at different periods of day, which is convenient for explaining the possible causes for the effects. Our results show that DST per se, by providing one more hour's visibility in the afternoon, can help reduce crashes during and after dusk; but it can also increase traffic. Overall, we find that DST, all else equal, is associated with fewer crashes for most dayparts; in addition, DST is also correlated with fewer fatal crashes than ST.
+
+#### Acknowledgements
+
+We thank Nathan Drews and Gene Hicks from the Minnesota Department of Transportation for providing the crash data set for this study.
+
+#### References
+
+- Adams, J. (1987). Smeed's Law: some further thoughts. Traffic Engineering and Control, 28, 70−73.
+- Adams, J. M. (2005). Year-round daylight saving and serious or fatal road traffic injuries in children in the Northeast of England. Journal of Public Health, 27(4), 316−317.
+- Broughton, J., Hazelton, M., & Stone, M. (1999). Influence of light level on the incidence of road casualties and the predicted effect of changing 'summertime'. Journal of the Royal Statistical Society: Series A, 162(2), 137−175.
+- Broughton, J., & Sedmand, R. J. (1989). The potential effects on road causalties of double British Summer Time, TRL report 228. Crowthorne: Transport Research Laboratory. Broughton, J., & Stone, M. (1998). A new assessment of the likely effects on road accidents of adopting SDST, TRL report 368. Crowthorne: Transport Research Laboratory.
+
+- Cameron, A., & Trivedi, P. (1998). Regression Analysis of Count Data. Cambridge: Cambridge University Press.
+- Coate, D., & Markowitz, S. (2003). The effects of daylight and daylight saving time on us pedestrian fatalities and motor vehicle occupant fatalities. Accident Analysis and Prevention, 36(3), 351−357.
+- Coren, S. (1996a). Acciental death and shift to daylight saving time. Perceptual and Motor Skills, 83, 921−922.
+- Coren, S. (1996b). Daylight savings time and traffic accidents. The New England Journal of Medicine, 334, 924.
+- Fatehi, S. (2009). Daylight Chart 3.0. <http://daylightchart.sourceforge.net/sunposition/>. Ferguson, S. A. (1995). Daylight savings time and motor vehicle crashes: the reduction
+- in pedestrian and vehicle occupant fatalities. American Journal of Public Health, 85 (1), 92−95. Hicks, G. J., Davis, J. W., & Hicks, R. A. (1998). Fatal alcohol-related traffic crashes
+- increase subsequent to changes to and from daylight savings time. Perceptual and Motor Skills, 86(3 Pt 1), 879−882.
+- Hicks, R. A., Lindseth, K., & Hawkins, J. (1983). Daylight savings time changes increase traffic accidents. Perceptual and Motor Skills, 56, 64−66.
+- Hillman, M. (1989). More daylight less accidents. Traffic Engineering and Control, 4(30), 191−193.
+- Lambe, M., & Cummings, P. (2000). The shift to and from daylight savings time and motor vehicle crashes. Accident Analysis and Prevention, 32, 609−611.
+- Ledger, D. (1994). The cost of sleep related accidents: a report for the national commission on sleep disorders research. Sleep, 17, 84−93.
+- Long, J. (1997). Regression Models for Categorical and Limited Dependent Variables. Thousand Oaks, CA: Sage Publications Inc.
+- Meyerhoff, N. J. (1978). The influence of daylight saving time on motor vehicle fatal traffic accidents. Accident Analysis and Prevention, 10, 207−221.
+- Sood, N., & Ghosh, A. (2007). The short and long run effects of daylight saving time on fatal automobile crashes.B.E. Journal of Economic Analysis & Policy, 7(1) (Article 11).
+- Stevens, C. R., & Lord, D. (2006). Evaluating safety effects of daylight savings time on fatal and nonfatal injury crashes in texas. Transportation Research Record, 1953, 147−155.
+- Varughese, J., & Allen, R. P. (2001). Fatal accidents following changes in daylight savings time: the American experience. Sleep Medicine, 2(1), 31−36.
+- Vincent, A. (1998). Effects of daylight savings time on collision rates. The New England Journal of Medicine, 339, 1167−1168.
+- Ward, H., Cave, J., Morrison, A., Allsop, R., Evans, A., Kuiper, C., et al. (1994). Pedestrian activity and accident risk, Technical report. London: Automobile Association Foundation for Road Safety Research.
+- Wooldridge, J. M. (2003). Introductory Econometrics: A Modern Approach. OH: South-Western Mason.
+
+Arthur Huang is a Ph.D. student at the Department of Civil Engineering at the University of Minnesota, Twin Cities. He is also enrolled in the Master of Urban and Regional Planning program at the Humphrey Institute of Public Affairs, University of Minnesota. His major research interests are the evolution of transportation and land use, business cluster analysis, and traffic safety. He is a student member of American Planning Association, Institute of Transportation Engineers, and Air & Waste Management Association.
+
+Dr. David Levinson is an Associate Professor in the Department of Civil Engineering at the University of Minnesota and Director of the Networks, Economics, and Urban Systems (NEXUS) research group. He currently holds the Richard P. Braun/CTS Chair in Transportation. Levinson has authored or edited several books and numerous peer reviewed articles. He is the editor of the Journal of Transport and Land Use.

Zotero/001_artiklid/Road Safety Indicators/ivandifferencescausalityfactorssingle1999.md

@@ -0,0 +1,238 @@
+---
+category: literaturenote
+citekey: ivandifferencescausalityfactorssingle1999
+title: Differences in causality factors for single and multi-vehicle crashes on two-lane roads
+authors: "Ivan, John N.; Pasupathy, Raghubhushan K.; Ossenbruggen, Paul J."
+year: 1999
+date: 1999-00-00 1999
+doi: 10.1016/s0001-4575(99)00030-5
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/s0001-4575(99)00030-5"
+zotero_key: KEEZ8SAF
+zotero_storage: NG72WZ99
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Ivan, John N."
+status: converted
+---
+on two-lane roads
+
+<sup>a</sup> *Department Ci*6*il and En*6*ironmental Engineering U*-37, *Uni*6*ersity of Connecticut*, *Connecticut Transportation Institute*, *Storrs*, *CT* 06269-2037, *USA* <sup>b</sup> *Edwards and Kelcey*, <sup>529</sup> *Main Street*, *Suite* <sup>203</sup>, *Charlestown*, *MA* <sup>02129</sup>, *USA* <sup>c</sup> *Department of Ci*6*il Engineering*, *Uni*6*ersity of New Hampshire*, *Durham*, *NH* <sup>03824</sup>, *USA* Received 1 October 1998; received in revised form 9 February 1999; accepted 26 March 1999
+
+Past research has found a non-linear relationship between traffic intensity or level of service (LOS) and highway crash rates. This paper investigates this relationship further by including the effects of site characteristics and estimating Poisson regression models for predicting single and multi-vehicle crashes separately. Analysis focuses on rural two-lane highways, with hourly LOS, traffic composition, and highway geometric characteristics as independent variables. The resulting models for single and multi-vehicle crashes have different explanatory variables. Single-vehicle crash rates decrease with increasing traffic intensity (lower LOS), shoulder width and sight distance. Multi-vehicle crash rates increase with the number of signals, the daily single-unit truck percentage, and the shoulder width, and decreased on principal arterials compared to other roadway classes. LOS does not significantly explain variation in the number of multi-vehicle crashes. Ongoing research by the authors is aimed at identifying other site factors, such as driveway density and intersection LOS, that can better explain the differing effects reported here and
+
+John N. Ivan a,\*, Raghubhushan K. Pasupathy <sup>b</sup>
+
+predict crash rates of both types better. © 1999 Elsevier Science Ltd. All rights reserved.
+
+*Keywords*: Highway crash rates; Level of service; Crash type; Two-lane roads; Daylight
+
+One of many important aspects of highway safety research is developing crash prediction models to quantify the relationship between site characteristics and the number of crashes observed. These prediction models can help identify the site characteristics tending to cause highway crashes. Crash prediction models, till now, have tended to be macroscopic in nature, meaning that they predict using summary statistics on traffic such as annual average daily traffic (AADT) rather than microscopic measures such as hourly volume counts. These macroscopic models also tended to consider all crashes together rather than separately by single and multi-vehicle crash types. Studies that have considered hourly volumes have generally not consid-
+
+\* Corresponding author. Tel.: +1-860-486-0352; fax: +1-860-486-
+
+0001-4575/99/\$ - see front matter © 1999 Elsevier Science Ltd. All rights reserved.
+
+*E*-*mail address*: johnivan@eng2.uconn.edu (J.N. Ivan)
+
+PII: S0001-4575(99)00030-5
+
+![](_page_0_Picture_1.jpeg)
+
+![](_page_0_Picture_2.jpeg)
+
+, Paul J. Ossenbruggen <sup>c</sup>
+
+ered geometric effects. In this paper, the significance of all of these factors—site location, highway geometry, daylight conditions, and traffic composition—are in-
+
+The focus of this research was to find differences in the effects of both traffic condition and site characteristics as independent variables for the expected single and multi-vehicle crash rates, measured in crashes per year. Eight different rural two-lane state highway locations across Connecticut with hourly volume counts between September 1990 and October 1996 were studied. Poisson regression was used to estimate and evaluate models for predicting the expected number of single and
+
+A number of researchers have investigated the effect of exposure on the number of crashes per year, or crash
+
+vestigated together.
+
+**2. Previous work**
+
+multi-vehicle crashes at these sites.
+
+Differences in causality factors for single and multi-vehicle crashes
+
+**Abstract**
+
+**1. Introduction**
+
+2298.
+
+![](_page_1_Picture_0.jpeg)
+
+![](_page_2_Picture_0.jpeg)
+
+![](_page_3_Picture_0.jpeg)
+
+![](_page_4_Picture_0.jpeg)
+
+![](_page_5_Picture_0.jpeg)
+
+Table 3 Preliminary models for multi-vehicle crashes<sup>a</sup>
+
+| Variable             | Model 1         | Model 2        | Model 3        | Model 4        |
+|----------------------|-----------------|----------------|----------------|----------------|
+| Intercept            | −10.08 (−11.53) | −9.88 (−12.04) | −9.98 (−10.72) | −11.31 (−7.33) |
+| Route 1              | Base            |                | Base           | Base           |
+| Route 5              | −2.03 (−3.72)   |                | −2.28 (−3.86)  | −2.30 (−3.78)  |
+| Route 7              | −1.86 (−4.19)   |                | −1.84 (−4.12)  | −1.81 (−3.99)  |
+| Route 8              | −1.17 (−2.10)   |                | −1.11 (−1.90)  | −0.94 (−1.50)  |
+| Route 12             | 0.88 (3.13)     |                | 0.86 (3.00)    | −1.01 (−3.18)  |
+| Route 66             | −1.95 (−5.70)   |                | −2.09 (−5.45)  | −2.21 (−5.35)  |
+| Route 81             | −2.59 (−3.15)   |                | −2.67 (−3.22)  | −2.63 (−3.05)  |
+| Route 124            | −0.99 (−4.00)   |                | −1.54 (−3.18)  | −1.66 (−3.13)  |
+| LOS A                | Base            | Base           | Base           | Base           |
+| LOS B                |                 | 0.64 (1.67)    | 0.15 (0.38)    | 0.14 (0.32)    |
+| LOS C                |                 | −0.14 (−0.32)  | 0.21 (0.49)    | 0.33 (0.67)    |
+| LOS D                |                 | −0.18 (−0.39)  | 0.63 (1.16)    | 0.76 (1.24)    |
+| LOS E                |                 | 0.46 (1.02)    | 0.83 (1.28)    | 0.86 (1.21)    |
+| Light                |                 |                |                | Base           |
+| Dark                 |                 |                |                | 0.26 (0.79)    |
+| Dusk                 |                 |                |                | 0.45 (1.27)    |
+| Exponent on exposure | 0.88 (12.20)    | 0.76 (10.37)   | 0.86 (10.73)   | 0.96 (7.96)    |
+| Dispersion           | 1.37            | 1.44           | 1.38           | 1.40           |
+| Null deviance        | 788.97          | 788.97         | 788.97         | 788.97         |
+| Residual deviance    | 373.28          | 504.44         | 370.26         | 367.97         |
+| AIC value            | 391.28          | 516.44         | 396.26         | 397.97         |
+
+<sup>a</sup> Values in parentheses are *t*-statistics after correction for dispersion.
+
+variables, is clearly the best. It has the smallest AIC value, and none of the LOS or light condition coefficients in the other models are significantly different from zero at 95% confidence. It seems that segment LOS and light conditions are not useful for predicting multi-vehicle crashes. This point will also be discussed later.
+
+### <sup>5</sup>.2. *Site characteristics*
+
+The preliminary analysis clearly showed that each site has a different average crash rate for both crash types. The next step is to replace the site variable with the actual site characteristics in the model estimations. These site characteristics are represented by a linear combination of specific continuous variables, as represented in the initial model form. The LOS and light condition variables were retained as categorical variables. The list of site variables and their abbreviations as used in the models are shown in Table 4. The base model (with respect to LOS and light conditions) was defined as before. All site characteristic, traffic, and light condition variables, and all two-way interactions between the categorical variable levels, were considered as predictors in estimating models for both types of crashes.
+
+Table 5 shows the results from estimating models for single-vehicle crashes. Model 5 includes the five LOS categories, along with three site-related variables: the percentage of the section with a sight distance of at least 1500 feet, the right shoulder width (feet), and the percentage of single-unit trucks in the peak traffic stream. All coefficients are significantly different from zero with at least 95% confidence; however, paired *t*-tests show that the coefficients on LOS D and LOS E are not significantly different from each other. Consequently, model 6 was run, which combines LOS D and E into a single category. The resulting increase in residual deviance is offset by the reduction in model complexity (one less parameter), so this model has a better (smaller) AIC value.
+
+Table 4 Site characteristic variables
+
+| Variable name                                       | Type of variable | Observed values                                                       |
+|-----------------------------------------------------|------------------|-----------------------------------------------------------------------|
+| Functional classifica-<br>tion                      | Categorical      | Principal arterial—<br>othera<br>; minor arterial;<br>major collector |
+| Right shoulder width<br>(feet)                      | Continuous       | 0–8                                                                   |
+| Sight distance over<br>1500 feet (% of sec<br>tion) |                  | 0, 10, 80                                                             |
+| Number of signals                                   | Continuous       | 0–1                                                                   |
+| % Single unit trucks<br>(daily)                     | Continuous       | 0–4                                                                   |
+| % Single unit trucks<br>(peak)                      | Continuous       | 0–2                                                                   |
+| Speed limit (mph)                                   | Continuous       | 35, 40, 45                                                            |
+
+<sup>a</sup> Non-freeway.
+
+Table 5 Final models for single vehicle crashes<sup>a</sup>
+
+| Variable                      | Model 5         | Model 6         | Model 7         | Model 8         | Model 9       |
+|-------------------------------|-----------------|-----------------|-----------------|-----------------|---------------|
+| Intercept                     | −14.78 (−11.74) | −14.88 (−12.22) | −13.06 (−10.48) | −13.06 (−11.84) | −8.91 (−4.13) |
+| LOS A                         | Base            | Base            | Base            | Base            | Base          |
+| LOS B                         | −1.65 (−5.22)   | −1.66 (−5.30)   | −1.66 (−4.89)   | −1.65 (−4.91)   | −1.59 (−4.90) |
+| LOS C                         | −2.81 (−6.27)   | −2.83 (−6.31)   | −2.53 (−5.36)   |                 |               |
+| LOS D                         | −3.90 (−5.72)   |                 |                 |                 |               |
+| LOS E                         | −3.70 (−6.31)   |                 |                 |                 |               |
+| LOS (D, E)                    |                 | −3.77 (−7.01)   | −2.59 (−5.90)   |                 |               |
+| LOS (C, D, E,)                |                 |                 |                 | −2.56 (−6.57)   | −2.76 (−6.86) |
+| Sight distance over 1500 feet | −0.02 (−2.30)   | −0.02 (−2.30)   | −0.02 (−2.31)   | −0.02 (−2.44)   | −0.02 (−2.49) |
+| Right shoulder width (feet)   | −0.27 (−4.23)   | −0.27 (−4.36)   | −0.12 (−2.32)   | −0.12 (−2.33)   | −0.15 (−2.84) |
+| % Single-unit trucks (peak)   | 1.47 (3.68)     | 1.49 (3.80)     |                 |                 |               |
+| % Single-unit trucks (daily)  |                 |                 | −0.01 (−0.06)   |                 |               |
+| Speed limit (mph)             |                 |                 |                 |                 | −0.09 (−2.10) |
+| Exponent on exposure          | 1.18 (11.71)    | 1.19 (12.13)    | 1.16 (11.26)    | 1.16 (11.50)    | 1.13 (11.53)  |
+| Dispersion                    | 0.64            | 0.64            | 0.75            | 0.75            | 0.71          |
+| Null deviance                 | 363.50          | 363.50          | 363.50          | 363.50          | 363.50        |
+| Residual deviance             | 185.95          | 186.01          | 194.94          | 194.96          | 191.44        |
+| AIC value                     | 203.95          | 202.01          | 210.94          | 206.96          | 205.44        |
+
+<sup>a</sup> Values in parentheses are *t*-statistics after correction for dispersion.
+
+The peak hour truck percentage variable seemed inappropriate, since each case potentially represents conditions aggregated over periods of time other than the peak. We therefore ran model 7, which replaces the peak truck variable with a corresponding daily variable; this model performs much worse; the coefficient on the daily truck percentage is not significantly different from zero at even 90% confidence. Furthermore, a paired *t*-test comparing the coefficients on LOS C and the combined LOS D/E category showed them to be statistically identical at 95% confidence (*t\**=1.96).
+
+In consideration of these findings, we then combined LOS C, D and E into one category and removed both truck variables: this was run as model 8. Because other research by the senior author (Ivan and O'Mara 1997) has found the posted speed limit to be a significant variable for predicting crash rates, model 9 was then run adding 'speed limit' to model 8. Model 9 performs better than model 8, having a smaller AIC, but not as well as model 6, which included the suspect peak singleunit truck percentage variable.
+
+Similar analysis was performed with the number of multi-vehicle crashes as the dependent variable; these results are shown on Table 6. In the first model (number 5), the best predictor variables were found to be the functional classification of the road, the number of traffic signals, the daily truck percentage, and the right shoulder width. Only the coefficient on the principal regional arterial roadway class was significantly different from the others (minor arterial or major collector) at 95% confidence; this class has a significantly lower average crash rate than the others. The positive coefficients on both the number of signals and the daily single unit truck percentage make intuitive sense; one would expect both of these variables to increase the intensity of conflicts between vehicles.
+
+However, the positive coefficient on right shoulder width is troubling; one normally expects a wider shoulder to be a safety feature. Consequently, we examined the data surrounding this variable more carefully. Figure 3 shows how the multi-vehicle crash rate varies with the right shoulder width: note how there is no clear linear relationship between the two quantities. We therefore examined several categorical representations of right shoulder width; those that performed best are shown as models 6, 7 and 8. These three models each divide the right shoulder width into two categories, with the lower category having shoulder widths up to 2, 3 and 4 feet, respectively. None of these models perform as well as model 5. The issue of right shoulder width will be discussed more in the next section.
+
+### **6. Discussion**
+
+Several general observations may be made about these results:
+
+- 1. The coefficient on shoulder width was *negati*6*e* for predicting single vehicle crashes, but *positi*6*e* for multi-vehicle crashes.
+- 2. For single-vehicle crashes, the most important predictor variables are the LOS and the percent sight distance over 1500 feet.
+- 3. For multi-vehicle crashes, the most important predictor variables are the class of roadway, number of signals and daily single unit truck percentage.
+
+Single vehicle crashes seem to occur when traffic volume is low and on roads with less forgiving geometry. This makes sense, since a single vehicle crash involves a driver losing control and colliding with a fixed object off the roadway (or an object on the roadway). Under low traffic volumes, there is less likely to be another vehicle to hit, and the less than ideal geometry (sharp horizontal curves and narrow shoulders) reduces the maneuvering room for the driver to recover without leaving the pavement, for example, if the driver had been speeding.
+
+Conversely, multi-vehicle crashes appear to occur in the presence of intense traffic conflicts such as would be observed at a traffic signal, or on roads serving local trips, or when there are many trucks in the traffic stream. Traffic signals tend to be placed in locations where intersecting streets have high traffic volumes. Minor arterials and major collector roads tend to serve local trips and thus have more driveways and local street intersections, increasing the potential for vehicle conflicts.
+
+Clearly, more analysis is required to properly interpret these findings. For one thing, the higher levels of LOS—which correspond to the higher numbers of single-vehicle crashes—are observed more typically at night rather than during the day. Consequently, it is not immediately clear from this analysis whether it is the low volumes, the darkness, or drowsiness of late night drivers that results in greater numbers of singlevehicle crashes. Each of these factors needs to be accounted for explicitly in order to fully understand this phenomenon.
+
+A second issue with these findings is that the LOS was computed for the analysis *segment*, not for the *intersections* on the segment. This might explain why LOS was not significant for multi-vehicle crashes, which occur where there is potential for vehicle conflicts, such as at intersections. If LOS were computed for the intersections on the segment, then it might become significant for predicting multi-vehicle crashes.
+
+Finally, it is difficult to explain the positive coefficient on right shoulder width in the multi-vehicle crash models. One interpretation is that with wider shoulders drivers are more likely to drive too fast and collide with other vehicles exiting or entering driveways. Another is that drivers might be tempted to use a wide shoulder to pass drivers waiting to make left turns, and thus not see a hazardous situation beyond the stopped vehicle. Both of these explanations are related to the presence of intersections on the segment; this issue is addressed in the next section.
+
+#### **7. Conclusions**
+
+In response to these findings, ongoing research by this research team is aimed at including new variables which attempt to clarify the effects of the predictive factors used in this study. Specifically, intersection volumes and the potential for conflicts leading to angle and turning collisions are incorporated into the analysis by considering the AADT on roads intersecting each analysis section. Second, the number of driveways of different types (e.g. residential, service station, retail, etc.) is included to account both for the land use intensity and for conflicts associated with driveway intersections. Third, the potential effect of time of day is being investigated by incorporating it as an additional variable along with daylight conditions.
+
+Table 6 Final models for multi-vehicle crashes<sup>a</sup>
+
+| Variable                           | Model 5         | Model 6         | Model 7         | Model 8         |
+|------------------------------------|-----------------|-----------------|-----------------|-----------------|
+| Intercept                          | −15.87 (−14.69) | −14.72 (−14.07) | −15.32 (−11.13) | −18.57 (−12.89) |
+| Minor arterial or major collector  | Base            | Base            | Base            | Base            |
+| Principal arterial                 | −5.79 (−9.83)   | −3.34 (−8.79)   | −4.05 (−7.55)   | −7.98 (−7.09)   |
+| Number of signals                  | 2.31 (7.78)     | 1.50 (4.40)     | 2.53 (6.39)     | 3.55 (8.53)     |
+| Daily single-unit truck percentage | 1.26 (6.84)     | 1.01 (5.39)     | 1.40 (4.92)     | 2.24 (8.17)     |
+| Right shoulder width (feet)        | 0.44 (5.15)     |                 |                 |                 |
+| Right shoulder width (52 feet)     |                 | Base            |                 |                 |
+| Right shoulder width (\2 feet)     |                 | 1.16 (5.07)     |                 |                 |
+| Right shoulder width (53 feet)     |                 |                 | Base            |                 |
+| Right shoulder width (\3 feet)     |                 |                 | 0.71 (1.84)     |                 |
+| Right shoulder width (54 feet)     |                 |                 |                 | Base            |
+| Right shoulder width (\4 feet)     |                 |                 |                 | 3.79 (4.48)     |
+| Exponent on exposure               | 0.88 (13.83)    | 0.90 (13.57)    | 0.87 (11.67)    | 0.91 (13.22)    |
+| Dispersion                         | 1.30            | 1.32            | 1.69            | 1.33            |
+| Null deviance                      | 788.97          | 788.97          | 788.97          | 788.97          |
+| Residual deviance                  | 370.88          | 374.07          | 400.26          | 384.67          |
+| AIC value                          | 382.88          | 386.07          | 412.26          | 396.67          |
+
+<sup>a</sup> Values in parentheses are *t*-statistics after correction for dispersion.
+
+![](_page_9_Figure_2.jpeg)
+
+Fig. 3. Single and multi-vehicle crashes per million vehicle miles by right shoulder width.
+
+Intersection traffic volumes and driveway counts are not as readily available as the other data used in this analysis. The permanent count stations only provide data for the main road; consequently, the precise exposure and volumes gathered for each analysis section are not available for the roads intersecting them. Instead, in this new research, these intersecting road volumes are estimated using an AADT observed for each road and the observed hourly volumes. Similarly, the HPMS and other roadway inventory datasets do not provide detailed information about the driveways along the section. This information is being collected using the ConnDOT photolog library, a unique data source comprised of driver's eye view images of the entire length of every state highway in Connecticut, updated once each year. Using the photolog library permits expanding our analysis beyond HPMS sites; consequently, we have identified ten new analysis sites in the vicinity of two-lane ATR stations, each 0.5 mile in length.
+
+This newly expanded dataset permits us to estimate additional models for predicting single and multi-vehicle crashes which consider these new variables. In addition, in this analysis we are including the time of day (divided into five categories), permitting us to isolate time of day, daylight and LOS effects and identify the value of each. We are also computing new variables for measuring exposure to multi-vehicle crashes which consider not only the traffic volume on the section being analyzed, but also the traffic volume on intersecting roads. We expect these new models to explain more variation than the models presented in this paper.
+
+# **Acknowledgements**
+
+The research described in this paper was funded by a grant from the United States Department of Transportation through the New England (Region 1) University Transportation Center and was conducted at the Connecticut Transportation Institute of the University of Connecticut.
+
+## **References**
+
+Agresti, A., 1990. Categorical Data Analysis. Wiley, New York.
+
+Bozdogan, H., 1987. Model selection and Akaike's Information Criterion (AIC): the general theory and its analytical extensions. Psychometrika 52, 345–370.
+
+Frantzeskakis, J., 1983. Accident analysis on two non-controlled access national highways in Greece. ITE Journal 65 (10), 32–34. Gwynn, D., 1967. Relationship of accident rates and accident involvements with hourly volumes. Traffic Quarterly 21, 407–418.
+
+TRB, 1994. Highway capacity manual. Special Report 209. National Research Council, Washington, DC.
+
+Ivan, J., O'Mara, P., 1997. Prediction of traffic accident rates using Poisson regression. Presented at 76th Annual Meeting of the Transportation Research Board, Washington, DC, paper no. 970861.
+
+Persaud, B., Mucsi, K., 1995. Microscopic accident potential models for two-lane rural roads. Transportation Research Record 1485, TRB, National Research Council, Washington, DC, pp. 134–139. S-PLUS User's Manual, 1995. Version 3.3 for Windows. Statistical
+
+Sciences Division, Mathsoft, Seattle. Valavanis, S., 1959. Econometrics: An Introduction to Maximum
+
+Likelihood Methods. McGraw-Hill, New York.
+
+Vogt. A., Bared, J., 1998. Accident Models for Two-Lane Rural Segments and Intersections. Transportation Research Record 1635, TRB, National Research Council, Washington, DC, 18–29.
+
+Zhou, M., Sisiopiku, V., 1997. Relationship Between Volume to Capacity Ratios and Accident Rates. Transportation Research Record 1581, TRB, National Research Council, Washington, DC, pp. 47–52.

Zotero/001_artiklid/Road Safety Indicators/jafariinvestigatinginfluencesocioeconomicfactors2025.md

@@ -0,0 +1,310 @@
+---
+category: literaturenote
+citekey: jafariinvestigatinginfluencesocioeconomicfactors2025
+title: Investigating the influence of socioeconomic factors on the relationships between road characteristics and traffic crash frequency and severity-- A hybrid structural equation modelling − artificial neural networks approach
+authors: "Jafari, Mahsa; Persaud, Bhagwant"
+year: 2025
+date: 2025-00-00 2025
+doi: 10.1016/j.aap.2025.108076
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2025.108076"
+zotero_key: GR6VQ68S
+zotero_storage: ASEB5UDE
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Jafari, Mahsa"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+Contents lists available at [ScienceDirect](www.sciencedirect.com/science/journal/00014575)
+
+# Accident Analysis and Prevention
+
+journal homepage: [www.elsevier.com/locate/aap](https://www.elsevier.com/locate/aap)
+
+![](_page_0_Picture_5.jpeg)
+
+![](_page_0_Picture_6.jpeg)
+
+Investigating the influence of socioeconomic factors on the relationships between road characteristics and traffic crash frequency and severity-- A hybrid structural equation modelling − artificial neural networks approach☆
+
+Mahsa Jafari \* , Bhagwant Persaud
+
+*Department of Civil Engineering, Toronto Metropolitan University, 350 Victoria St, Toronto, ON M5B 2K3, Canada*
+
+#### ARTICLE INFO
+
+*Keywords:* Crash severity Traffic safety Structural equation modeling Moderation Artificial neural networks
+
+## ABSTRACT
+
+Traffic crashes result from complex interactions between driver, roadway, and environmental factors, which traditional methods often fail to capture. This paper investigates the influence of road, weather, and socioeconomic factors on traffic crashes, using a two-stage hybrid Structural Equation Modelling (SEM)-Artificial Neural Networks (ANN) approach to capture the complex relationships between these factors and crash intensity, a variable that jointly captures the frequency and severity of crashes. A database from Ohio collector road segments served as the case study in this novel hybrid approach, which utilized SEM to analyze the complex and moderating relationships between different factors and crash intensity. SEM revealed significant relationships between crash intensity and factors such as "Horizontal Curve," "Road" (AADT and surface width index), "Segment Length," "Speed Limit," "Vertical Curve," and "Vehicle Possession." Based on the SEM results, "Vehicle Possession" significantly moderated the relationship between "Horizontal Curve" and crash intensity. In the next step, ANN further identified key predictors, including "Segment Length," "Road," the interactions of "Vehicle Possession-Speed Limit," "Vehicle Possession-Vertical Curve," and "Age-Road." The findings highlight the advantage of the complementary application of linear and nonlinear methods in providing invaluable theoretical and methodological insights for crash data analysis.
+
+## **1. Introduction**
+
+Globally, traffic fatalities are unacceptably high ([Sheykhfard et al.,](#page-6-0)  [2023\)](#page-6-0). According to statistics published by the World Health Organization (WHO), traffic crashes are projected to rank as the fifth leading cause of death in the world by 2030 ([World Health Organization, 2018](#page-6-0)). The WHO revealed that despite various efforts to reduce road crashes, the reported results were not even close to the 50 % reduction target for 2020 ([World Health Organization, 2018\)](#page-6-0). In the United States, for example, 40,990 people were killed in highway crashes in 2023, with a fatality rate of 1.26 per 100 million vehicle miles ([National Highway](#page-5-0)  [Traffic Safety Administration, 2024\)](#page-5-0). These numbers suggest that investigating traffic crashes and evaluating their influencing factors are essential elements of road safety research. Such investigations typically involve analyzing multiple interrelated factors while considering their inherent complexity [\(Haghshenas et al., 2023\)](#page-5-0). Key contributing factors to traffic crashes can be grouped into categories pertaining to driver traits, road conditions, vehicle characteristics, and crash attributes (Chang & [Wang, 2006\)](#page-5-0). Additionally, socioeconomic elements can also play a significant role in explaining traffic collision patterns and disparities across diverse populations ([Liu et al., 2024\)](#page-5-0). While previous literature has explored direct relationships between crashes and socioeconomic factors ([Gehrke et al., 2023; Mahmoudi et al., 2023; Osama](#page-5-0) & [Sayed, 2017; Saleem et al., 2020](#page-5-0)), limited attention has been given to potential mediator or moderator impacts that might be exerted on underlying relationships between crashes and contributing factors.
+
+Various statistical approaches have been employed to study the influence of different factors on crash frequency and severity (See ([Savolainen et al., 2011](#page-6-0)) and (Lord & [Mannering, 2010\)](#page-5-0) for detailed reviews.); these encompass regression models, Structural Equation
+
+*E-mail address:* [mahsa.jafari@torontomu.ca](mailto:mahsa.jafari@torontomu.ca) (M. Jafari).
+
+<sup>☆</sup> This article is part of a special issue entitled: 'RSS 2024' published in Accident Analysis and Prevention.
+
+<sup>\*</sup> Corresponding author.
+
+Models (SEM), and machine learning techniques. It is important to note that some of these methodologies (e.g., regressions models) have limitations, including issues related to non-normal distributions of data, over- or under-dispersion, and endogeneity (Lord & [Mannering, 2010;](#page-5-0)  [Savolainen et al., 2011](#page-5-0)), that may influence the outcome, potentially making them biased (Wang & [Wang, 2012](#page-6-0)). SEM offers distinct benefits compared to conventional methods through enhanced capacity to model complex interactions among variables, evaluate latent constructs, concurrently test multiple dependent and independent variables, and accommodate both normal and non-normal datasets [\(Hassan, 2016;](#page-5-0)  [Kashani et al., 2021](#page-5-0)).
+
+Generally, in some SEMs, relationships imply that independent constructs directly influence dependent constructs without any systematic influences from other variables. This assumption, however, is only sometimes valid. The understanding of the nature of direct relationships can change dramatically when a third variable is introduced into the analysis. What is referred to as "moderation" occurs when a third variable (i.e., the moderator) influences the strength or direction of a relationship between two constructs ([Hair et al., 2022\)](#page-5-0). The moderation analysis has been employed in a few studies on transportation [\(Guo](#page-5-0)  [et al., 2023; Haerani et al., 2019; Jing et al., 2023; Taiwo et al., 2024;](#page-5-0)  [Zhang et al., 2019](#page-5-0)). However, there has been little or no emphasis on exploring socioeconomic modifiers affecting both crash frequency and severity.
+
+Despite the strengths of SEM, critics point to its incapacity to account for nonlinear variable impacts ([Lee et al., 2020](#page-5-0)), thereby simplifying the complex decision-making process [\(Leong et al., 2013\)](#page-5-0). To address this issue, artificial neural networks (ANNs), a form of artificial intelligence (AI), can be applied to model complex problems with linear and nonlinear relationships ([Parhi et al., 2022\)](#page-5-0). Thus, it seems of interest to complement linear SEM methods with nonlinear ANN methods (Li´[ebana-Cabanillas et al., 2018](#page-5-0)) in that ANNs can be used to deal with these complex problems by modelling linear and nonlinear relationships. It is, however, inappropriate to use ANN methodology to test hypotheses since it is a black-box approach [\(Leong et al., 2013](#page-5-0)). Consequently, it is always recommended that a linear SEM model be complemented with a nonlinear ANN model (Li´[ebana-Cabanillas et al.,](#page-5-0)  [2018\)](#page-5-0).
+
+Among the notable applications of this hybrid method beyond the traffic safety area are those in the fields of business (Arpaci & [Bahari,](#page-5-0)  2023; Xia & [Zhang, 2022](#page-5-0)), health ([Sun et al., 2022](#page-6-0)), data analysis [\(Leong](#page-5-0)  et al., 2020; Li´[ebana-Cabanillas et al., 2018](#page-5-0)), hospitality and tourism (Xia & [Zhang, 2022\)](#page-6-0), structural engineering [\(Elseufy et al., 2022\)](#page-5-0), and urban planning [\(Zhou et al., 2021\)](#page-6-0). In one study in the latter category, the annoyance caused by traffic noise was assessed using the "two-stage structural equation modelling with artificial neural networks" approach. PLS-SEM was utilized to determine the weight and importance of each latent factor, test hypotheses and evaluate the reliability and validity of measurements. Then, ANN was used to uncover nonlinear relationships within the research model ([Das et al., 2022](#page-5-0)).
+
+To the best of the authors' knowledge, no study has thoroughly analyzed crashes in the traffic safety area using a hybrid method such as SEM-ANN to jointly consider both crash frequency and severity level while analyzing the moderation effect of socioeconomic variables. Typically, socioeconomic variables represent differences in travel behaviour, driving behaviour, and driving capabilities that are important for safety analyses but are not usually incorporated in traffic safety analysis. To help fill this void, the primary goal, and the central novelty, of this study is the investigation of crash frequency and severity using the hybrid SEM-ANN method to tease out influential road and environmental factors and the moderation effects of socioeconomic variables. The amalgamated measure of crash frequency and severity in the context of this investigation is referred to as crash intensity. Thus, the higher the crash frequency and the greater the severity of these crashes, the larger will be the crash intensity. A database of Ohio collector road segments was used as a case study for the investigation. In the process,
+
+the research sought to address the following questions: 1) Which factors significantly affect crash intensity? 2) Do socioeconomic factors significantly moderate the relationships between road characteristics and crash intensity? 3) What are the most important variables in predicting crash intensity, considering nonlinear and more complex relationships? To address these questions, PLS-SEM was first employed to analyze and capture the moderating effect of certain variables. Then, ANN was applied to evaluate the potentially nonlinear relationships between latent variables and crash intensity.
+
+#### **2. Methodology**
+
+As noted above, a two-stage hybrid SEM-ANN approach was adopted. The first stage tested the hypotheses and identified factor scores for the ANN input to the second stage. The Smart-PLS 4.1 software [\(Ringle](#page-5-0)  [et al., 2024\)](#page-5-0) was used to develop the PLS-SEM model. The PLS-SEM algorithm involves combining two models: the measurement model, which links latent variables to their observed variables, and the structural model, which maps how the latent variables relate to each other. The process begins by estimating initial scores for all available latent variables, usually with ordinary regression. It then refines these estimates by calculating how the observed data contribute to each latent variable and by analyzing the relationships between the latent variables. This iterative process ensures that the model maximizes the explained variance in the dependent constructs [\(Hair et al., 2022](#page-5-0)).
+
+The next step after estimating the model parameters was to evaluate the PLS-SEM results. The first stage of evaluating the PLS-SEM model was to assess the measurement models. For this purpose, indicator reliability, internal consistency, convergent validity, and discriminant validity were evaluated for each factor following specifications recommended by Hair et al. ([Hair et al., 2022](#page-5-0)). The indicator reliability criterion states that the factor loadings between the indicators and the construct should be at least 0.708. For internal consistency, Cronbach's alpha is a traditional measure, which should be larger than 0.7. However, Cronbach's alpha assumes that the reliability of all indicators is equal and is sensitive to the number of indicators within each construct. Therefore, the reliability coefficient (*ρA*) was used instead. For internal consistency, values below 0.6 indicate a lack of consistency. Convergent validity measures the share of the variance of the indicators explained by the construct, denoted by Average Variance Extracted (AVE). AVE should be 0.5 or higher, meaning that at least 50 % of the variance is explained by the latent variable. Discriminant validity measures how distinct the construct is from other constructs, which is assessed using the "Fornell-Larcker" criteria. According to the "Fornell-Larcker" criterion, the square root of the Average Variance Extracted (AVE) should be greater than the correlation coefficient with other constructs in the model.
+
+For the second stage, to measure relationships between the predictors and the output, SPSS 29.0 software (IBM [Corp, 2023\)](#page-5-0) was used to develop the ANN model, employing a multi-layer perceptron with a Feedforward-Back Propagation algorithm. To specify the model, all independent and moderator variable scores derived from the preceding SEM analysis formed inputs within the initial layer. At the same time, crash intensity served as the ultimate output target. A 10-fold crossvalidation with a 90:10 training–testing data partition was applied to avoid overfitting [\(Parhi et al., 2022; Sim et al., 2014\)](#page-5-0). The modelling process is shown in [Fig. 1](#page-2-0).
+
+# **3. Data**
+
+The data set, which was assembled for a study conducted by Saleem et al. ([Saleem et al., 2020](#page-5-0)), combines data from three sources: crash and roadway inventory data ([FHWA, 2018](#page-5-0)), climate data [\(National Oceanic](#page-5-0)  [and Atmospheric Administration, 2018\)](#page-5-0), and socioeconomic census data (U.S. [Census Bureau, 2018](#page-6-0)). The data pertain to 6,971 traffic crashes on Ohio collector road segments during a 6-year period. In creating that
+
+<span id="page-2-0"></span>![](_page_2_Figure_2.jpeg)
+
+**Fig. 1.** Conceptual Framework of the study.
+
+dataset, road segments were linked to climate and census data by representing each road segment as a single point at its midpoint; each midpoint was then assigned to the corresponding census-block group polygon that contained it, and a polygon file was used to map each census-block group as a unique geographic area. This process facilitated the spatial matching of road segments with relevant climate and census data ([Saleem et al., 2020\)](#page-5-0). Collector road segments were selected for use in this study because drivers from surrounding neighborhoods most likely use them; as a result, the socioeconomic characteristics of those neighborhoods logically become pertinent.
+
+Table 1 summarizes the descriptive statistics for each variable in the dataset. Descriptive labels for the indicator variables used in the analysis, which are found in the last column, were sourced from the original dataset [\(Saleem et al., 2020](#page-5-0)). According to this Table, over half of the curves and grades fall into the "tangent" and "level" categories. Similarly, more than half of the population hold a diploma or higher degree and earn over \$50,000 annually. Table 1 also summarizes data on the indicators of crash intensity, for which frequency was modelled as the count of fatal plus severe lane departure crashes, while severity was considered as an ordinal variable, with 1 indicating property damage only (PDO) crashes, 2 indicating minor injury crashes, and 3 indicating fatal and severe injury crashes combined.
+
+# **4. Analysis and results**
+
+As depicted in [Fig. 2,](#page-3-0) the SEM model incorporates independent latent variable indicators comprising horizontal curvature, Ln(AADT), surface width, Ln(segment length), speed limits categorized as 40–50 mph and 50–60 mph, vertical grade, rainfall, and temperatures. Furthermore, socioeconomic variables considered as independent and moderator variables consist of age, the percentage of the population holding a higher degree of education, the percentage of household earnings surpassing \$50,000, and the percentage of the population owning more than one vehicle. Ultimately, crash intensity, the integrated measure of crash frequency and severity level, as defined earlier, was considered as the dependent variable.
+
+In developing the SEM model, the construct reliability and validity were evaluated using criteria and threshold values presented earlier for indicator reliability, consistency reliability, convergent validity, and discriminant validity, which were gleaned from the book published by Hair et al. ([Hair et al., 2022](#page-5-0)). Indicator reliability was assessed using the loadings of indicators on the corresponding latent variables. According to [Table 2,](#page-3-0) construct validity is achieved in the model since the absolute loading values are higher than 0.7. In addition, consistency reliability and convergent validity criteria were also met, as the reliability coefficient (*ρA*) and Average Variance Extracted (AVE) exceeded 0.7 and 0.5, respectively [\(Hair et al., 2022](#page-5-0)).
+
+The discriminant validity of the identified constructs was evaluated using the "Fornell-Larcker" criterion. According to [Table 3](#page-3-0), discriminant validity of latent variables was also achieved as the square root of AVE values (the diagonal values) are greater than the inter-construct correlation coefficients values (the non-diagonal values).
+
+Lastly, the structural model was evaluated for its explanatory power
+
+**Table 1**  Descriptive Statistics and Variable Names.
+
+| Variable<br>description                                                             | mean           | std          | Min         | max         | Variable name/<br>category<br>used in the analysis                                    |
+|-------------------------------------------------------------------------------------|----------------|--------------|-------------|-------------|---------------------------------------------------------------------------------------|
+| Segment length<br>(mi)                                                              | 0.58           | 0.94         | 0.001       | 8.03        | Ln(Segment Length)                                                                    |
+| Curve radius (ft)                                                                   | 69.6           | 195.6        | 0           | 1311        | radiusft_moderate<br>(500–1000 ft)<br>radiusft_tangent<br>(>1000 ft and no<br>radius) |
+| Grade percentage                                                                    | 2.22           | 3.78         | 0           | 20          | grade steepness_level<br>(0–3)<br>grade<br>steepness_steep (≥ 6)                      |
+| Annual Average<br>Daily Traffic                                                     | 2132           | 1648         | 92          | 12,360      | Ln(AADT)                                                                              |
+| Surface width (ft)<br>Speed limit (mph)                                             | 20.88<br>52.78 | 2.58<br>5.33 | 16.00<br>25 | 46.00<br>60 | surf_wid<br>speedlimit 40–50<br>mph<br>speedlimit 50–60<br>mph                        |
+| Total average<br>rainfall and<br>snowfall (in)                                      | 69.7           | 21.73        | 48.2        | 158.64      | average<br>precipitation                                                              |
+| Average annual<br>maximum<br>temperature (F)                                        | 61.09          | 2.54         | 57.00       | 69.00       | Tempmaxavg                                                                            |
+| Percentage of<br>population ages<br>25 + with a<br>diploma and<br>university degree | 0.56           | 0.10         | 0.14        | 0.89        | diploma and<br>universityplus                                                         |
+| Percentage of<br>households with<br>income<br>>\$50,000                             | 0.51           | 0.14         | 0.03        | 0.96        | income more than<br>50 k                                                              |
+| Percentage of<br>households with<br>at least one<br>vehicle                         | 0.94           | 0.09         | 0.12        | 1.00        | vehicle 1 and more                                                                    |
+| Percentage of<br>population ages<br>45+                                             | 0.46           | 0.1          | 0.04        | 0.83        | Age45plus                                                                             |
+| Fatal plus severe<br>lane departure<br>crashes                                      | 0.59           | 0.88         | 0           | 10          | LNDP_KAB                                                                              |
+| Minor lane<br>departure<br>crashes                                                  | 0.16           | 0.42         | 0           | 5           | n/a                                                                                   |
+| PDO lane departure<br>crashes                                                       | 1.06           | 1.19         | 0           | 18          | n/a                                                                                   |
+
+using the coefficient of determination (R2 ), the traditional measure of model fit. According to the output of PLS-SEM, the value of R2 was 0.085, indicating a weak explanatory power, possibly due to complex relationships between variables. This outcome suggested the need for further exploration using the hybrid SEM-ANN approach, which is described later.
+
+It is noteworthy that SmartPLS, the software used for PLS-SEM,
+
+<span id="page-3-0"></span>![](_page_3_Figure_2.jpeg)
+
+**Fig. 2.** The Developed SEM Model.
+
+**Table 2**  Loadings, Reliability coefficients and Average Variance Extracted values of constructs.
+
+| Constructs          | Indicators                                           | Loading          | Composite<br>reliability<br>(ρA) | Average<br>variance<br>extracted<br>(AVE) |
+|---------------------|------------------------------------------------------|------------------|----------------------------------|-------------------------------------------|
+| Horizontal<br>curve | Radius ft_tangent<br>Radiusft_moderate               | 0.981<br>− 0.806 | 1.184                            | 0.805                                     |
+| Road                | AADT<br>Surface Width                                | 0.943<br>0.681   | 0.818                            | 0.677                                     |
+| Intensity           | LNDP_KAB<br>Severity                                 | 0.952<br>0.884   | 0.932                            | 0.844                                     |
+| Speedlimit          | speedlimit 40–50<br>speedlimit 50–60                 | − 0.938<br>0.977 | 1.003                            | 0.917                                     |
+| Vertical<br>curve   | Grade steepness<br>level<br>Grade steepness<br>steep | 0.977<br>− 0.957 | 0.971                            | 0.935                                     |
+| Weather             | average<br>precipitation<br>Tempmaxavg               | 0.935<br>− 0.836 | 0.838                            | 0.786                                     |
+
+provides various fit measures, including SRMR, Exact fit criteria d\_ULS and d\_G, NFI, and Chi<sup>2</sup> . Nevertheless, Hair et al. emphasize that PLS-SEM focuses on prediction rather than explanatory modeling. Consequently, fit measures have limited utility in PLS-SEM models and can even be detrimental, as researchers may compromise predictive power in pursuit of a better fit [\(Hair et al., 2022](#page-5-0)).
+
+After confirming the validity and reliability of the constructs, the path coefficients and p-values for all relationships were evaluated, as shown in [Table 4,](#page-4-0) which reveals that six key factors − horizontal curve, road, segment length, speed limits, vertical grade, and vehicle possession − significantly influence crash intensity.
+
+Considering road alignments, the SEM results revealed that decreasing radii are associated with more severe and frequent crashes, i. e., crash intensity (estimate = -0.233, p *<* 0.05); this is consistent with previous studies that indicated that sharper curves are associated with increased severity and frequency [\(Himes et al., 2019; Wu et al., 2017\)](#page-5-0). In terms of vertical grade, the results show that level vertical alignments are associated with increasing crash intensity and that steep grades are related to lower intensity (estimate = -0.091, p = 0.001), which is consistent with the results of Saleem et al. ([Saleem et al., 2020\)](#page-5-0). These results can potentially be attributed to higher operating speeds on straight and level segments. For the road characteristics, the results showed that increasing AADT and surface width (which comprise the "road" variable) are associated with increasing crash intensity (estimate = 0.064, p *<* 0.05). The results also show that higher speed limits are associated with higher intensity (estimate = 0.102, p = 0.002), as indicated by Yu et al. (Yu & [Xu, 2018](#page-6-0)).
+
+Among the socioeconomic factors, vehicle possession positively impacts crash intensity, indicating that an increasing percentage of the population with more than one vehicle per household results in higher crash intensity (estimate = 0.066, p = 0.027), which is consistent with the results of Forrest et al. [\(Forrest et al., 2023](#page-5-0)). Regarding the interactions of socioeconomic variables on road and weather characteristics, the results show that vehicle possession moderates the impact of the horizontal curve on the intensity with a coefficient of − 0.072 (p = 0.04). The results further show that as the percentage of the higherincome population rises, the impact of horizontal grade on intensity
+
+**Table 3**  Discriminant Validity using AVE Square Roots and Correlation Coefficients.
+
+|                    | Age    | Intensity | education | horizontal curve | income | road   | segment length | Spd<br>limt | vehicle possession | vertical curve | weather |
+|--------------------|--------|-----------|-----------|------------------|--------|--------|----------------|-------------|--------------------|----------------|---------|
+| Age                | 1.00   |           |           |                  |        |        |                |             |                    |                |         |
+| Intensity          | 0.00   | 0.92      |           |                  |        |        |                |             |                    |                |         |
+| education          | 0.30   | 0.00      | 1.00      |                  |        |        |                |             |                    |                |         |
+| horizontal curve   | 0.02   | 0.05      | 0.01      | 0.90             |        |        |                |             |                    |                |         |
+| income             | 0.06   | 0.05      | 0.29      | 0.09             | 1.00   |        |                |             |                    |                |         |
+| road               | − 0.01 | 0.09      | 0.02      | 0.19             | 0.22   | 0.82   |                |             |                    |                |         |
+| segment length     | 0.00   | 0.27      | 0.05      | 0.39             | 0.15   | 0.16   | 1.00           |             |                    |                |         |
+| spdlimt            | − 0.01 | 0.07      | 0.07      | 0.02             | 0.03   | − 0.05 | 0.13           | 0.96        |                    |                |         |
+| vehicle possession | 0.36   | 0.03      | 0.50      | 0.02             | 0.21   | − 0.02 | 0.06           | 0.03        | 1.00               |                |         |
+| vertical curve     | − 0.01 | 0.09      | 0.05      | 0.11             | 0.06   | 0.08   | 0.44           | 0.04        | 0.05               | 0.97           |         |
+| weather            | 0.03   | 0.06      | − 0.01    | 0.14             | 0.17   | 0.24   | 0.15           | 0.06        | − 0.10             | 0.04           | 0.89    |
+
+<span id="page-4-0"></span>**Table 4**  Path coefficients of SEM model and their significance.
+
+| Relationships                                       | Coefficient | P values |
+|-----------------------------------------------------|-------------|----------|
+| Age − > Intensity                                   | − 0.013     | 0.691    |
+| education − > Intensity                             | − 0.051     | 0.155    |
+| horizontal curve − > Intensity                      | − 0.233     | <0.001   |
+| income − > Intensity                                | − 0.016     | 0.597    |
+| road − > Intensity                                  | 0.064       | <0.001   |
+| segment length − > Intensity                        | 0.3         | <0.001   |
+| spdlimt − > Intensity                               | 0.102       | 0.002    |
+| vehicle possession − > Intensity                    | 0.066       | 0.027    |
+| vertical curve − > Intensity                        | − 0.091     | 0.001    |
+| weather − > Intensity                               | 0.016       | 0.274    |
+| Age − vertical curve − > Intensity                  | − 0.04      | 0.226    |
+| vehicle possession − vertical curve − > Intensity   | − 0.014     | 0.608    |
+| education − road − > Intensity                      | − 0.009     | 0.57     |
+| income − spdlimt − > Intensity                      | − 0.047     | 0.188    |
+| income − horizontal curve − > Intensity             | 0.022       | 0.532    |
+| education − horizontal curve − > Intensity          | 0.013       | 0.734    |
+| income − vertical curve − > Intensity               | 0.047       | 0.093    |
+| Age − spdlimt − > Intensity                         | − 0.022     | 0.577    |
+| education − spdlimt − > Intensity                   | − 0.001     | 0.99     |
+| vehicle possession − spdlimt − > Intensity          | 0.018       | 0.605    |
+| vehicle possession − horizontal curve − > Intensity | − 0.072     | 0.04     |
+| vehicle possession − road − > Intensity             | 0.013       | 0.354    |
+| income − road − > Intensity                         | 0.024       | 0.069    |
+| education − vertical curve − > Intensity            | 0.032       | 0.315    |
+| Age − horizontal curve − > Intensity                | 0.058       | 0.093    |
+| Age − road − > Intensity                            | − 0.002     | 0.916    |
+
+gets lower. Based on the SEM results, most moderation effects appear negligible and insignificant, likely due to unobserved nonlinearity among relationships. Applying Artificial Neural Networks (ANN) facilitated the exploration of such nonlinear relationships. The ANN investigation is described next.
+
+For the ANN investigation, a 10-fold cross-validation with a 90:10 training–testing data partition was used to avoid overfitting. The 10-fold cross-validation assesses the performance of the ANN model by dividing data into ten subsets. This process helps to reduce the variability in the model performance and provides a more robust evaluation. As shown in Table 5, nine folds train the model in each iteration or network (the training sample), while the remaining fold validates it (the test sample). This process is repeated ten times. The performance metrics of the model are then averaged, providing a robust estimate of its expected performance on unseen data. Root Mean Square Error (RMSE), a measure that reflects the difference between predicted and actual observed values, was then used to evaluate the ANN models as a proxy for prediction accuracy (Li´[ebana-Cabanillas et al., 2017\)](#page-5-0). Ideally, minimal values should be achieved in the RMSE scores attained from train or test samples. Table 5 shows the sample sizes of training and test samples in each iteration and their corresponding RMSE. It can be seen that the
+
+**Table 5**  ANN validation results (RMSE values).
+
+| Network                       | Sample<br>size<br>(Training) | Sample<br>size (Test) | RMSE<br>(Training) | RMSE<br>(Test) | RMSE<br>(Training<br>Test) |
+|-------------------------------|------------------------------|-----------------------|--------------------|----------------|----------------------------|
+| 1                             | 6278                         | 693                   | 0.077              | 0.078          | 0.000                      |
+| 2                             | 6225                         | 746                   | 0.081              | 0.075          | 0.005                      |
+| 3                             | 6215                         | 756                   | 0.078              | 0.079          | 0.001                      |
+| 4                             | 6291                         | 680                   | 0.077              | 0.077          | 0.000                      |
+| 5                             | 6242                         | 729                   | 0.084              | 0.094          | 0.010                      |
+| 6                             | 6274                         | 697                   | 0.077              | 0.077          | 0.000                      |
+| 7                             | 6221                         | 750                   | 0.078              | 0.079          | 0.000                      |
+| 8                             | 6253                         | 718                   | 0.079              | 0.085          | 0.006                      |
+| 9                             | 6272                         | 699                   | 0.079              | 0.075          | 0.005                      |
+| 10                            | 6249                         | 722                   | 0.079              | 0.077          | 0.002                      |
+| Mean                          |                              |                       | 0.079              | 0.080          | 0.003                      |
+| Standard<br>Deviation<br>(SD) |                              |                       | 0.002              | 0.006          | 0.003                      |
+
+RMSE values ranged from 0.07 to 0.09 for the outcome variable. In addition, there is a very slight average difference between training and testing RMSE values in the model (Mean = 0.003, SD = 0.003), further corroborating the substantial relative prediction accuracy and predictive power of the approach used in this study.
+
+Following the optimal ANN architecture configuration, a sensitivity analysis ensued to discern the comparative significance of each variable. Sensitivity analysis in the ANN model shows the normalized importance scores as percentages that represent the variation in the dependent variable with changes in the independent variables associated with it. The sensitivity analysis results are shown in [Table 6. Table 6](#page-5-0) shows the ten most important variables obtained through sensitivity analysis. The normalized ranks from the sensitivity analysis indicate that the segment length is the most important predictor of intensity, followed by "Road" factor (an index of AADT and surface width), "Education-Road", "Vehicle Possession-Vertical Curve", and "Income-Road". These results seem to be intuitively logical and reasonable.
+
+#### **5. Summary and Conclusions**
+
+The paper aimed to apply a novel hybrid technique to predict the intensity of traffic crashes, which is an amalgamated measure of the frequency of crashes and their severity level. To accomplish this objective, a dataset containing crash and roadway inventory data, climate data and socioeconomic census data for Ohio collector road segments was used. A hybrid PLS-SEM-ANN technique was employed to analyze the factors affecting the intensity of traffic crashes. Using SEM, the influence of socioeconomic variables as moderators on the impact of road and weather characteristics on crash intensity was assessed, and, using ANN, the nonlinear complex relationships between independent, moderating and dependent variables were captured. SEM results showed that "Horizontal Curve", "Road", "Speed Limit", "Segment Length", "Vertical Curve", and "Vehicle Possession" have significant linear impacts on crash intensity. Still, most of the moderating effects were insignificant. However, ANN results revealed that the interactions are among the most important predictors of crash predictors, suggesting that using ANN to enrich SEM analyses offers unique insights into model relationships that would otherwise remain undetected.
+
+Combining these complementary approaches can provide a more holistic understanding of contributing factors to traffic crashes, allowing for better managerial decisions. The in-depth analyses resulting from the hybrid approaches described can lead to designing appropriate countermeasures to mitigate crash intensity. Although SEM-ANN modelling may be considered complex, the approach can potentially help safety professionals identify key crash factors, prioritize them, and allocate resources effectively, thereby enabling a more strategic approach to improving traffic safety. In so doing, it offers several benefits, including enhanced accuracy, a deeper understanding of non-linear relationships, greater robustness to outliers, and improved model interpretability through prioritized factor analysis. Nevertheless, there are some limitations to this research. First, this research was only conducted for a database from Ohio; hence, the findings themselves cannot be generalized for application elsewhere. To further extend the scope of the current study, future studies may involve cross-national or cross-cultural perspectives. Second, although several variables are considered in the current study, more comprehensive information would be helpful, for example, more information about neighborhoods, such as religion and ethnicity, driver sociodemographic and behavioral characteristics, and characteristics of the vehicles involved in crashes. Third, a hybrid framework can be developed using more advanced machine learning algorithms to enhance the investigation of factors associated with crashes and their severity.
+
+#### **CRediT authorship contribution statement**
+
+**Mahsa Jafari:** Conceptualization, Data curation, Investigation,
+
+<span id="page-5-0"></span>**Table 6**  ANN Sensitivity Analysis Results.
+
+| Network<br>Variables                | Normalized Importance |      |      |      |      |      |      |      |      |      |                 |
+|-------------------------------------|-----------------------|------|------|------|------|------|------|------|------|------|-----------------|
+|                                     | 1                     | 2    | 3    | 4    | 5    | 6    | 7    | 8    | 9    | 10   | Normalized rank |
+| segment length                      | 100                   | 100  | 100  | 100  | 100  | 100  | 91.8 | 100  | 100  | 100  | 1               |
+| road                                | 46.5                  | 29.2 | 32.2 | 47.3 | 48.3 | 41.4 | 21.9 | 94.9 | 64.1 | 13.7 | 0.44            |
+| vehicle possession − spdlimt        | 18.9                  | 72.2 | 25.8 | 13.5 | 14.8 | 31.8 | 100  | 68.8 | 24.2 | 46.5 | 0.42            |
+| vehicle possession − vertical curve | 10.9                  | 21.5 | 53.1 | 17.2 | 16.6 | 14.2 | 47.7 | 49.2 | 48.1 | 70.2 | 0.35            |
+| Age − road                          | 13.5                  | 36.4 | 47.1 | 15.4 | 13.1 | 15.7 | 10.8 | 62.9 | 50.2 | 83.3 | 0.35            |
+| education − road                    | 21.4                  | 58.6 | 12.0 | 14.7 | 12.4 | 26.8 | 9.30 | 62.2 | 47.9 | 63.9 | 0.33            |
+| income − road                       | 11.9                  | 68.0 | 24.3 | 18.4 | 17.9 | 17.4 | 41.0 | 43.3 | 24.9 | 37.6 | 0.31            |
+| vehicle possession − road           | 24.5                  | 41.0 | 17.2 | 16.6 | 35.4 | 13.2 | 44.9 | 36.8 | 53.6 | 12.8 | 0.29            |
+| income                              | 13.2                  | 21.1 | 17.3 | 23.3 | 34.1 | 26.2 | 7.80 | 67.3 | 26.2 | 43.8 | 0.28            |
+| education- horizontal curve         | 16.9                  | 53.4 | 17.4 | 9.30 | 43.0 | 17.4 | 18.3 | 22.4 | 11.8 | 46.9 | 0.25            |
+
+Methodology, Software, Writing – original draft. **Bhagwant Persaud:**  Conceptualization, Funding acquisition, Supervision, Writing – review & editing.
+
+## **Declaration of Generative AI and AI-assisted technologies in the writing process**
+
+No generative AI or AI-assisted technologies are used in this manuscript.
+
+#### **Declaration of competing interest**
+
+The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
+
+#### **Acknowledgment**
+
+The financial support provided by a discovery grant (Appl. ID RGPIN-2023-03787) from the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged, as is the provision of the data set by the University of North Carolina Highway Safety Research Center.
+
+#### **Appendix A. Supplementary data**
+
+Supplementary data to this article can be found online at [https://doi.](https://doi.org/10.1016/j.aap.2025.108076)  [org/10.1016/j.aap.2025.108076.](https://doi.org/10.1016/j.aap.2025.108076)
+
+## **Data availability**
+
+Data will be made available on request.
+
+# **References**
+
+- [Arpaci, I., Bahari, M., 2023. Investigating the role of psychological needs in predicting](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0005) [the educational sustainability of Metaverse using a deep learning-based hybrid SEM-](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0005)ANN technique. *[Interactive Learning Environments](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0005)* 1–13.
+- [Chang, L.-Y., Wang, H.-W., 2006. Analysis of traffic injury severity: An application of](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0010) [non-parametric classification tree techniques.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0010) *Accident Analysis & Prevention* 38 (5),
+- [Das, C.P., Goswami, S., Das, M., 2022. Prediction of traffic noise induced annoyance of](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0015)  [vendors through noise indices using structural equation modeling: Artificial neural](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0015)
+- network model. *[Environmental Quality Management](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0015)* 32 (2), 29–44. [Elseufy, S.M., Hussein, A., Badawy, M., 2022.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0020) *A Hybrid SEM-ANN Model for Predicting [Overall Rework Impact on the Performance of Bridge Construction Projects](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0020)*. 46, 713–724.
+- FHWA. (2018). *Highway Safety Information System (HSIS)*. https://www.hsisinfo.org. [Forrest, M., Heydari, S., Cherrett, T., 2023. Examining the impact of exposure, built](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0030)  [environment and socio-demographics on pedestrian safety: A case study of Greater](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0030)  London. *[Safety Science](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0030)* 159, 106015.
+- [Gehrke, S.R., Russo, B.J., Huff, M.P., Smaglik, E.J., 2023. Multiscale spatial analysis of](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0035) [macro-level determinants of bicycle crash frequencies in the Phoenix metro region.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0035)  *[Journal of Transportation Safety](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0035) & Security* 1–19.
+
+- [Guo, M., Yuan, Z., Janson, B., Peng, Y., Yue, R., Zhang, G., 2023. Do factors associated](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0040)  [with older pedestrian crash severity differ? A causal factor analysis based on](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0040)  [exposure level of pedestrians.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0040) *Traffic Injury Prevention* 24 (4), 321–330.
+- [Haerani, S., Parmitasari, R.D.A., Aponno, E.H., Aunalal, Z.I., 2019. Moderating effects of](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0045)  [age on personality, driving behavior towards driving outcomes.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0045) *International Journal [of Human Rights in Healthcare](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0045)* 12 (2), 91–104.
+- [Haghshenas, S.S., Guido, G., Vitale, A., Astarita, V., 2023. Assessment of the level of road](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0050)  [crash severity: Comparison of intelligence studies.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0050) *Expert Systems with Applications*  [234, 121118.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0050)
+- [Hair, J.F., Hult, M., Th, G., Ringle, C.M., Sarstedt, M., 2022.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0055) *A Primer on Partial least [squares structural equation modeling \(PLS-SEM\)](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0055)*. (Third). SAGE Publications.
+- [Hassan, H.M., 2016. Investigation of the self-reported aberrant driving behavior of young](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0060)  [male Saudi drivers: A survey-based study.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0060) *Journal of Transportation Safety & Security*  [8 \(2\), 113](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0060)–128.
+- [Himes, S., Porter, R.J., Hamilton, I., Donnell, E., 2019. Safety evaluation of geometric](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0065) [design criteria: Horizontal curve radius and side friction demand on rural, two-lane](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0065)
+- highways. *[Transportation Research Record](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0065)* 2673 (3), 516–525. Corp, I.B.M., 2023. *IBM SPSS Statistics for Windows* [\(Version Version 20.0\) \[Computer](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0070) [software\]. IBM Corp, Armonk, NY.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0070)
+- [Jing, L., Shan, W., Zhang, Y., 2023. Risk preference, risk perception as predictors of risky](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0075)  [driving behaviors: The moderating effects of gender, age, and driving experience.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0075) *[Journal of Transportation Safety](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0075) & Security* 15 (5), 467–492.
+- [Kashani, A.T., Jafari, M., Bondarabadi, M.A., Dabirinejad, S., 2021. Factors affecting the](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0080)  [accident size of motorcycle-involved crashes: A structural equation modeling](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0080)
+- approach. *[International Journal of Injury Control and Safety Promotion](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0080)* 28 (1), 16–21. [Lee, V.-H., Hew, J.-J., Leong, L.-Y., Tan, G.-W.-H., Ooi, K.-B., 2020. Wearable payment: A](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0085)  [deep learning-based dual-stage SEM-ANN analysis.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0085) *Expert Systems with Applications*  [157, 113477.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0085)
+- [Leong, L.-Y., Hew, T.-S., Ooi, K.-B., Wei, J., 2020. Predicting mobile wallet resistance: A](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0090)  [two-staged structural equation modeling-artificial neural network approach.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0090) *[International Journal of Information Management](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0090)* 51, 102047.
+- [Leong, L.-Y., Hew, T.-S., Tan, G.-W.-H., Ooi, K.-B., 2013. Predicting the determinants of](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0095)  [the NFC-enabled mobile credit card acceptance: A neural networks approach.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0095) *Expert [Systems with Applications](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0095)* 40 (14), 5604–5620.
+- Li´[ebana-Cabanillas, F., Marinkovic, V., De Luna, I.R., Kalinic, Z., 2018. Predicting the](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0100) [determinants of mobile payment acceptance: A hybrid SEM-neural network](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0100)  approach. *[Technological Forecasting and Social Change](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0100)* 129, 117–130.
+- Li´ebana-Cabanillas, F., Marinkovi´c, V., Kalini´[c, Z., 2017. A SEM-neural network](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0105) [approach for predicting antecedents of m-commerce acceptance.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0105) *International [Journal of Information Management](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0105)* 37 (2), 14–24.
+- [Liu, J., Das, S., Khan, M.N., 2024. Decoding the impacts of contributory factors and](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0110) [addressing social disparities in crash frequency analysis.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0110) *Accident Analysis & Prevention* [194, 107375.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0110)
+- Lord, D., Mannering, F., 2010. The statistical analysis of crash-frequency data: A review and assessment of methodological alternatives. *Transportation Research Part a: Policy and Practice* 44 (5), 291–305.<https://doi.org/10.1016/j.tra.2010.02.001>.
+- [Mahmoudi, J., Xiong, C., Yang, M., Luo, W., 2023. Modeling the frequency of pedestrian](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0120)  [and bicyclist crashes at intersections: Big data-driven evidence from Maryland.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0120)  *[Transportation Research Record](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0120)* 2677 (3), 1245–1260.
+- National Highway Traffic Safety Administration Early estimate of motor vehicle traffic fatalities in 2023 (DOT HS 813 561) 2024 https://crashstats.nhtsa.dot.gov/Api/ Public/ViewPublication/813561.
+- N. Oceanic A. Administration "National Oceanic and Atmospheric Administration" (website) 2018 https://www.noaa.gov.
+- [Osama, A., Sayed, T., 2017. Evaluating the impact of socioeconomics, land use, built](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0135) [environment, and road facility on cyclist safety.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0135) *Transportation Research Record* 2659 [\(1\), 33](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0135)–42.
+- [Parhi, S., Joshi, K., Wuest, T., Akarte, M., 2022. Factors affecting Industry 4.0](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0140)  adoption–[A hybrid SEM-ANN approach.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0140) *Computers & Industrial Engineering* 168, [108062](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0140).
+- [Ringle, C.M., Wende, S., Becker, J.-M., 2024.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0145) *SmartPLS 4* [Computer software]. SmartPLS, Bonningstedt ¨ [https://www.smartpls.com](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0145).
+- [Saleem, T., Porter, R.J., Srinivasan, R., Carter, D., Himes, S., Le, T., 2020.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0150) *Contributing [Factors for Focus Crash and Facility Types](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0150)*. United States, Federal Highway [Administration.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0150)
+
+- <span id="page-6-0"></span>Savolainen, P.T., Mannering, F.L., Lord, D., Quddus, M.A., 2011. The statistical analysis of highway crash-injury severities: A review and assessment of methodological alternatives. *Accident Analysis & Prevention* 43 (5), 1666–1676. [https://doi.org/](https://doi.org/10.1016/j.aap.2011.03.025)  [10.1016/j.aap.2011.03.025](https://doi.org/10.1016/j.aap.2011.03.025).
+- [Sheykhfard, A., Haghighi, F., Fountas, G., Das, S., Khanpour, A., 2023. How do driving](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0160) [behavior and attitudes toward road safety vary between developed and developing](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0160)  [countries? Evidence from Iran and the Netherlands.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0160) *Journal of Safety Research*.
+- [Sim, J.-J., Tan, G.-W.-H., Wong, J.C., Ooi, K.-B., Hew, T.-S., 2014. Understanding and](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0165)  [predicting the motivators of mobile music acceptance](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0165)–a multi-stage MRA-artificial [neural network approach.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0165) *Telematics and Informatics* 31 (4), 569–584.
+- [Sun, Y., Hamedani, M.F., Javidi, G., Sheybani, E., Hao, F., 2022. Examining COVID-19](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0170) [vaccine attitude using SEM-Artificial Neural Networks approach: A case from Reddit](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0170)  community. *[Health Promotion International](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0170)* 37 (6), daac157.
+- [Taiwo, O.A., Mahmud, N., Hassan, S.A., Mohsin, R.B., 2024. Influence of commercial](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0175) drivers' [risky behavior on accident involvement: Moderating effect of positive](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0175) driving behavior. *[Journal of Engineering and Applied Science](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0175)* 71 (1), 1–19.
+- U.S. Census Bureau. (2018). *"Socioeconomic Census Data" (website)*. https://www.census. gov/data.html.
+
+- Wang, J., Wang, X., 2012. *[Structural equation modeling: Applications using Mplus](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0185)*. John [Wiley](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0185) & Sons.
+- World health organization. (2018). *GLOBAL STATUS REPORT ON ROAD SAFETY*.
+- [Wu, L., Lord, D., Geedipally, S.R., 2017. Developing crash modification factors for](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0195)  [horizontal curves on rural two-lane undivided highways using a cross-sectional](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0195)  study. *[Transportation Research Record](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0195)* 2636 (1), 53–61.
+- [Xia, M., Zhang, Y., 2022. Linear and nonlinear relationships: A hybrid SEM-neural](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0200)  [network approach to verify the links of online experience with luxury hotel](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0200)  branding. *[Journal of Hospitality and Tourism Insights](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0200)* 5 (5), 1062–1079.
+- [Yu, C.-Y., Xu, M., 2018. Local variations in the impacts of built environments on traffic](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0205)  safety. *[Journal of Planning Education and Research](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0205)* 38 (3), 314–328.
+- [Zhang, T., Chan, A.H., Xue, H., Zhang, X., Tao, D., 2019. Driving anger, aberrant driving](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0210)  [behaviors, and road crash risk: Testing of a mediated model.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0210) *International Journal of [Environmental Research and Public Health](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0210)* 16 (3), 297.
+- [Zhou, Y., Zhang, L., Chiaradia, A.J., 2021. An adaptation of reference class forecasting](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0215) [for the assessment of large-scale urban planning vision, a SEM-ANN approach to the](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0215)  [case of Hong Kong Lantau tomorrow.](http://refhub.elsevier.com/S0001-4575(25)00162-9/h0215) *Land Use Policy* 109, 105701.

Zotero/001_artiklid/Road Safety Indicators/jigeographicallyweightedpoissonregression2021.md

@@ -0,0 +1,649 @@
+---
+category: literaturenote
+citekey: jigeographicallyweightedpoissonregression2021
+title: "Geographically weighted poisson regression under linear model of coregionalization assistance: Application to a bicycle crash study"
+authors: "Ji, Shujuan; Wang, Yuanqing; Wang, Yao"
+year: 2021
+date: 2021-00-00 2021
+doi: 10.1016/j.aap.2021.106230
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2021.106230"
+zotero_key: RVGQ646R
+zotero_storage: ZI2S6HDF
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Ji, Shujuan"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+Contents lists available at [ScienceDirect](www.sciencedirect.com/science/journal/00014575)
+
+# Accident Analysis and Prevention
+
+journal homepage: [www.elsevier.com/locate/aap](https://www.elsevier.com/locate/aap)
+
+![](_page_0_Picture_5.jpeg)
+
+![](_page_0_Picture_6.jpeg)
+
+# Geographically weighted poisson regression under linear model of coregionalization assistance: Application to a bicycle crash study
+
+Shujuan Ji , Yuanqing Wang \* , Yao Wang
+
+*Key Laboratory of Transport Industry of Management, Control and Cycle Repair Technology for Traffic Network Facilities in Ecological Security Barrier Area, Chang'an University, P.O. Box 487, Xi'an 710064, China* 
+
+# ARTICLE INFO
+
+*Keywords:*  Semi-parametric GWPR Spatial non-stationary Spatial correlation Bicycle lane
+
+#### ABSTRACT
+
+While cycling benefits individuals and society, cyclists are vulnerable road users, and their safety concerns arouse more macro-level spatial crash studies. Our study intends to investigate the spatial effects of population, land use, and bicycle lane infrastructures on bicycle crashes. This was done by developing a semi-parametric Geographically Weighted Poisson Regression (sGWPR) model which deals with the issue of spatial correlation and spatial non-stationarity simultaneously. It is a model that combines both constant and geographically varying parameters. To determine which parameter is fixed or non-stationary, previous studies suggest monitoring the Akaike Information Criterion (AICc). Yet, relying only on AICc might bury some spatial associations. So, in this study, we propose a Linear Model of Coregionalization (LMC) to assist the decision. Here, we use bicycle crash data across the metropolitan area of Greater Melbourne to establish sGWPR models suggested by AICc and LMC, respectively. Comparing the two sGWPR models, we found the sGWPR model under LMC results performs as well as sGWPR models suggested by AICc from the AICc perspective, and a 22.5% improvement in the mean squared error (MSE). It also uncovers more details about the spatial relationship between bicycle crashes and bicycle lane intersection density (BLID), an effect not suggested under AICc results. The parameters of BLID, a new measurement of bicycle lane facilities proposed by us, vary over space across analysis zones in Greater Melbourne.
+
+### **1. Introduction**
+
+Cycling can deliver health, environmental, economic, and social benefits. Bicycle use in Australia has steadily grown. For example, in Melbourne City and Greater Melbourne, about 41% and 37% of residents participated in cycling respectively, due to the implementation of a series of Bicycle Plan in 2012 ([Melbourne, 2013](#page-14-0)). Despite the benefits of cycling, cyclists are one of vulnerable road users who are exposed to a higher risk of crashes in traffic. In the world, approximately 1.35 million people die each year as a result of road traffic crashes, and 26% of all road traffic deaths are among cyclists, [\(WHO \(World Health Organiza](#page-14-0)[tion\), 2018](#page-14-0)). The high traffic death rate for cyclists impels micro-level and macro-level spatial crash studies. This study focuses on the latter.
+
+An important issue is spatial dependence which is the propensity for nearby locations to influence each other and to possess similar attributes. Bicycle crash occurrence is more likely to be similar at points two meters apart than at points two kilometers apart. Another possible issue is spatial non-stationarity, a condition where the relationships between the independent and dependent variables vary over space
+
+([Fotheringham et al., 2002](#page-13-0)). Ignorance of spatially non-stationarity causes biased parameter estimates and misleading significance tests ([Anselin and Griffith, 1988](#page-13-0)). The Geographically Weighted Poisson Regression (GWPR) technique has been used to fully capture spatial dependence in the residuals and spatial non-stationarity simultaneously in macro-level crash count analysis ([Huang et al., 2018, Ariannezhad](#page-14-0)  [et al., 2020, Li et al., 2013, Xu and Huang, 2015, Zhang et al., 2015,](#page-14-0)  [Gomes et al., 2019, Bao et al., 2017, Briz-Redon et al., 2019, Arvin et al.,](#page-14-0)  [2019, Hezaveh et al., 2019, Yao et al., 2015](#page-14-0)). For example, [Huang et al.](#page-14-0)  [\(2018\)](#page-14-0) used the GWPR model to find that both the strength and direction (either positive or negative relationship) of relationships between crashes and the built environment vary over space.
+
+Semi-parametric Geographically Weighted Poisson Regression (sGWPR) is one of the GWPR techniques. It is a mixed model that combines both geographical varying and constant parameters to accommodate situations in which some parameters may not vary geographically ([Nakaya et al., 2005\)](#page-14-0). Nakaya and his colleagues (2005) suggested the decision whether a parameter varies geographically or not should follow the stepwise regression model selection process by
+
+*E-mail address:* [wyqing@chd.edu.cn](mailto:wyqing@chd.edu.cn) (Y. Wang).
+
+<sup>\*</sup> Corresponding author.
+
+monitoring Akaike Information Criterion (AICc). The comparison is between a full GWPR with all varying parameters and a sGWPR model in which all but one is allowed to vary geographically. Then select the one with a smaller AICc value in each comparison only if the AICc difference between two models is greater than 2. However, model selection or variable selection (it is a variable selection of which the variable's parameter should be fixed or varied) exclusively depending on the AICc might ignore the important details in the spatial relationships ([Ribeiro](#page-14-0)  [et al., 2016\)](#page-14-0).
+
+To overcome this, our study aims to establish an sGWPR technique under Linear Model of Coregionalization (LMC) to provide an insight in appropriate model frameworks for macro-level crash analysis. LMC is a geostatistical tool broadly used in applied geology, ecology, epidemiology and other geosciences [\(Wackernagel, 2003, Pelletier et al., 2008,](#page-14-0)  [Ribeiro et al., 2013, Zawadzki and Badura, 2013\)](#page-14-0) but less used in crash studies. Since it is a convenient tool to estimate the spatial relationship between variables and to quantify the spatial dependence of variables ([Goulard and Voltz, 1992, Isaaks and Srivastava, 1989](#page-14-0)), a LMC estimation, instead of an AICc measure, is applied to the model selection. In addition, under the framework, we also tend to assess bicycle safety implications socio-economic, road network, and traffic characteristics. During the assessment, we develop a new variable, bicycle lane intersection density (BLID) which no previous studies have mentioned, to measure the effect of intersections between bicycle lanes and other types of lanes.
+
+The remainder of this article is organized as follows: A literature review for macro-level bicycle crashes is provided in the second part. The next section describes data and the study area. The GWPR, sGWPR and LMC used in our study are presented in the third part. This is followed by results and discussions. The last section is the conclusion.
+
+## **2. Literature review of cashes analyses at Macro level**
+
+# *2.1. Built environment factors contributing to bicycle crashes*
+
+One of the road characteristic factors is the attribute of bicycle facilities or road intersections, such as intersection density [\(Siddiqui et al.,](#page-14-0)  [2012, Raihan et al., 2019, Osama and Sayed, 2016, Lee et al., 2015](#page-14-0)), bicycle lane length ([Wei and Lovegrove, 2013](#page-14-0)), bicycle lane types ([Morrison et al., 2019](#page-14-0)) and bike network connectivity [\(Zhang et al.,](#page-15-0)  [2015, Osama and Sayed, 2016](#page-15-0)). Some studies found road intersection density and bicycle lane length are all positive to bicycle crashes. [Mor](#page-14-0)[rison et al. \(2019\)](#page-14-0) point out that bicycle lane effect on bicycle risks depends on the bicycle lane types and the roadway conditions or characteristics. For example, the crash odds on roadway sections with bicycle lanes increase (compared to the roadway sections without bicycle lanes) while bicycle lanes were associated with reduced crash odds where bus routes and tram stops were present.
+
+The pattern of land use is an important predictor of crashes. Land use is classified as commercial, residential, industrial etc. Different patterns of commercial use have different effects on bicycle crashes. In studies of [Narayanamoorthy et al. \(2013\) and Osama and Sayed \(2016\),](#page-14-0) the commercial use is insignificant in bicycle crashes. Retail land use is not always significantly associated with an increase in cyclist casualties ([Priyantha Wedagama et al., 2006, Cai et al., 2016](#page-14-0)). However, the commercial use [\(Zhang et al., 2015\)](#page-15-0), big box stores (a retail land use with a building area of 50,000 square feet or more) ([Dumbaugh and Li,](#page-13-0)  [2010\)](#page-13-0) and "log of rooms of hotel, motel, and time shares per square mile" are positive in non-motorist crashes ([Lee et al., 2015](#page-14-0)).
+
+Residential use has a positive effect on bicycle crashes in most studies. The number of dwelling units ([Siddiqui et al., 2012, Osama and](#page-14-0)  [Sayed, 2016, Guo et al., 2018](#page-14-0)) and percentage of residential use [\(Amoh-](#page-13-0)[Gyimah et al., 2016](#page-13-0)) are associated with an increase of cyclist crashes. Only in the study of [Zhang et al. \(2015\)](#page-15-0) was number of dwelling units an insignificant factor. Some studies explored the impact of mixed land use development on bicycle crashes. Different calculations of a land use mix index have different effects: the land use balance index [\(Amoh-Gyimah](#page-13-0)  [et al., 2017\)](#page-13-0) was associated with an increase of bicycle crashes; land use mixture was negatively associated with the increase ([Chen and Shen,](#page-13-0)  [2016\)](#page-13-0).
+
+# *2.2. Other factors contributing to bicycle crashes*
+
+Commonly, bicycle crashes are predicted by socio-economic and demographic factors such as percentage of households without motor vehicles, median household income, and age. [Amoh-Gyimah et al.](#page-13-0)  [\(2016\)](#page-13-0) found that a young population (under 19 years old) is less likely to be involved in bicycle crashes while an elderly population (above 65 years old) has a positive relationship with total and minor-level crashes. [Saha et al. \(2018\)](#page-14-0) discovered that a population aged from 18 to 64 all has a positive association with cyclist crashes. The variable of a household's car ownership is positive with more cycling crahes ([Saha et al.,](#page-14-0)  [2018, Siddiqui et al., 2012](#page-14-0)).
+
+Previous studies have explained the risk or exposure characteristics on macro-level bicycle crash studies. Traffic exposure measures of vehicle traffic volume ([Dumbaugh and Li, 2010, Guo et al., 2018, Zhang](#page-13-0)  [et al., 2015](#page-13-0)), and bike traffic volume ([Saha et al., 2018, Guo et al., 2018](#page-14-0)), or measures of heavy vehicles ([Cai et al. 2016\)](#page-13-0) are significant factors in bicycle crashes. Other significant exposure variables include measurements of commuters with different transportation modes (cycling, transit riding or driving) ([Zhang et al., 2015, Lee et al., 2017, Jacobsen,](#page-15-0)  [2003\)](#page-15-0) and population ([Amoh-Gyimah et al., 2016, Siddiqui et al., 2012](#page-13-0)).
+
+### *2.3. Crash prediction models*
+
+The appropriate analytical modelling approaches and effective explanatory variables are essential to the accuracy of crash prediction. Numerous crash spatial analysis methods have been explored in the previous literature and [Ziakopoulos and Yannis \(2020\)](#page-15-0) discussed those different modelling approaches. Some advanced approaches such as zero inflation [\(Cai et al., 2016, Raihan et al., 2019, Qin et al., 2004](#page-13-0)), autoregressive prior models ([Besag et al., 1991, Zhai et al., 2018b,](#page-13-0)  [Huang and Darwiche, 2010](#page-13-0)), latent class [\(Yasmin and Eluru, 2016,](#page-15-0)  [Heydari et al., 2017, Eluru et al., 2012](#page-15-0)), Bayesian ([Miaou. 2003, Li et al.,](#page-14-0)  [2019, MacNab, 2004](#page-14-0)) and machine learning([Cai et al., 2019, Bao et al.,](#page-13-0)  [2019\)](#page-13-0) have been widely applied to deal with the data problems of excess zeros, spatial autocorrelations, spatial heterogeneity, spatial spillover, and spatiotemporal effects.
+
+Apart from the GWR techniques mentioned in the Introduction, random parameter techniques (e.g. random parameter negative binomial (RPNB)) ([Xu and Huang, 2015, Amoh-Gyimah et al., 2016, Zhai](#page-14-0)  [et al., 2018b, Wang et al., 2017](#page-14-0)) and Bayesian spatially varying coefficient (BSVC) models also can capture the variation relationships over space ([Congdon, 1997, Congdon, 2003, Xu et al., 2017\)](#page-13-0). The two approaches are different from GWR techniques: the former is statistical models drawn only from some mathematical distribution and structures, while the latter assumes local spatial regression models in each spatial unit. From this view, GWR techniques could provide powerful and intuitive descriptions and explanations for researchers and policy makers ([Pirdavani et al., 2014\)](#page-14-0). On the other hand, [Xu and Huang](#page-14-0)  [\(2015\)](#page-14-0) employ RPNB and sGWPR approaches to account for the spatial variation's relationships between the number of crashes and traffic, road, socio-demographic characteristics, and found sGWPR is the more appropriate tool for zonal crash analysis than RPNB in terms of mean absolute and AIC values.
+
+Another issue in crash data may be overdispersion, so the Poissonbased models (e.g., negative binomial (NB)) have been used to deal with this issue. Some studies proved the GWPR has a better model performance than the NB model. Recently, Geographically Weighted Negative Binomial Regression Models (GWNBR) was proposed by [da](#page-13-0)  [Silva and Rodrigues \(2014\)](#page-13-0) for capturing overdispersion. The GWNBR model incorporates the negative binomial distribution instead of the
+
+<span id="page-2-0"></span>Poisson distribution in the GWR framework. [Gomes et al. \(2017\), Tang](#page-14-0)  [et al.\(2020\) and Obelheiro et al.\(2020\)](#page-14-0) applied the GWNBR using % gwnbr macro in SAS [\(da Silva and Rodrigues, 2016\)](#page-13-0) to estimate the spatial effect of traffic exposure, socio-economics, road networks, and land use on crash occurrence.
+
+[Gomes et al. \(2017\)](#page-14-0) compared the performance of the GWPR and GWNBR models. In terms of AICc value, the GWNBR has a better model performance than the GWPR and the NB, while based on the root mean squared error, the GWPR has a more powerful prediction. Further, as mentioned in the Introduction, the sGWPR can fit the situation where some parameters may not vary geographically, but the GWNBR cannot accommodate this kind of situation. Finally, SAS can estimate GWNBR rather than sGWPR, and GWR 4.0 cannot support the estimation of GWNBR but can be used for the estimation of sGWPR instead. Given all these considerations, we focus on the sGWPR results in our study and the GWNBR is ready for future research.
+
+Overall, the literature indicates the built environment is an important factor of bicycle crash prediction. Road networks (especially intersections), bicycle lane infrastructures and land use are the most frequent measurements of a built environment. However, there is a lack of consistent findings between bicycle crashes and land use, especially commercial use. [Siddiqui et al. \(2012\)](#page-14-0) also stated that research highlighting the effects of a built environment on bicycle crashes is limited. Therefore, more research is still needed to capture the interactions
+
+between built environment and bicycle crashes. We presented a new measurement—bicycle lane intersection density, for bicycle lane infrastructure characteristics. This study mainly focuses on the GWPR techniques to explore the spatial effect of commercial use, exposure characteristics and bicycle infrastructures on bicycle crashes.
+
+## **3. Data Description**
+
+# *3.1. Traffic analysis zones*
+
+Based on the 2016 Australian Census by the Australian Bureau of Statistics (ABS, 2017), Great Melbourne consists of 9 Statistical Area level 4 (SA4) zones, 309 Statistical Area level 2 (SA2) zones and 10,849 Statistical Area level 1 (SA1) zones. Comparing with using SA1 as units, SA2 zones are used as spatial analysis units in this study because more bicycle crash data can be aggregated in each unit, and several units without bicycle crashes can be avoided.
+
+# *3.2. Bicycle crash data*
+
+Bicycle crash data from three years (from 2015 to 2017) in Great Melbourne, Australian, were collected[\(VicRoads, 2019a](#page-14-0)). Our study integrates the number of bicycle crashes over three years into SA2s as a dependent variable to investigate the association with contributing
+
+![](_page_2_Figure_11.jpeg)
+
+**Fig. 1.** Study area and spatial distribution of total bicycle crashes (2015–2017).
+
+<span id="page-3-0"></span>factors [\(Amoh-Gyimah et al., 2016, Gomes et al., 2019\)](#page-13-0). We found 51.6% of bicycle crashes occurred in an inner area. The spatial distribution of SA2 zones, an inner area (one of SA4s), and bicycle crashes are shown in [Fig. 1](#page-2-0).
+
+### *3.3. Bicycle lane intersection density (BLID) at macro level*
+
+We have created a new measurement, bicycle lane intersection density (BLID), combining the bicycle lane and road intersection characteristics simultaneously to counter the effect of bicycle lanes on bicycle crashes. Two reasons are provided to support the variable development. First, bicycle lanes are attractive to cyclists and benefit their safety [\(Schepers et al., 2017\)](#page-14-0). Roadway segments with bicycle lanes could decrease crash risks, compared to those without [\(Lusk et al.,](#page-14-0)  [2011\)](#page-14-0). But the effectiveness of bicycle lanes differs according to other roadway characteristics or conditions. [Morrison et al. \(2019\)](#page-14-0) underline bicycle lanes with the roadway conditions or characteristics (i.e., speed limit, bus routes, tram routes, bridges, one-way flow, traffic lane width), which are associated with bicycle crash risks. They, however, did not consider the intersection characteristic, which is one of the important roadway characteristics in the impact of bicycle crashes. Second, the length of bicycle lanes [\(Wei and Lovegrove, 2013\)](#page-14-0) used in bicycle crash analysis did not consider roadway characteristics and could not measure the risks of bicycle lanes when the bicycle lane encounters intersections.
+
+A bicycle lane link is a road link with an on-road bicycle lane or offroad bicycle lane. Bicycle lane intersection (BLI) is the endpoint of a bicycle lane link that connects to other road links. BLID in SA2 *i* is calculated by:
+
+$$BLID_i = \frac{BLI_i}{tLBL_i \times area_i} \tag{1}$$
+
+Where*BLIi*is the number of bicycle lane intersections and*tLBLi*is the total length of bicycle lane links within SA2 *i* and*areai*area of SA2 *i*.
+
+The bicycle lane intersections were identified by three steps based on two shape files: principal bicycle network ([VicRoads, 2016](#page-14-0)) and road network (Open Street Map [\(OSM \(Open Street Map\), 2019\)](#page-14-0)) in Software ArcGIS 9.2:
+
+Step 1: The buffers of bicycle lane links with a 20-meter buffer distance were established using the Buffer tool. The bicycle lane links are
+
+usually drawn on the road central line (shown in Fig. 2) and a 20-meter radius is half of the maximum width of roads in Great Melbourne ([VicRoads, 2019b](#page-14-0)). Therefore, the buffer with a 20 m distance could cover the intersections connecting bicycle lanes and other roads;
+
+Step 2: The point file of road intersections was generated by applying the Feature Vertices To Points tool of GIS to the road network data;
+
+Step 3: Using the Intersect tool, a new point file of bicycle lane intersections was generated by intersecting the point file of road intersections created by Step 2 and the buffers created by Step 1. The bicycle lane intersections might repeat, since buffers were overlapped in the end of bicycle lane links. So, the points data of road intersections was coded with unique numbers. When summarizing the identified bicycle lane intersections within each spatial unit, the repeated bicycle lane intersections were excluded in terms of a unique number. Most SA2s have bicycle lanes. For the 32 SA2s without any bicycle lanes, the values of BLID are equal to zeros.
+
+### *3.4. Land use*
+
+Residential and commercial areas are important trip productions and attractions; the land use is considered as crash generation variable. In this study commercial use is grouped into large-scale commercial use and small-scale commercial use in this study. Land use data was collected from Victorian Resources Online ([VRO \(Victorian Resources](#page-14-0)  [Online\), 2017\)](#page-14-0). A typical size of shops along the roadside or in a community is less than a threshold, 80,000 square feet (about 7432 square meters). About 96% of the commercial areas are less than the threshold and only account for about 33.4% of all commercial use. Thus, the area of the commercial use less than 80,000 square feet is defined as smallscale commercial use in our study.
+
+# *3.5. Other variables*
+
+Socio-economic and risk or exposure variables [\(Table 1\)](#page-4-0) suggested in most prior studies are also considered in this study. Socio-economic data come from the 2016 Census ([ABS \(Australian Bureau of Statistics\).,](#page-13-0)  [2017\)](#page-13-0). Daily vehicle kilometers traveled (DVKT) is calculated by the Annual Average Daily Traffic (AADT) data and the roadway length within each SA2 ([VicRoads, 2019c\)](#page-14-0).
+
+![](_page_3_Picture_17.jpeg)
+
+**Fig. 2.** Identification of bicycle lane intersections.
+
+<span id="page-4-0"></span>**Table 1**  Descriptive statistics of spatial units (309 SA2s).
+
+| Variables           | Description                                         | Min | Max    | Mean  | S.D.  |
+|---------------------|-----------------------------------------------------|-----|--------|-------|-------|
+| Bicycle             | Total number of bicycle                             | 0   | 236.0  | 11.9  | 22.1  |
+| crashes             | crashes                                             |     |        |       |       |
+| BCR                 | Bicycle crash rate per                              | 0   | 1.05   | 0.06  | 0.12  |
+|                     | thousand vehicle<br>kilometers travelled            |     |        |       |       |
+| Socio-economic      |                                                     |     |        |       |       |
+| Per14               | Percentage of                                       | 0.0 | 39.1   | 18.5  | 5.0   |
+|                     | population below 14                                 |     |        |       |       |
+|                     | years old %                                         |     |        |       |       |
+| Per15_24            | Percentage of                                       | 0.0 | 45.9   | 13.1  | 4.4   |
+|                     | population between 15<br>and 24 years old %         |     |        |       |       |
+| Per25_34            | Percentage of                                       | 0.0 | 40.7   | 15.7  | 6.6   |
+|                     | population between 25                               |     |        |       |       |
+|                     | and 34 years old %                                  |     |        |       |       |
+| Per35_44            | Percentage of                                       | 0.0 | 33.3   | 14.3  | 3.0   |
+|                     | population between 35                               |     |        |       |       |
+|                     | and 44 years old %                                  |     |        |       |       |
+| Per45_64            | Percentage of<br>population between 45              | 0.0 | 35.3   | 23.9  | 5.3   |
+|                     | and 64 years old %                                  |     |        |       |       |
+| Per64               | Percentage of                                       | 0.0 | 32.5   | 13.5  | 5.2   |
+|                     | population above 64                                 |     |        |       |       |
+|                     | years old %                                         |     |        |       |       |
+| Mincome             | Mediate personal                                    | 0.0 | 1.6    | 0.7   | 0.2   |
+|                     | income weekly /1000\$                               |     |        |       |       |
+| PerNC               | Percentage of household                             | 0.0 | 75.0   | 8.0   | 8.0   |
+| Risk or exposure    | with no cars%                                       |     |        |       |       |
+| Ln(pop)             | Log of population(pop)                              | 0   | 10.5   | 9.4   | 1.2   |
+| Perbyk              | Percentage of                                       | 0.0 | 33.3   | 0.8   | 2.3   |
+|                     | commuters cycling to                                |     |        |       |       |
+|                     | work %                                              |     |        |       |       |
+| DVKT                | Daily vehicle kilometers                            | 3.0 | 1534.3 | 298.9 | 255.8 |
+|                     | traveled in thousands                               |     |        |       |       |
+| PerPT               | Percentage of<br>commuters to work by               | 0.0 | 36.9   | 11.5  | 8.0   |
+|                     | public transit (Tram,                               |     |        |       |       |
+|                     | transit, and light rail) %                          |     |        |       |       |
+| Road infrastructure |                                                     |     |        |       |       |
+| Road network        | Density of road network                             | 0.5 | 41.2   | 9.0   | 4.0   |
+| density             | [km/km2<br>]                                        |     |        |       |       |
+| Sidewalk            | Density of sidewalk                                 | 0.0 | 60.9   | 1.7   | 4.2   |
+| density             | [km/km2<br>]                                        |     |        |       |       |
+| Residential<br>road | Density of residential<br>road [km/km2<br>]         | 0.1 | 31.8   | 7.1   | 4.1   |
+| density             |                                                     |     |        |       |       |
+| Arterial            | Density of Arterial                                 | 0.0 | 14.1   | 0.7   | 1.2   |
+| Density             | mileage [km/km2<br>]                                |     |        |       |       |
+| Secondary           | Density of Secondary                                | 0.0 | 15.8   | 0.4   | 1.0   |
+| density             | road mileage [km/km2<br>]                           |     |        |       |       |
+| Tertiary            | Density of Tertiary road                            | 0.0 | 14.1   | 1.3   | 1.4   |
+| Density             | mileage [km/km2<br>]                                |     |        |       |       |
+| BLID                | Bicycle lane intersection<br>density [/(km*km2<br>] | 0.0 | 19.2   | 1.5   | 2.1   |
+| Land use            |                                                     |     |        |       |       |
+| PerLC               | Percentage of large                                 | 0.0 | 44.8   | 2.0   | 3.78  |
+|                     | scale commercial area                               |     |        |       |       |
+|                     | [/km2<br>]                                          |     |        |       |       |
+| PerSC               | Percentage of small                                 | 0.0 | 28.7   | 2.0   | 3.33  |
+|                     | scale commercial use                                |     |        |       |       |
+|                     | area[/km2<br>]                                      |     |        |       |       |
+| PerSR               | Percentage of single                                | 0.0 | 74.6   | 37.2  | 15.2  |
+|                     | family residential area<br>[/km2<br>]               |     |        |       |       |
+| PerMR               | Percentage of multiple                              | 0.0 | 25.1   | 3.2   | 3.5   |
+|                     | family residential area                             |     |        |       |       |
+|                     | [/km2<br>]                                          |     |        |       |       |
+| SA2 area            | The area of SA2[km2<br>]                            | 1.3 | 729.8  | 32.3  | 84.1  |
+
+*Note:* All variables in this study are continuous except Crashes. S.D. is standard deviation.
+
+#### **4. Methodology**
+
+We intend to establish a Semi-parametric Geographically Weighted Poisson Regression model (sGWPR) to fit spatial variations in associations between contributing factors and bicycle crashes. The model and the variable selection procedure are shown in Fig. 3. The potential factors were selected after possible multicollinearity identification by Pearson correlation and variance inflation factor (VIF) analysis. Then we put the potential factors into full Geographically Weighted Passion Regression (full GWPR) and monitored the AICc to estimate which best owns the lowest AICc value and all its parameters vary over study area. After that, we tried two approaches to establish the best sGWPR model under the selection of a smaller AICc value and the results of the Linear Model of Coregionalization (LMC), respectively. Finally, we compared the goodness of fit between these two sGWPR models using the Bayesian Information Criterion (BIC) and mean squared error (MSE). GWPR and sGWPR models were performed by the GWR 4.0, and the LMC results are from R 4.0.3 "gstat" package [\(Pebesma, 2004](#page-14-0)).
+
+### *4.1. Linear model of coregionalization (LMC)*
+
+[Nakaya et al. \(2005\)](#page-14-0) point out that because the meaning of covariates may change depending on different geographical contexts, and some relationships might be intrinsically different across space, we need to consider geographical variations in the associations of analysis. In our study, we apply the LMC to explain changed covariates of variables based on an analysis of auto-variogram and cross-variogram between variable pairs.
+
+The general statistical framework of LMC is:
+
+![](_page_4_Figure_10.jpeg)
+
+**Fig. 3.** sGWPR modeling selection procedure. *Note*s: Pearson correlation coefficient larger than 0.6 indicates high multicollinearity ([Evans, 1996](#page-13-0)) ; VIF larger than 5 indicates a significant multicollinearity between these variables [\(Neter](#page-14-0)  [et al., 1996; O](#page-14-0)'brien, 2007; [Xu and Huang, 2015; Hezaveh et al., 2019](#page-14-0)).
+
+<span id="page-5-0"></span>
+$$\gamma(\mathbf{h}) = \begin{bmatrix} \gamma_{11}(h) & \cdots & \gamma_{1p}(h) \\ \vdots & \ddots & \vdots \\ \gamma_{p1}(h) & \cdots & \gamma_{pp}(h) \end{bmatrix} = \sum_{d=1}^{D} C^{d} * g^{d}(\mathbf{h}) \\
+= \sum_{d=1}^{D} \begin{bmatrix} c_{11}^{d} & \cdots & c_{1p}^{d} \\ \vdots & \ddots & \vdots \\ c_{p1}^{d} & \cdots & c_{pp}^{d} \end{bmatrix} * g^{d}(\mathbf{h}) \tag{2}$$
+
+*γ*(*h*)represents a simple variogram matrix of two sets of variables, and it can be calculated by (3):
+
+$$\gamma_{ij}(\mathbf{h}) = \frac{1}{2N(\mathbf{h})} \sum_{(i|\hat{\mathbf{h}})=1}^{N(\mathbf{h})} (x_i(s) - x_i(s+\mathbf{h}))(x_j(s) - x_j(s+\mathbf{h}))$$
+(3)
+
+*h* and *s* are both vectors. If*j* ∕= *i*, *γij*(*h*)is for cross variograms ofXiandXj; if*j* = *i*,*γij*(*h*) is for the auto-variogram of one variableXi. *xi*and*xj*are observations of the variablesXiandXj(i, j = 1,…,*N*(*h*)).*N*(*h*) means the number of SA2s separated by a distance*h*, *s*and*s* + *h*are two SA2s pairs separated by a distance*h*.
+
+The LMC model is defined by*C<sup>d</sup>*(coregionalization matrix) and*g<sup>d</sup>*(*h*)variogram function at spatial scale *d* (d = 1,2…D)*.C<sup>d</sup>*are unknown non-negative matrices and corresponding coefficient matrices of variogram functions at different spatial scale *d. cd ij* is known as partial sill (psill).
+
+In this study, we considered 3 spatial scales (*d* = 0,1,2)—the distance effect without the effect of anisotropy influence. We set a linear combination of three basic functions, nugget effect (4), periodic (5) and Gaussian (6) to fit LMC model and the same basic model must appear in each auto- and cross variogram ([Isaaks and Srivastava, 1989,](#page-14-0) p390).
+
+$$g^{0}(\mathbf{h}) = \begin{cases} 0, & \text{if } \mathbf{h} = 0 \\ 1, & \text{otherwise} \end{cases} \tag{4}$$
+
+$$g^{1}(\boldsymbol{h}) = 1 - \cos(-2\pi \boldsymbol{h}/a) \tag{5}$$
+
+$$g^2(\mathbf{h}) = 1 - \exp(-3\mathbf{h}^2/b^2)$$
+ (6)
+
+*a* and *b* are spatial ranges.
+
+Based on coregionalization matrices*C<sup>d</sup>*, we can calculate overall spatial variance or spatial variability*Qd*in each spatial scale *d* by (2) ([Isaaks and Srivastava, 1989](#page-14-0)). From (7), we could estimate the contribution of each scale to overall variation, which could provide the contribution details of each scale to overall variability.
+
+$$Q^{d} = \sum_{d=0}^{D=2} C^{d} \tag{7}$$
+
+The spatial covariance*Qd xij*between variablesXiandXj(*j* ∕= *i*) can be calculated by summing the partial sills of cross variograms:
+
+$$Q_{x_{ij}}^{d} = \sum_{i=0}^{D-2} c_{x_{ij}}^{d}$$
+ (8)
+
+If*j* = *i*, (8) presents the variance of univariate variableXi, which is the sum of partial sills of Xi. Based on coregionalization matrices, we can decompose spatial variability of each variable and estimate the relative contribution (percentage) to each spatial scale *d.* The ratio of the nugget effect can present the degree of spatial dependence [\(Isaaks and Srivas](#page-14-0)[tava, 1989](#page-14-0)). For example, a LMC model can be exclusively defined by a nugget effect variogram function entailing a complete no spatial dependence (no spatial correlation). The contribution of two variables at each scale *d* could provide the importance of geographical effect in the relationship between explanatory variables and bicycle crashes.
+
+# *4.2. GWPR and sGWPR model*
+
+GWPR, allowing each parameter to vary across study area, can
+
+clearly capture some important local variations in the relationship between bicycle crash counts and explanatory variables with spatial data about socio-economic, traffic, road infrastructure and land use. The model specification of the GWPR:
+
+$$O_i \sim P\widetilde{o}isson\left[E_i exp\left(\beta_0\left(u_i, v_i\right) + \sum_{k=2}^p \beta_k\left(u_i, v_i\right) x_{ik}\right)\right] \tag{9}$$
+
+Where*Oi*is the total crashes took place 2015–2017 in SA2 *i*, (*ui,vi*) is a center coordinate of SA2 *i*. *xik* are explanatory factors. *Ei* is the exposure variable (i.e., daily vehicle kilometers traveled (DVKT) in the current study) of SA2 *i*. *p* is the number of expanding factors; The parameters*βk*(*ui, vi*)at each SA2 can be estimated by solving a maximization problem of the geographically weighted log-likelihood ([Nakaya et al.,](#page-14-0)  [2005\)](#page-14-0):
+
+$$maxL(u_i, v_i) = \sum_{i=1}^{N} (- \hat{O}_j(\widehat{\beta}(u_i, v_i)) + O_j log \hat{O}_j(\widehat{\beta}(u_i, v_i))) \times w_{ij} d_{ij}$$
+(10)
+
+where *Oj* is the number of crashes occurring in SA2 *j*, and *O* (̂*β*(*ui, vi*) is the predicted number of crashes in SA2 *j* at regression point SA2 *i*:
+
+$$\bigwedge_{j}^{\hat{\rho}}(\widehat{\beta}(u_{i}, v_{i}) = E_{j}exp(\beta_{0}(u_{i}, v_{i}) + \sum_{k=0}^{p} \beta_{k}(u_{i}, v_{i})x_{jk})$$
+(11)
+
+*wij*(*j* = 1,2,…,*n*) is the weight of SA2 *j* in calibration of the model for SA2 *i*.*wij*can be calculated by adaptive bi-square as suggested by ([Li](#page-14-0)  [et al., 2013\)](#page-14-0):
+
+$$w_{ij} = \begin{cases} \left(1 - d_{ij}^2 / \theta_{i(n)}^2\right)^2 d_{ij} < \theta_{i(n)}, d_{ij} > \theta_{i(n)} \\ 0 \text{ otherwise} \end{cases}$$
+ (12)
+
+Here,*dij*represents the Euclidean distance between SA2 *i* and SA2 *j*; *θi*(*n*)is an adaptive bandwidth size defined as the longest distance between SA2 *i* and SA2 *n*. *n* is number of near SA2s incorporated into the model in calibration for SA2 *i*.
+
+An important extension of GWPR is semi-parametric Geographically Weighted Poisson Regression (sGWPR). The model specification is:
+
+$$O_i \sim Poisson \left[ \mathbb{E}_i exp(\beta_0(u_i, v_i) + \sum_{j=2}^l \beta_j x_{ij} + \sum_{k=l+1}^p \beta_k(u_i, v_i) x_{ik}) \right] \tag{13}$$
+
+where*βj*the *j*th global parameter, which indicates no significant geographical variation across the study area. We described the selection of variable whose parameter is global in the Introduction section and more details can be found in [Nakaya et al.\(2005\).](#page-14-0)
+
+The Akaike information criterion (AIC) can evaluate prediction error and thereby it also can be a model selection indicator. When the sample size is small, however, AIC would overfit: it could select models with too many parameters. To iron out such potential overfitting, the secondorder AIC—AICc is recommended in this study. More details can be found in [Nakaya et al. \(2005\).](#page-14-0)
+
+### *4.3. Model performance assessment*
+
+The goodness of fit of the GWR models was assessed through the Akaike information criterion (AICc) ([Fotheringham et al., 2002, Gomes](#page-13-0)  [et al., 2019, Hezaveh et al., 2019](#page-13-0)). Unlike the AIC, the Bayesian Information Criterion (BIC) penalizes the model more for its complexity. The lower the BIC, the less complex the model. The BIC is calculated as follow:
+
+$$BIC = -2LL + log(N)K (14)$$
+
+where *LL* is the log-likelihood of the model, *N* is the number of SA2s, *K* is the number of parameters in the model.
+
+Mean Squared Error (MSE), assessing the estimation error, is typi-
+
+cally used in other traffic crash studies for model comparison [\(Arvin](#page-13-0)  [et al., 2019, Amoh-Gyimah et al., 2016, Lu et al., 2014](#page-13-0)). We also used MSE to compare the performance between sGWPR models under monitoring AICc results and under LMC results, respectively. The smaller value of MSE indicates better performance of model estimation. MSE has the form:
+
+$$MSE = \frac{\sum_{i=1}^{N} |\hat{y}_{i} - y_{i}|^{2}}{N}$$
+ (15)
+
+where*yi*is the crash count of SA2 *i*, *yi* is the expected crash count of each SA2 *i* obtained by GWR models, and *N* is the number of SA2s.
+
+# **5. Results**
+
+# *5.1. Variable selection and model selection for a full GWPR*
+
+Through Pearson correlation analysis, if the correlation of two variables is higher than 0.6, then the two were maintained in separate models but not together. After the Pearson correlation analysis, we monitored the AICc value, we selected a full GWPR model with smallest AICc = 905.44 [\(Table 3\)](#page-7-0) and incorporated 10 explanatory variables (Table 2). The VIF values range from 1.21 to 2.02 and the Pearson correlations fall into the range of − 0.49 to 0.50, which indicates no significant multicollinearity in our models (shown in Table 2).
+
+# *5.2. Variable selection for sGWPR model with monitoring AICc values*
+
+When applying GWPR models, we tried adaptive bi-square to determine the best bandwidth. The best bandwidth from the adaptive bisquare method represents the number of SA2s which were incorporated to estimate the GWPR models. Now, we determine which parameter of variable should vary geographically by comparing the AICc values of two models: one is the full GWPR model (M0), another is a sGWPR model in which all parameters except one vary geographically (M1- M10). The comparison is shown in [Table 3](#page-7-0).
+
+By comparing the AICc value of the full GWPR model M0 with that of the sGWPR model (M1-M10), we can find a sGWPR model with only invariant Sidewalk density (Δ*AIC*cof M5 = -3.03), Bicycle lane intersection density (BLID,Δ*AIC*cof M7 = -3.81 or PerSC (Δ*AIC*cof M8 = -5.25) would get better than the full GWPR model (AICc of M0 = 905.44) ; the sGWPR model with only one variable constant (M1-M4, M6 and M9-M10) is a worse model, respectively (the AICc ranges from 916.80 to 1145.07). According to the AICc results and the specification of variable choice for the sGWPR model in [Nakaya et al. \(2005\),](#page-14-0) if the sGWPR model with only invariant variable (M1-M10) could reduce the AICc value at least 2 comparing with the AICc value of full GWPR M0, the invariant variable should be incorporated into the final optimal model M11. Thus, the optimal model (M11 with AICc = 896.65 in [Table 3](#page-7-0)) is a semi-parametric one, in which parameters associated with three variables (Sidewalk density, BLID and PerSC) are held constant and other parameters associated with the remaining explanatory variables are
+
+allowed to vary geographically.
+
+## *5.3. Variable selection for sGWPR model with LMC results*
+
+Due to different scale of variable measurements, the scale of psills in a coregionalization matrix from LMC can be dramatically different. That leads to no general metric to estimate the relative spatial variability contribution on overall variability, mentioned in section 4.1. Thus, when estimating a spatial variability contribution, all explanatory variables here are standardized to have zero mean and unit variance in the LMC estimations.
+
+89.1% of bicycle crashes involves vehicles in our dataset. In addition, [Fournier et al. \(2019\)](#page-13-0) claim the bicycle crash rate is related to vehicle and bicycle exposure simultaneously; but we lack bicycle exposure, so Bicycle crashes is dealt with the bicycle crash rate per vehicle per kilometer travelled (BCR) in the LMC estimation. BCR in SA2 *i* is calculated by:
+
+$$BCR_i = \frac{y_i}{E_i} \tag{16}$$
+
+where*yi*is an observed number of bicycle crashes in SA2 *i* and*Ei*is an exposure variable i.e., daily vehicle kilometers traveled (DVKT) of SA2 *i*.
+
+In this study, we applied sample variograms to describe how the spatial continuity changes (*γ*(*h*)) with distance, along with the accompanying details in [Fig. 4](#page-7-0) (dots). We can find the periodic trends are tense and dramatic in our data set, and these trends are different variogram shapes from the general one where the*γ*(*h*)increases slightly and erratically as distance [\(Isaaks and Srivastava, 1989](#page-14-0)).
+
+Due to the periodic behavior in the simple variograms, we introduce the periodic function to capture this behavior. The linear combination of LMC in this study is:
+
+$$\gamma(h) = Nug(0) + Per(40) + Gau(59)$$
+ (17)
+
+*Nug*(0)is the nugget effect,*Per*(40)is a smaller spatial scale where the LMC is fitted with periodic variogram function within a 40 km range, and*Gau*(59)is a larger spatial scale where the LMC is fitted with a Gaussian variogram function within the 40 km and 59 km range. The fitted LMC model is depicted in [Fig. 4](#page-7-0) (continuous solid line). Based on coregionalization matrices, the spatial variability of overall and each variable was decomposed at different scales and presented in the [Table 4](#page-9-0). The higher ratio of the nugget effect can present the lower spatial dependence ([Isaaks and Srivastava, 1989](#page-14-0)), thus a variable with high nugget effect should be invariant.
+
+The first row shows the nugget, periodic and gaussian effects capture at 48.7%, 12.0%, 39.3% of the overall variation respectively. The 51.3% overall variation is explained by a variance appearing in the nearby SA2s up to 59 km. The second row shows that 30.8% of BCR (bicycle crash rate) variability is explained by the nugget effect; the remaining variability, 10.0% is explained by periodic behavior within a smaller scale (up to 40 km), and 59.2% is interpreted by the variance in the range of 40–59 km. BCR variance represents greater spatial dependence, which implies that the GWPR is an appropriate method to capture the variable
+
+**Table 2**  Pearson correlation matrix and VIF for the explanatory variables used in the study.
+
+| Type                        | Pearson correlation |      |        |        |        |      |        |        |      |      | VIF  |
+|-----------------------------|---------------------|------|--------|--------|--------|------|--------|--------|------|------|------|
+| Variables                   | A                   | B    | C      | D      | E      | F    | G      | H      | I    | J    |      |
+| Per14 (A)                   | 1.00                |      |        |        |        |      |        |        |      |      | 1.92 |
+| Mincome (B)                 | − 0.02              | 1.00 |        |        |        |      |        |        |      |      | 1.21 |
+| Ln(pop) (C)                 | 0.31                | 0.12 | 1.00   |        |        |      |        |        |      |      | 1.60 |
+| Perbyk (D)                  | − 0.44              | 0.43 | 0.05   | 1.00   |        |      |        |        |      |      | 1.45 |
+| Sidewalk density (E)        | − 0.27              | 0.21 | 0.04   | 0.33   | 1.00   |      |        |        |      |      | 1.31 |
+| Residential road density(F) | − 0.27              | 0.12 | 0.12   | 0.50   | 0.32   | 1.00 |        |        |      |      | 1.67 |
+| BLID (G)                    | − 0.39              | 0.31 | − 0.02 | 0.31   | 0.33   | 0.49 | 1.00   |        |      |      | 1.91 |
+| PerSC (H)                   | − 0.49              | 0.28 | 0.12   | 0.26   | 0.32   | 0.36 | 0.50   | 1.00   |      |      | 2.02 |
+| PerSR (I)                   | 0.37                | 0.02 | 0.33   | − 0.18 | − 0.20 | 0.16 | − 0.10 | − 0.21 | 1.00 |      | 1.42 |
+| PerMR (J)                   | − 0.22              | 0.17 | 0.18   | 0.21   | 0.06   | 0.40 | 0.31   | 0.26   | 0.10 | 1.00 | 1.32 |
+
+<span id="page-7-0"></span>**Table 3**  Comparison of Negative binomial (NB), Full GWPR (M0) and sGWPR (M1-M12) models.
+
+| Model   | Explanation                         | Best bandwidth size n/θ | AICc    | ΔAICc   | MSE    | BIC     |
+|---------|-------------------------------------|-------------------------|---------|---------|--------|---------|
+| Poisson | Poisson regression                  | None                    | 2202.27 | 1296.83 | 65.56  | 2246.60 |
+| NB      | Negative binomial                   | None                    | 1790.10 | 884.66  | 159.55 | 1830.28 |
+| M0      | Full GWPR                           | 58/56.6 km              | 905.44  | 0       | 36.17  | 1180.47 |
+| M1      | sGWPR_ Per14                        | 110                     | 1145.07 | 239.63  | /      | /       |
+| M2      | sGWPR_Mincome                       | 58                      | 938.28  | 32.84   | /      | /       |
+| M3      | sGWPR_Ln(pop)                       | 47                      | 1093.67 | 188.23  | /      | /       |
+| M4      | sGWPR_Perbyk                        | 54                      | 916.80  | 11.36   | /      | /       |
+| M5      | sGWPR_Sidewalk density              | 57                      | 902.41  | − 3.03  | /      | /       |
+| M6      | sGWPR_Residential road density      | 57                      | 904.83  | − 0.61  | /      | /       |
+| M7      | sGWPR_BLID                          | 57                      | 901.63  | − 3.81  | /      | /       |
+| M8      | sGWPR_PerSC                         | 58                      | 900.19  | − 5.25  | /      | /       |
+| M9      | sGWPR_PerSR                         | 62                      | 955.58  | 50.14   | /      | /       |
+| M10     | sGWPR_PerMR                         | 54                      | 991.37  | 85.93   | /      | /       |
+| M11#    | sGWPR_Sidewalk density, BLID, PerSC | 46/55.1 km              | 896.65  | − 8.79  | 37.34  | 1168.45 |
+| M12#    | sGWPR_Sidewalk density, PerSC       | 54/55.7 km              | 895.98  | − 9.46  | 28.94  | 1163.22 |
+
+*Notes:* sGWPR\_X indicates a semi-parametric geographically weighted Poisson regression model in which only the parameter of the variable X is invariant. Δ*AIC*cis the difference of AICc of a model from that of the full GWPR(M0). M11# is the model under AICc results, while M12# is the model under LMC results. the third column *n/ θ*represents the number of samples incorporated into models and an adaptive bandwidth size defined as the longest distance between two samples (in equation [\(12\)\)](#page-5-0).
+
+![](_page_7_Figure_5.jpeg)
+
+**Fig. 4.** Plots of auto-variograms of variables(dots) and the LMC fitted line. *Note:* The horizonal axes represent distance *h* ranging from 0 to 59 km.
+
+![](_page_8_Figure_2.jpeg)
+
+**Fig. 4.** (*continued*).
+
+variance.
+
+For the explanatory variables, we can see the Sidewalk density and PerSC are more monopolized by the nugget effect (higher ratio over 60%) while remaining variables are less dominated by the nugget effect (lower ratio less than 40%). For the variables with a lower nugget ratio, the spatial dependence of Per14, Mincome and Ln (pop) is more staged in the smaller spatial scale, ranging up to 40 km (over 40%). The spatial dependence of Residential road density, BLID, PerSR (Percentage of
+
+<span id="page-9-0"></span>**Table 4**  Spatial variability contribution (%) of variable on overall variability at each scale.
+
+| Variable                    | Nugget         | Periodic(0,40<br>km) | Gaussian(40 km,59<br>km) |
+|-----------------------------|----------------|----------------------|--------------------------|
+| Overall                     | 48.7           | 12.0                 | 39.3                     |
+| BCR (bicycle crash<br>rate) | 30.8           | 10.0                 | 59.2                     |
+| Per14                       | 29.1(5.0)      | 41.2(41.8)           | 29.7(53.2)               |
+| Mincome                     | 39.6<br>(12.3) | 49.4(65.9)           | 11.0(21.8)               |
+| Ln(pop)                     | 24.4<br>(36.9) | 74.9(29.2)           | 0.7(33.9)                |
+| Perbyk                      | 31.0<br>(34.5) | 37.1(14.9)           | 31.9(50.6)               |
+| Sidewalk density            | 69.2<br>(23.0) | 30.2(65.8)           | 0.6(11.2)                |
+| Residential road<br>density | 33.5<br>(86.0) | 2.5(3.3)             | 64.0(10.7)               |
+| BLID                        | 31.6<br>(36.2) | 6.0(15.2)            | 62.4(48.6)               |
+| PerSC                       | 62.6<br>(12.2) | 9.8(9.2)             | 27.6(78.6)               |
+| PerSR                       | 38.7<br>(16.6) | 11.4(38.0)           | 49.9(45.4)               |
+| PerMR                       | 33.9(7.1)      | 17.9(6.2)            | 48.2(86.7)               |
+
+*Note:* Values in brackets represent spatial variability contribution (%) between explanatory variables at each scale.
+
+single residential area) and PerMR (Percentage of multiple residential area) is mainly presented in the range of 40–59 km (over 48%), and the spatial dependence of Perbyk equally appears in these two spatial scales (37.1% in the smaller spatial scale and 31.9% in the larger spatial scale).
+
+The size of the non-nugget effect (ranging up to a 40 km and range of 40 km-59 km) represents spatial dependence. The lower non-nugget effect is in Sidewalk density and PerSC (30.8% and 37.4%, respectively); the higher non-nugget effect is in the remaining variables: Per14, Mincome, Perbyk, Ln (pop), Residential road density, BLID, PerSR and PerMR (from 61% to 75%). These variables with higher non-nugget effect should be allowed to vary and the ones with lower non-nugget effect should be held constant in the sGWPR model to explain the geographical variation in BCR. From the spatial variances between BCR and explanatory variables (the values in brackets in Table 4), we can find non-nugget effect ranges from 14% (variance with Residential road density) to 95% (variance with Per14).
+
+Thus, based on the nugget effect, we fitted a sGWPR model (M12 in [Table 3\)](#page-7-0) in which the parameters associated with Sidewalk density and PerSC are held constant, and the parameters associated with remaining variables vary spatially.
+
+# **6. Discussion**
+
+# *6.1. Model performance comparation and parameter estimation*
+
+Before the comparison of sGWPR under AICc values and under LMC results, we run the non-spatial Poisson model (Negative binomial model, NB and Poisson model) for comparison. From [Table 3](#page-7-0), the traditional NB and Poisson model perform worser than the GWR models since they have higher AICc, MSE, and BIC values. The worst performance of NB indicates the adaptability of spatial models (GWPR and sGWPR models), which is consistent with many previous studies, such as [Xu and Huang](#page-14-0)  [\(2015\).](#page-14-0)
+
+The current research relied on the AICc value to identify the spatial association in the sGWPR model. Based on AICc results, it suggested a sGWPR in which Sidewalk density, BLID and PerSC are held constant and the remaining variables vary spatially in this study. No evidence suggests that the spatial relationship between BLID and bicycle crashes would vary over space. However, LMC results provide such evidence suggesting the spatial varying association between BLID and bicycle crashes, since 69.4% BLID variability and 63.8% variance between BLID and bicycle crashes can be explained by spatial dependence at two spatial scales (ranging up to 40 km and a range of 40 km-59 km). The difference of M11 and M12 in AICc values is less than 2 [\(Table 3\)](#page-7-0) and that means the M12 (sGWPR under LMC results) does not reduce the performance from AICc perspective. We also applied BIC and MSE to measure model performance of M11 and M12; however, their results show the M12 has lower BIC (1163.22, − 5.23) with less complexity, lower MSE (278.94, − 8.4) than M11 and its accuracy improves by 22.5% ((37.34–28.94/37.34) (see in [Table 3](#page-7-0)).
+
+Table 5 summarizes model estimation results of the Full GWPR(M0) and sGWPR (M11 and M12). In Full GWPR and sGWPR models, where different crash prediction models are developed for each individual SA2, the parameters are described by the mean, minimum (Min), lower quartile (Lower), median, upper quartile (Upr), and maximum (Max) of values.
+
+Two general observations are worthy of notice: [\(1\)](#page-3-0) With respect to the five statistics indicators (Mean, Min, Lower, Median, Upr and Max), the direction (either positive or negative relationship) of relationships between crashes and each explanatory variable across SA2s is consistent in the corresponding counterparts in M0, M11and M12. For example, the parameters of Per14 in M0, M11and M12 are all negative or negative in each SA2, although the signs of a few parameters changed from negative to positive or from positive to negative; [\(2\)](#page-5-0) The varying parameters of each variable in M0 always fall into the range of
+
+**Table 5**  Model estimations of Full GWPR(M0) and sGWPR (M11 and M12).
+
+| M0-Full GWPR      |              |          |        |        |        |        |
+|-------------------|--------------|----------|--------|--------|--------|--------|
+| Variable          | Mean         | Min      | Lower  | Median | Upr    | Max    |
+| Intercept         | − 6.45       | − 20.03  | − 9.11 | − 7.26 | − 3.97 | 10.53  |
+| Per14             | − 1.70       | − 24.95  | − 8.34 | − 0.14 | 3.27   | 25.75  |
+| Mincome           | 1.22         | − 2.91   | 0.18   | 0.97   | 1.97   | 11.54  |
+| Ln(pop)           | 0.10         | − 1.43   | 0.02   | 0.15   | 0.35   | 1.20   |
+| Perbyk            | 29.13        | − 34.87  | 10.25  | 18.87  | 37.97  | 317.08 |
+| Sidewalk density  | 0.03         | − 1.05   | − 0.05 | 0.03   | 0.15   | 0.71   |
+| Residential road  | 0.06         | − 0.32   | 0.02   | 0.06   | 0.12   | 0.28   |
+| density           |              |          |        |        |        |        |
+| BLID              | 0.01         | − 0.74   | − 0.06 | 0.02   | 0.07   | 0.76   |
+| PerSC             | 7.24         | − 42.96  | 2.13   | 6.56   | 13.32  | 35.27  |
+| PerSR             | 0.53         | − 3.09   | − 0.62 | 0.93   | 1.82   | 5.88   |
+| PerMR             | 5.88         | − 15.23  | − 1.40 | 3.32   | 14.06  | 32.31  |
+| M11-sGWPR         |              |          |        |        |        |        |
+| Variable          | Mean         | Min      | Lower  | Median | Upr    | Max    |
+| Intercept         | − 6.39       | − 16.74  | − 9.95 | − 6.83 | − 3.50 | 12.36  |
+| Per14             | − 2.25       | − 28.45  | − 9.29 | − 1.28 | 3.23   | 33.98  |
+| Mincome           | 1.28         | − 5.02   | − 0.04 | 1.11   | 2.28   | 9.79   |
+| Ln(pop)           | 0.10         | − 1.81   | − 0.01 | 0.18   | 0.42   | 0.97   |
+| Perbyk            | 22.48        | − 302.04 | 7.71   | 18.17  | 47.81  | 253.73 |
+| Sidewalk density# | 0.04* (5.83) |          |        |        |        |        |
+| Residential road  | 0.05         | − 0.35   | 0.01   | 0.05   | 0.11   | 0.27   |
+| density           |              |          |        |        |        |        |
+| BLID#             | 0.05* (5.48) |          |        |        |        |        |
+| PerSC#            | 5.08* (8.38) |          |        |        |        |        |
+| PerSR             | 0.56         | − 4.45   | − 0.66 | 0.67   | 1.98   | 6.12   |
+| PerMR             | 5.95         | − 20.45  | − 3.30 | 4.70   | 15.22  | 65.14  |
+| M12- sGWPR        |              |          |        |        |        |        |
+| Variable          | Mean         | Min      | Lower  | Median | Upr    | Max    |
+| Intercept         | − 6.71       | − 21.07  | − 9.47 | − 7.09 | − 3.98 | 10.38  |
+| Per14             | − 1.86       | − 20.65  | − 7.67 | − 0.58 | 2.72   | 29.07  |
+| Mincome           | 1.34         | − 3.33   | 0.10   | 0.98   | 2.16   | 10.58  |
+| Ln(pop)           | 0.13         | − 1.65   | 0.03   | 0.17   | 0.39   | 1.28   |
+| Perbyk            | 27.44        | − 102.55 | 9.75   | 17.95  | 43.63  | 233.29 |
+| Sidewalk density# | 0.02*(4.16)  |          |        |        |        |        |
+| Residential road  | 0.06         | − 0.36   | 0.02   | 0.06   | 0.12   | 0.24   |
+| density           |              |          |        |        |        |        |
+| BLID              | 0.02         | − 0.64   | − 0.07 | 0.03   | 0.11   | 1.20   |
+| PerSC#            | 5.75* (8.90) |          |        |        |        |        |
+| PerSR             | 0.47         | − 3.94   | − 0.75 | 0.65   | 1.80   | 5.61   |
+| PerMR             | 5.94         | − 16.55  | − 1.97 | 3.64   | 13.41  | 36.34  |
+
+*Note*: #The variable holds a constant coefficient across the study area. \*The coefficient is significant at 10% level.
+
+<span id="page-10-0"></span>corresponding counterparts in M11 and M12; similarly, the varying parameters of each variable in M12 always fall into the range of corresponding counterparts in M11. This might indicate as the number of variables with fixed parameters increases, the effects of the variables with varying parameters on crash occurrence estimated by the GWPR technique would be enhanced.
+
+The reversal of relationship is a common counterintuitive signs problem in GWPR ([Chow et al., 2006, Li et al., 2013, Huang et al., 2018](#page-13-0)). One reason may be that counterintuitive signs of variables are insignificant in some zones due to the possible basis of calibrating sGWPR[\(Xu](#page-14-0)  [and Huang, 2015, Munira and Sener, 2020, Hezaveh et al., 2019, Zhai](#page-14-0) 
+
+[et al., 2018a\)](#page-14-0). This can be measured by mapping t-value with 1.65 at a 10% confidence level. For avoiding false positives or negatives and the Simpon's Paradox [\(Tu et al., 2008\)](#page-14-0), we corrected the t-value at a 10% confidence level as proposed by [da Silva and Fotheringham \(2016\)](#page-13-0). *α*can be calculated by equation [\(14\):](#page-5-0)
+
+$$\alpha = \frac{\zeta}{p_e} p \tag{18}$$
+
+*α*is the corrected*α*, *ζ*is the usual*α*, *pe*is the effective number of independent parameters, *p*is the is the number of parameters. For the results of M13, *ζ*,*pe*and*p*is 0.10, 81.73, and 9, so the corrected*α*is 0.011. So we
+
+![](_page_10_Figure_7.jpeg)
+
+**Fig. 5.** Maps of local parameters and the significance of (a) Per14, (b) Mincome, (c) Ln (pop), (d) Perbyk, (e) Residential road density, (f) Bicycle lane intersection density (BLID), (g) Percentage of single-family residential area (PerSR) and (h) Percentage of multiple-family residential area (PerMR) in sGWPR M13.
+
+![](_page_11_Figure_2.jpeg)
+
+**Fig. 5.** (*continued*).
+
+mapped the t-value with 2.575 at the 10% level of confidence instead of the value of 1.65 in [Fig. 5.](#page-10-0) In this study, we only discuss the significant relationships.
+
+After the correction, some significant parameters still exist with reversal signs for a variable. [Hadayeghi et al. \(2010\)](#page-14-0) ascribed the issue to the local multicollinearity, however, effective diagnostic tools have not been developed to examine this issue up to now. Another explanation might be that unobserved factors which affect the traffic conditions are not considered. This explanation for each counterintuitive relationship with bicycle crashes is discussed in section 6.3.
+
+*6.2. Specification of spatial scale analysis in bicycle crash variance by LMC* 
+
+In this study, we explained the bicycle crash variance based on an analysis of auto-variograms and cross-variograms between variable pairs ([Fig. 4\)](#page-7-0). For the spatial scale dependence analysis, we used the Mean Center tool in ArcGIS 10.2 to produce location of mean center of bicycle crashes occurring in each SA2 and considered the distance between mean centers in estimating auto-variograms and crossvariograms. We simplified the distance calculation, and it may make some bias in the estimation of variograms.
+
+We established the Linear Model of Coregionalization (LMC) to fit the variogram and estimated the coregionalization matrix or psill in R 4.0.3 with an R "gstat" package suggested by [Pebesma \(2004\)](#page-14-0). Here we do a succinct introduction for this estimation: first, we fitted each variogram model to a direct or cross variogram (there are 20 basic variogram models in "gstst" package); then each of the partial sill coefficient matrices was approached by its (in a least square sense) closest positive definite matrices; finally, the "best reasonable" LMC was chosen in this way.
+
+There are some practical difficulties in the choice of reasonable variogram models and spatial ranges to fit the spatial patterns. We found the periodic behavior is tense and dramatic in almost variable pairs ([Fig. 4\)](#page-7-0). We applied each variogram function to fit the periodic behavior, and found the periodic function was the reasonable one which could capture this behavior. This periodic behavior appearance may associate with the geographical distribution of population, road facilities and land use, such as urban centers. Applying the *fit. variogram* model in the "gstst" package, spatial range a (equation [\(6\)\)](#page-5-0) in the LMC estimation was determined, and we found the range of 40 km is more reasonable in the LMC estimation. Similarly, in Gaussian function, 59 km was chosen as spatial range b (equation [\(7\)\)](#page-5-0). Apart from this, an adaptive bandwidth size of three models M0, M11, M12 (56.6 km, 55.1 km, 55.7 km, respectively in [Table 4](#page-9-0)) is less than 59 km. Thus, we argued the range up to 59 km could reasonably capture the spatial dependence of variable pairs.
+
+## *6.3. Specification of the sGWPR model with LMC results*
+
+The constant parameters of Sidewalk density and Percentage of small-scale commercial use area (PerSC) are 0.02 (t-value = 4.16) and 2.75 (t-value = 8.90), respectively. We can find as these two variables increase, the number of bicycle crashes increase (significant at 10% level). Higher Sidewalk density potentially means the more walking spaces where there might be more cycling mixing with walking; thus, it is better to separate bicycles and pedestrians reducing the deterrence of real or perceived safety concerns since 60 percent of Victorians are curious about cycling (according to the report of Transport for [Victoria](#page-14-0)  [\(2018\)](#page-14-0).The effect of PerSC on bicycle crashes is in accordance with conclusions of studies of [Dumbaugh and Li \(2010\) and Cai et al. \(2016\)](#page-13-0). Shopping strips or sidewalk alongside arterial roads has intense pedestrian activity, and it is a potential conflict area between bicycles and pedestrians, or even other vehicles.
+
+The results for the varied parameters associated with the remaining explanatory variables in sGWPR model under LMC results is shown in [Fig. 5.](#page-10-0) According to the Metropolitan Transport Plan, the Greater Melbourne is divided into three areas: the inner area (shown in [Fig. 5](#page-10-0)), middle area, and outer area. The significant parameters of Per14 are mainly concentrated in the inner city and partial outer areas (Melbourne West and Mornington Peninsula). The increase in percentage of the young (under 14 years old) decreases the number of bicycle crashes of the inner city or outer area. This is consistent with and supports the findings in which an increase in percentage of the younger would reduce traffic crashes ([Boufous et al., 2012, Ladron](#page-13-0) ´ de Guevara et al., 2004). It is mainly because those under 14 years old do not cycle for commuting as much as other age cohorts [\(Amoh-Gyimah et al., 2016\)](#page-13-0).
+
+The significant parameters of mediate personal weekly income (Mincome) are concentrated in the middle area and part of the outer area while parameters of Mincome are insignificant in the inner area ([Fig. 5](#page-10-0) (b)). From the Australian Bureau of Statistics 2013, we compared the median household income between the inner area and the rest of the area, and found the inner area has a higher median household income. lower income areas have a higher probability of bicycle crashes. Perhaps people with lower income are more likely to cycle. In a higher income area, there appears to be an insignificant relationship between income and bicycle crashes. [Siddiqui et al \(2012\)](#page-14-0). Estimated a Bayesian model of fixed global parameters to accounting for spatial correlation and found median household income has an insignificant effect on bicycle crashes. Maybe such a global method covers the spatially varying relationships. Since all the significant parameters are positive, we claim the Mincome would increase bicycle crash occurrence.
+
+The two exposure variables log of population (Ln (pop)) and percentage of commuters cycling to work (Perbyk) both have a positive and significant association with bicycle crashes in majority SA2s: 94% and 89% SA2s with significant effect are positive ([Fig. 5](#page-10-0) (c) and (d)). Our findings support the previous studies ([Amoh-Gyimah et al., 2016, Kim](#page-13-0)  [et al., 2007\)](#page-13-0).
+
+Residential road density displays a significantly positive effect in more than half of the study area, and 96% of SA2s with significant relationships are positive [\(Fig. 5](#page-10-0) (e)). This result is consistent with the finding of [Dumbaugh and Li \(2010\)](#page-13-0), which states that residential street increases crash incidence because wider rights-of-way in residential streets lead to higher speeds ([Smith and Appleyard, 1981\)](#page-14-0). Research on residential street design reducing bicycle crash occurrence is further needed.
+
+The sGWPR model with AICc measurement does not suggest that the relationship between bicycle lane intersection density (BLID) and bicycle crashes should vary over space. According to the suggestion of our approach, however, the association of BLID and bicycle crashes should vary over space, because the variability of BLID can be explained by 68.4% of spatial variability patterns, and the variance between BLID and bicycle crashes can also be explained by 63.8% of spatial variability. Our approach applies a geostatistical tool capturing details in the spatial variations of associations in bicycle crash analysis, which cannot be captured by only measuring the AICc values. BLID, a newly defined measurement proposed by us, could better explain the bicycle crash occurrence. Intersection density (the ratio of the number of intersections and road length in each unit) is a common variable to explain crash occurrence. We incorporate intersection density, instead of BLID, into the original model M12 and find that it performs worse than the original model M12 (AICc of 913.42 vs 895.98).
+
+BLID is significantly associated with bicycle crashes in the Inner area and the east of Inner area (78% of SA2s with significant relationships) (Shown in [Fig. 5](#page-10-0) (f)). As the number of bicycle lane intersections increases, bicycle crashes are likely to increase. Bicycle lane intersections are potential conflict areas between bicycles and other vehicles. According to the Metropolitan Transport Plan, the government plans to continue building the Principal Bicycle Network (PBN) on arterial roads. We suggest the government should also establish a program to improve visibility using green paving or improve the standards for bicycle facilities by separating cyclists from traffic streams at the intersections of arterial roads where the PBN is built.
+
+In some SA2s which are located in the Mornington Peninsula and the south of the Greater Melbourne, however, BLID is significantly associated with a decrease in bicycle crashes. It seems counterintuitive but is also observed in [Huang et al \(2018\)](#page-14-0) and is reasonable. The reason might be that the data of unobserved factors that affect the traffic or infrastructure condition of intersections in each SA2s were not collected. For example, better traffic control signals may help ease traffic conflicts at intersections to reduce bicycle crashes. Other factors may include intersection configuration or intersection sight distance, etc. More detailed traffic control data and other related data are needed to help confirm these hypotheses. Another possibility is the effect of BLID on bicycle crashes might intrinsically vary across space. For instance, people's preferences and attitudes for crossing intersections might vary over space due to factors which are enormously complex and impossible to identify in the form of a single explanatory variable, such as cycling culture, habits and customs.
+
+The significant effect of PerSR or PerMR on bicycle crashes vary over space both only in strength but also in direction: the relationships are mainly positive. The PerSR (percentage of single-family residential areas) has a positive effect on bicycle crash occurrence in 60.8% of SA2s. 75% of SA2s with significant relationships between PerSR and bicycle <span id="page-13-0"></span>crashes are positive. We observe the distribution of SA2s with significantly positive effect. The majority of these SA2s are concentrated on the Inner area and part of Western Melbourne and Eastern Melbourne (shown in [Fig. 5](#page-10-0) (g)). The PerMR (percentage of multiple-family residential area) shows a positive relationship with bicycle crashes in 67% of SA2s. 72% of SA2s with significantly positive relationships between PerMR and bicycle crashes are positive. These SA2s are mainly located in the Southern Melbourne, shown in [Fig. 5](#page-10-0) (h). These findings support the [Kim and Yamashita \(2002\)](#page-14-0), who showed that crashes occur in areas with mixed residential and commercial land use. We also observe that the PerMR or PerSR has a significant influence on bicycle crashes in a small part of the study area. It might be that there is more occurrence of disconnected residential subdivisions in some SA2s. Disconnected residential subdivisions are reported fewer crashes than residential communities with an accessibility to heavy traffic ([Marks, 1957, Merlin](#page-14-0)  [et al., 2020\)](#page-14-0).
+
+# **7. Conclusion**
+
+Our study of crashes aims to identify the social economic and built environment factors in the relationship with bicycle crashes over a Metropolitan area. We modeled spatial scale patterns by the Linear Model of Coregionalization (LMC) method. Its results provided evidence, combined with the sGWPR approach, to determine what spatial local (its parameters varying) or global (its parameters held constant) association is in the bicycle crash occurrence. We found that there is no performance degradation between GWPR models enhanced by LMC results and under AICc values, but there are yields some insights into spatial association and model specification in crash studies. This model uncovers spatial and significant associations between bicycle crashes and bicycle lane intersection density (BLID) as well as characteristics of traffic exposure, social economics, road networks and land use in Greater Melbourne. The effect of residential road density, percentage of single-family area, percentage of multiple-family area, and BLID positively varies across the space, and Sidewalk density and Percentage of small-scale commercial use area (PerSC) with constant parameters are positively associated with bicycle crashes.
+
+### **8. Limitations**
+
+Due to the limited methods and access to data sources, we must acknowledge two limitations. One is that the Traffic volume data (2017) does not fully cover the crash data (2015–2017). The possible change of traffic flow during the three years might influence the accuracy of estimation. Another limitation is that [Fig. 4](#page-7-0) shows different variogram shapes with tense and dramatic erratic in this study to the general trend with visible increase of *γ*(*h*)with distance, with slight periodic trends ([Isaaks and Srivastava, 1989](#page-14-0)). [Marcotte \(2012\)](#page-14-0) argued that more variogram function structures are needed to fit different variogram shapes in the LMC estimation. That would be our further research.
+
+### **CRediT authorship contribution statement**
+
+**Shujuan Ji:** Conceptualization, Methodology, Software, Writing original draft, Visualization. **Yuanqing Wang:** Supervision, Validation. **Yao Wang:** Visualization, Writing - review & editing.
+
+# **Declaration of Competing Interest**
+
+The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
+
+### **Acknowledgements**
+
+The authors are grateful to Australian Bureau of Statistics (ABS) for
+
+sharing data and we also thank to Runing Ye, PhD, for her contributions at the early stages of this research. The authors are also grateful for the support from National Natural Science Foundation of China (51878062, 51908462), the Natural Science Foundation of Shaanxi Province (2020JM-246), the 111 project of Sustainable Development of Transportation in Western Urban Agglomeration (No. B20035) and the Fundamental Research Funds for the Central Universities, Chang'an University (300102210214).
+
+# **References**
+
+- ABS (Australian Bureau of Statistics). (2017). 2016 census. https://datapacks. censusdata.abs.gov.au/datapacks/.
+- Amoh-Gyimah, R., Saberi, M., Sarvi, M., 2016. Macroscopic modeling of pedestrian and bicycle crashes: A cross-comparison of estimation methods. Accid. Anal. Prev. 93, 147–159. <https://doi.org/10.1016/j.aap.2016.05.001>.
+- Amoh-Gyimah, R., Saberi, M., Sarvi, M., 2017. The effect of variations in spatial units on unobserved heterogeneity in macroscopic crash models. Analytic Methods in Accident Research 13, 28–51. [https://doi.org/10.1016/j.amar.2016.11.001.](https://doi.org/10.1016/j.amar.2016.11.001)
+- [Anselin, L., Griffith, D.A., 1988. Do spatial effects really matter in regression analysis?](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0020)  [Papers in Regional Science 65 \(1\), 11](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0020)–34.
+- Ariannezhad, A., Karimpour, A., Wu, Y.-J., 2020. Incorporating mode choices into safety analysis at the macroscopic level. Journal of Transportation Engineering, Part A: Systems 146. <https://doi.org/10.1061/jtepbs.0000337>.
+- Arvin, R., Kamrani, M., Khattak, A.J., 2019. How instantaneous driving behavior contributes to crashes at intersections: Extracting useful information from connected vehicle message data. Accid. Anal. Prev. 127, 118–133. [https://doi.org/10.1016/j.](https://doi.org/10.1016/j.aap.2019.01.014) [aap.2019.01.014](https://doi.org/10.1016/j.aap.2019.01.014).
+- Bao, J., Liu, P., Ukkusuri, S.V., 2019. A spatiotemporal deep learning approach for citywide short-term crash risk prediction with multi-source data. Accid. Anal. Prev. 122, 239–254. <https://doi.org/10.1016/j.aap.2018.10.015>.
+- Bao, J., Liu, P., Yu, H., Xu, C., 2017. Incorporating twitter-based human activity information in spatial analysis of crashes in urban areas. Accid. Anal. Prev. 106,
+- 358–369. <https://doi.org/10.1016/j.aap.2017.06.012>. Besag, J., York, J., Molli´[e, A., 1991. Bayesian image restoration, with two applications in](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0045)  [spatial statistics. Ann. Inst. Stat. Math. 43 \(1\), 1](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0045)–20.
+- Boufous, S., de Rome, L., Senserrick, T., Ivers, R., 2012. Risk factors for severe injury in cyclists involved in traffic crashes in victoria, australia. Accid. Anal. Prev. 49, 404–409. <https://doi.org/10.1016/j.aap.2012.03.011>.
+- Briz-Redon, A., Martinez-Ruiz, F., Montes, F., 2019. Investigation of the consequences of the modifiable areal unit problem in macroscopic traffic safety analysis: A case study accounting for scale and zoning. Accid. Anal. Prev. 132, 105276 [https://doi.org/](https://doi.org/10.1016/j.aap.2019.105276) [10.1016/j.aap.2019.105276](https://doi.org/10.1016/j.aap.2019.105276).
+- Cai, Q., Abdel-Aty, M., Sun, Y., Lee, J., Yuan, J., 2019. Applying a deep learning approach for transportation safety planning by using high-resolution transportation and land use data. Transportation Research Part A: Policy and Practice 127, 71–85. [https://doi.org/10.1016/j.tra.2019.07.010.](https://doi.org/10.1016/j.tra.2019.07.010)
+- Cai, Q., Lee, J., Eluru, N., Abdel-Aty, M., 2016. Macro-level pedestrian and bicycle crash analysis: Incorporating spatial spillover effects in dual state count models. Accid.
+- Anal. Prev. 93, 14–22. [https://doi.org/10.1016/j.aap.2016.04.018.](https://doi.org/10.1016/j.aap.2016.04.018) Chen, P., Shen, Q., 2016. Built environment effects on cyclist injury severity in automobile-involved bicycle crashes. Accid. Anal. Prev. 86, 239–246. [https://doi.](https://doi.org/10.1016/j.aap.2015.11.002)
+- [org/10.1016/j.aap.2015.11.002.](https://doi.org/10.1016/j.aap.2015.11.002) [Chow, L.F., Zhao, F., Liu, X., Li, M., Ubaka, I., 2006. Transit ridership model based on](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0080) [geographically weighted regression. Transp. Res. Rec. 1972, 105](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0080)–114.
+- [Congdon, P., 1997. Bayesian models for spatial incidence: A case study of suicideusing](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0085) [the bugs program. Health Place 3 \(4\), 229](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0085)–247.
+- [Congdon, P., 2003. Modelling spatially varying impacts of socioeconomicpredictors on](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0090)  [mortality outcomes. J. Geogr. Syst. 5, 161](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0090)–184.
+- da Silva, A.R., Fotheringham, A.S., 2016. The multiple testing issue in geographically weighted regression. Geographical Analysis 48, 233–247. [https://doi.org/10.1111/](https://doi.org/10.1111/gean.12084)  [gean.12084.](https://doi.org/10.1111/gean.12084)
+- da Silva, A.R., Rodrigues, T.C.V., 2014. Geographically weighted negative binomial regression—incorporating overdispersion. Statistics and Computing 24 (5), 769–783. <https://doi.org/10.1007/s11222-013-9401-9>.
+- [da Silva, A.R., Rodrigues, T.C.V., 2016. 2016. A sas macro for geographically weighted](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0105)  [negative binomial regression, AS Global Forum](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0105).
+- Dumbaugh E. and Li W. (2010). Designing for the safety of pedestrians, cyclists, and motorists in urban environments. Journal of the American Planning Association, 77, 69-88.10.1080/01944363.2011.536101.
+- Eluru, N., Bagheri, M., Miranda-Moreno, L.F., Fu, L., 2012. A latent class modeling approach for identifying vehicle driver injury severity factors at highway-railway crossings. Accid. Anal. Prev. 47, 119–127. [https://doi.org/10.1016/j.](https://doi.org/10.1016/j.aap.2012.01.027) [aap.2012.01.027](https://doi.org/10.1016/j.aap.2012.01.027).
+- [Evans, J.D., 1996. Straightforward statistics for the behavioral sciences. Brooks/Cole](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0120) [Publishing Company, Pacific Grove, California.](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0120)
+- [Fotheringham, A.S., Brunsdon, C., Charlton, M., 2002. Geographically weighted](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0125) [regression: The analysis of spatially varying relationships. John Wiley](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0125) & Sons Inc., [New York.](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0125)
+- N. Fournier E. Christofa M.A. Knodler Jr. A mixed methods investigation of bicycle exposure in crash rates Accident Analysis & Prevention 130 2019 54 61 10.1016/j. aap.2017.02.004.
+
+- <span id="page-14-0"></span>Gomes, M., Cunto, F., da Silva, A.R., 2017. Geographically weighted negative binomial regression applied to zonal level safety performance models. Accid. Anal. Prev. 106, 254–261. <https://doi.org/10.1016/j.aap.2017.06.011>.
+- [Gomes, M.M., Pirdavani, A., Brijs, T., Pitombo, C.S., 2019. Assessing the impacts of](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0140) [enriched information on crash prediction performance. Accid. Anal. Prev.](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0140)
+- Goulard, M., Voltz, M., 1992. Linear coregionalization model: Tools for estimation and choice of cross-variogram matrix. Math. Geol. 24, 269–286. [https://doi.org/](https://doi.org/10.1007/bf00893750)  [10.1007/bf00893750](https://doi.org/10.1007/bf00893750).
+- Guo, Y., Osama, A., Sayed, T., 2018. A cross-comparison of different techniques for modeling macro-level cyclist crashes. Accid. Anal. Prev. 113, 38–46. [https://doi.org/](https://doi.org/10.1016/j.aap.2018.01.015)  [10.1016/j.aap.2018.01.015](https://doi.org/10.1016/j.aap.2018.01.015).
+- Hadayeghi, A., Shalaby, A.S., Persaud, B.N., 2010. Development of planning level transportation safety tools using geographically weighted poisson regression. Accid. Anal. Prev. 42, 676–688. <https://doi.org/10.1016/j.aap.2009.10.016>.
+- Heydari, S., Fu, L., Miranda-Moreno, L.F., Jopseph, L., 2017. Using a flexible multivariate latent class approach to model correlated outcomes: A joint analysis of pedestrian and cyclist injuries. Analytic Methods in Accident Research 13, 16–27. [https://doi.](https://doi.org/10.1016/j.amar.2016.12.002) [org/10.1016/j.amar.2016.12.002](https://doi.org/10.1016/j.amar.2016.12.002).
+- Hezaveh, A.M., Arvin, R., Cherry, C.R., 2019. A geographically weighted regression to estimate the comprehensive cost of traffic crashes at a zonal level. Accid. Anal. Prev. 131, 15–24. [https://doi.org/10.1016/j.aap.2019.05.028.](https://doi.org/10.1016/j.aap.2019.05.028)
+- Huang, H.A.-A.M.A., Darwiche, A.L., 2010. County-level crash risk analysis in florida: Bayesian spatial modeling. Transp. Res. Rec. 2148 (1), 27–37. [https://doi.org/](https://doi.org/10.3141/2148-04) [10.3141/2148-04.](https://doi.org/10.3141/2148-04)
+- Huang, Y., Wang, X., Patton, D., 2018. Examining spatial relationships between crashes and the built environment: A geographically weighted regression approach.
+- J. Transp. Geogr. 69, 221–233. [https://doi.org/10.1016/j.jtrangeo.2018.04.027.](https://doi.org/10.1016/j.jtrangeo.2018.04.027) [Isaaks, E.H., Srivastava, R.M., 1989. An introduction to applied geostatistics. Oxford](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0180)
+- [University Press, New York](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0180). Jacobsen L. P. (2003). Safety in numbers: More walkers and bicyclists, safer walking and bicycling[j]. Injury prevention, 2003, 9(3):205-209. Injury Prevention, 9(3):205- 209.
+- [Kim, J.-K., Kim, S., Ulfarsson, G.F., Porrello, L.A., 2007. Bicyclist injury severities in](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0190) bicycle–[motor vehicle accidents. Accid. Anal. Prev. 39 \(2\), 238](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0190)–251.
+- [Kim, K., Yamashita, E., 2002. Motor vehicle crashes and land use: Empirical analysis](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0195)  [from hawaii. Transp. Res. Rec. 1784, 73](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0195)–79.
+- Ladron ´ [de Guevara, F., Washington, S., Oh, J., 2004. Forecasting crashes at the planning](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0200)  [level: Simultaneous negative binomial crash model applied in tucson, arizona.](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0200)  [Transp. Res. Rec. 1897, 191](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0200)–199.
+- Lee, J., Abdel-Aty, M., Cai, Q., 2017. Intersection crash prediction modeling with macrolevel data from various geographic units. Accid. Anal. Prev. 102, 213–226. [https://](https://doi.org/10.1016/j.aap.2017.03.009)  [doi.org/10.1016/j.aap.2017.03.009.](https://doi.org/10.1016/j.aap.2017.03.009)
+- Lee, J., Abdel-Aty, M., Jiang, X., 2015. Multivariate crash modeling for motor vehicle and non-motorized modes at the macroscopic level. Accid. Anal. Prev. 78, 146–154. [https://doi.org/10.1016/j.aap.2015.03.003.](https://doi.org/10.1016/j.aap.2015.03.003)
+- Li, Z., Chen, X., Ci, Y., Chen, C., Zhang, G., 2019. A hierarchical bayesian spatiotemporal random parameters approach for alcohol/drug impaired-driving crash frequency analysis. Analytic Methods in Accident Research 21, 44–61. [https://doi.org/](https://doi.org/10.1016/j.amar.2019.01.002)  [10.1016/j.amar.2019.01.002.](https://doi.org/10.1016/j.amar.2019.01.002)
+- Li, Z., Wang, W., Liu, P., Bigham, J.M., Ragland, D.R., 2013. Using geographically weighted poisson regression for county-level crash modeling in california. Saf. Sci.
+- 58, 89–97. <https://doi.org/10.1016/j.ssci.2013.04.005>. Lu B., Charlton M., Harris P. and Fotheringham A. S. (2014). Geographically weighted regression with a non-euclidean distance metric: A case study using hedonic house price data. International Journal of Geographical Information Science, 28, 660-
+- 681.10.1080/13658816.2013.865739. [Lusk, A.C., Furth, P.G., Morency, P., Miranda-Moreno, L.F., Willett, W.C., et al., 2011.](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0230)  [Risk of injury for bicycling on cycle tracks versus in the street. Injury Prevention. 17](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0230)  [\(2\), 131](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0230)–135.
+- MacNab, Y.C., 2004. Bayesian spatial and ecological models for small-area accident and injury analysis. Accid. Anal. Prev. 36, 1019–1028. [https://doi.org/10.1016/j.](https://doi.org/10.1016/j.aap.2002.05.001) [aap.2002.05.001](https://doi.org/10.1016/j.aap.2002.05.001).
+- [Marcotte, D., 2012. Revisiting the linear model of coregionalization. Springer,](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0240)  [Netherlands.](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0240)
+- [Marks, H., 1957. Subdividing for traffic safety.](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0245) *Traffic Quarterly* 11 (3), 308–325.
+- Melbourne C. o. (2013). Melbourne bicycle account: Cycling census 2013. Merlin, L.A., Cherry, C.R., Mohamadi-Hezaveh, A., Dumbaugh, E., 2020. Residential accessibility's relationships with crash rates per capita. Journal of Transport and Land Use 13, 113–128. <https://doi.org/10.5198/jtlu.2020.1626>.
+- Miaou, S.-P.L.D., 2003. Modeling traffic crash-flow relationships for intersections: Dispersion parameter, functional form, and bayes versus empirical bayes methods. Transp. Res. Rec. 1840 (1), 31–40. <https://doi.org/10.3141/1840-04>.
+- Morrison, C.N., Thompson, J., Kondo, M.C., Beck, B., 2019. On-road bicycle lane types, roadway characteristics, and risks for bicycle crashes. Accid. Anal. Prev. 123, 123–131. <https://doi.org/10.1016/j.aap.2018.11.017>.
+- Munira, S., Sener, I.N., 2020. A geographically weighted regression model to examine the spatial variation of the socioeconomic and land-use factors associated with strava bike activity in austin, texas. J. Transp. Geogr. 88 [https://doi.org/10.1016/j.](https://doi.org/10.1016/j.jtrangeo.2020.102865)  [jtrangeo.2020.102865](https://doi.org/10.1016/j.jtrangeo.2020.102865).
+- Nakaya, T., Fotheringham, A.S., Brunsdon, C., Charlton, M., 2005. Geographically weighted poisson regression for disease association mapping. Stat. Med. 24, 2695–2717.<https://doi.org/10.1002/sim.2129>.
+- [Narayanamoorthy, S., Paleti, R., Bhat, C.R., 2013. On accommodating spatial](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0280) [dependence in bicycle and pedestrian injury counts by severity level. Transportation](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0280)  [Research Part B: Methodological 55, 245](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0280)–264.
+
+- [Neter, J., Kutner, M.H., Nachtsheim, C.J., Wasserman, W., 1996. Applied linear statistical](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0285)  [models. McGraw-Hill Education \(Irwin Chicago\).](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0285)
+- O'brien, R.M., 2007. A caution regarding rules of thumb for variance inflation factors. Qual. Quant. 41, 673–690. <https://doi.org/10.1007/s11135-006-9018-6>.
+- Obelheiro, M.R., da Silva, A.R., Nodari, C.T., Cybis, H.B.B., Lindau, L.A., 2020. A new zone system to analyze the spatial relationships between the built environment and traffic safety. J. Transp. Geogr. 84 [https://doi.org/10.1016/j.jtrangeo.2020.102699.](https://doi.org/10.1016/j.jtrangeo.2020.102699)
+- Osama, A., Sayed, T., 2016. Evaluating the impact of bike network indicators on cyclist safety using macro-level collision prediction models. Accid. Anal. Prev. 97, 28–37. [https://doi.org/10.1016/j.aap.2016.08.010.](https://doi.org/10.1016/j.aap.2016.08.010)
+- OSM (Open Street Map) (2019). Road network. https://www.openstreetmap.org/ #map=15/-37.6859/144.5712,.
+- [Pebesma, E.J., 2004. Multivariable geostatistics in s: The gstat package. Comput. Geosci.](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0310)  [30, 683](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0310)–691.
+- B.D. Pelletier P. Larocque G. Fyles JW, Coregionalization analysis with a drift for multiscale assessment of spatial relationships be- tween ecological variables 2. Estimation of correlations and coeffi- cients of determination Environmental and Ecological Statistics 16 4 2008 467 94.
+- Pirdavani, A., Bellemans, T., Brijs, T., Kochan, B., Wets, G., 2014. Assessing the road safety impacts of a teleworking policy by means of geographically weighted regression method. J. Transp. Geogr. 39, 96–110. [https://doi.org/10.1016/j.](https://doi.org/10.1016/j.jtrangeo.2014.06.021) [jtrangeo.2014.06.021.](https://doi.org/10.1016/j.jtrangeo.2014.06.021)
+- Wedagama D. Priyantha M., Bird R. N. and Metcalfe A. V. The influence of urban landuse on non-motorised transport casualties Accident Analysis & Prevention 38 2006 1049 57 10.1016/j.aap.2006.01.006.
+- Qin, X., Ivan, J.N., Ravishanker, N., 2004. Selecting exposure measures in crash rate prediction for two-lane highway segments. Accid. Anal. Prev. 36, 183–191. [https://](https://doi.org/10.1016/s0001-4575(02)00148-3)  [doi.org/10.1016/s0001-4575\(02\)00148-3](https://doi.org/10.1016/s0001-4575(02)00148-3).
+- Raihan, M.A., Alluri, P., Wu, W., Gan, A., 2019. Estimation of bicycle crash modification factors (cmfs) on urban facilities using zero inflated negative binomial models. Accid. Anal. Prev. 123, 303–313. <https://doi.org/10.1016/j.aap.2018.12.009>.
+- Ribeiro, M.C., Sousa, A.J., Pereira, M.J., 2016. A coregionalization model can assist specification of geographically weighted poisson regression: Application to an ecological study. Spat Spatiotemporal Epidemiol 17, 1–13. [https://doi.org/10.1016/](https://doi.org/10.1016/j.sste.2016.02.001)  [j.sste.2016.02.001](https://doi.org/10.1016/j.sste.2016.02.001).
+- M.P. Ribeiro P. Llop E. Branquinho C. Sousa A.J. Pereira MJ, Multivariate geostatistical methods for analysis of relationships between ecological indicators and environmental factors at multiple spatial scales Ecological Indicators 29 2013 339 47.
+- Saha, D., Alluri, P., Gan, A., Wu, W., 2018. Spatial analysis of macro-level bicycle crashes using the class of conditional autoregressive models. Accid. Anal. Prev. 118, 166–177. <https://doi.org/10.1016/j.aap.2018.02.014>.
+- [Schepers, P., Twisk, D., Fishman, E., Fyhri, A., Jensen, A., 2017. The dutch road to a high](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0355)  [level of cycling safety. Saf. Sci. 92, 264](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0355)–273.
+- Siddiqui, C., Abdel-Aty, M., Choi, K., 2012. Macroscopic spatial analysis of pedestrian and bicycle crashes. Accid. Anal. Prev. 45, 382–391. [https://doi.org/10.1016/j.](https://doi.org/10.1016/j.aap.2011.08.003)
+- Smith D. T. and Appleyard D. (1981). Improving the residential street environment. (report fhwa/rd-81/031). Washington DC: Federal Highway Administration, U.S. Department of Transportation.
+- Tang, J., Gao, F., Liu, F., Han, C., Lee, J., 2020. Spatial heterogeneity analysis of macrolevel crashes using geographically weighted poisson quantile regression. Accid. Anal. Prev. 148, 105833<https://doi.org/10.1016/j.aap.2020.105833>.
+- Y.-K. Tu D. Gunnell MS. G. Simpson's paradox, lord's paradox, and suppression effects are the same phenomenon – the reversal paradox Emerg Themes Epidemiol. 2008.
+- VicRoads (2016). 2016 principal bicycle network data. https://vicroadsopendatavicroadsmaps.opendata.arcgis.com/datasets/principal-bicycle-network-v2.
+- VicRoads (2019a). Crashes last five years. https://services2.arcgis.com/ 18ajPSI0b3ppsmMt/arcgis/rest/services/Crashes\_Last\_Five\_Years/FeatureServer/0.
+- VicRoads (2019b). Road width and number of lanes. https://vicroadsopendatavicroadsmaps.opendata.arcgis.com/datasets/24ccad5c745e4addabfcfb32c400ee83\_ 0/geoservice.
+- VicRoads (2019c). Traffic volume. https://vicdata.vicroads.vic.gov.au/server/rest/ services/Operations\_Traffic/FeatureServer/0.
+- [Victoria, T.f., 2018. Victorian cycling strategy 2018](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0400)–28: Increasing cycling for transport. [Transport for Victoria Department of Economic Development, Jobs, Transport and](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0400) [Resources, Melbourne Victoria http://www.economicdevelopment.vic.gov.au](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0400).
+- VRO (Victorian Resources Online) (2017). Victorian land use information system 2016. https://discover.data.vic.gov.au/dataset/victorian-land-cover-mapping-2016. DOI: 10.4226/92/590aba615a44d.
+- [Wackernagel, H., 2003. Multivariate geostatistics,3rd ed. Springer.](http://refhub.elsevier.com/S0001-4575(21)00261-X/h0410)
+- Wang, J., Huang, H., Zeng, Q., 2017. The effect of zonal factors in estimating crash risks by transportation modes: Motor vehicle, bicycle and pedestrian. Accid. Anal. Prev. 98, 223–231.<https://doi.org/10.1016/j.aap.2016.10.018>.
+- Wei, F., Lovegrove, G., 2013. An empirical tool to evaluate the safety of cyclists: Community based, macro-level collision prediction models using negative binomial regression. Accid. Anal. Prev. 61, 129–137. [https://doi.org/10.1016/j.](https://doi.org/10.1016/j.aap.2012.05.018)  [aap.2012.05.018](https://doi.org/10.1016/j.aap.2012.05.018).
+- WHO (World Health Organization) (2018). Global status report on road safety 2018 summary.Https://www.Who.Int/publications/i/item/9789241565684.
+- Xu, P., Huang, H., 2015. Modeling crash spatial heterogeneity: Random parameter versus geographically weighting. Accid. Anal. Prev. 75, 16–25. [https://doi.org/10.1016/j.](https://doi.org/10.1016/j.aap.2014.10.020)
+- Xu, P., Huang, H., Dong, N., Wong, S.C., 2017. Revisiting crash spatial heterogeneity: A bayesian spatially varying coefficients approach. Accid. Anal. Prev. 98, 330–337. [https://doi.org/10.1016/j.aap.2016.10.015.](https://doi.org/10.1016/j.aap.2016.10.015)
+
+- <span id="page-15-0"></span>Yao, S., Loo, B.P., Lam, W.W., 2015. Measures of activity-based pedestrian exposure to the risk of vehicle-pedestrian collisions: Space-time path vs. Potential path tree methods. Accid. Anal. Prev. 75, 320–332. [https://doi.org/10.1016/j.](https://doi.org/10.1016/j.aap.2014.12.005)  [aap.2014.12.005](https://doi.org/10.1016/j.aap.2014.12.005).
+- Yasmin, S., Eluru, N., 2016. Latent segmentation based count models: Analysis of bicycle safety in montreal and toronto. Accid. Anal. Prev. 95, 157–171. [https://doi.org/](https://doi.org/10.1016/j.aap.2016.07.015) [10.1016/j.aap.2016.07.015](https://doi.org/10.1016/j.aap.2016.07.015).
+- J.F. Zawadzki n., P Badura, H, Estimation of methane content in coal mines using supplementary physical measurements and multivariable geostatistics International Journal of Coal Geology 118 2013 33 44.
+- Zhai, L., Li, S., Zou, B., Sang, H., Fang, X., et al., 2018a. An improved geographically weighted regression model for pm 2.5 concentration estimation in large areas. Atmos. Environ. 181, 145–154. [https://doi.org/10.1016/j.atmosenv.2018.03.017.](https://doi.org/10.1016/j.atmosenv.2018.03.017)
+- Zhai, X., Huang, H., Gao, M., Dong, N., Sze, N.N., 2018b. Boundary crash data assignment in zonal safety analysis: An iterative approach based on data augmentation and bayesian spatial model. Accid. Anal. Prev. 121, 231–237. [https://](https://doi.org/10.1016/j.aap.2018.09.010)  [doi.org/10.1016/j.aap.2018.09.010.](https://doi.org/10.1016/j.aap.2018.09.010)
+- Zhang, Y., Bigham, J., Ragland, D., Chen, X., 2015. Investigating the associations between road network structure and non-motorist accidents. J. Transp. Geogr. 42, 34–47. <https://doi.org/10.1016/j.jtrangeo.2014.10.010>.
+- Ziakopoulos, A., Yannis, G., 2020. A review of spatial approaches in road safety. Accid. Anal. Prev. 135, 105323<https://doi.org/10.1016/j.aap.2019.105323>.

Zotero/001_artiklid/Road Safety Indicators/johnstonbeyondbestpracticeroad2010.md

@@ -0,0 +1,254 @@
+---
+category: literaturenote
+citekey: johnstonbeyondbestpracticeroad2010
+title: Beyond “best practice” road safety thinking and systems management – A case for culture change research
+authors: "Johnston, Ian"
+year: 2010
+date: 2010-00-00 2010
+doi: 10.1016/j.ssci.2009.12.003
+publication: Safety Science
+url: "https://doi.org/10.1016/j.ssci.2009.12.003"
+zotero_key: PR9F7VPX
+zotero_storage: KNXMMH43
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Johnston, Ian"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+Contents lists available at [ScienceDirect](http://www.sciencedirect.com/science/journal/09257535)
+
+# Safety Science
+
+![](_page_0_Picture_4.jpeg)
+
+![](_page_0_Picture_5.jpeg)
+
+# Beyond ''best practice" road safety thinking and systems management – A case for culture change research
+
+# Ian Johnston \*
+
+Monash University Accident Research Centre, Monash University, Victoria, Australia
+
+#### article info
+
+Article history: Received 1 September 2009 Received in revised form 26 November 2009 Accepted 12 December 2009
+
+Keywords: Road use culture Institutional behaviour Safe system model Road safety best practice Road safety management system
+
+#### abstract
+
+Forecasts of an ''explosion" in the level of road trauma as many of the world's most populous nations rapidly motorise have contributed to the urgency of documenting best practice thinking and management systems to improve road safety performance. By examining the success stories of nations with either the lowest fatailty rates per head of population or with the largest recent improvements a number of ''lessons" have emerged. Data-driven problem identification and the development of evidence-driven countermeasure packages formalised in a strategy for effective implementation, with ambitious, quantitative targets and transparent lines of institutional accountability are the critical success factors. Basing this planning upon the cornerstones of the safe system conceptual model constitutes best practice thinking. This paper argues that critical elements of the safe system model are in discord with behavioural mores in the cultures of many western motorised nations and that this hampers the adoption of the most effective safety programs within key institutions and within political systems. A case is made for a systematic examination of car use and safety cultures in order to add a further dimension to best practice learnings. -2009 Elsevier Ltd. All rights reserved.
+
+# 1. Introduction
+
+Most westernmotorised nations have made substantial advances in reducing death and serious injury from road crashes over the last few decades. In some, but by no means in all, the gains have been in the absolute number of fatalities, not just in fatality rates, with substantial international variation in the degree of progress. [Table 1](#page-1-0) illustrates the performance gulf between Australia and the United States of America over the last decade or so. It is an update of data the author has previously published [\(FHWA, 2006](#page-6-0)).
+
+Performance variations of this kind, a rising sense of urgency resulting from a decade-long period of little progress in most nations ([FlA Foundation, 2006\)](#page-6-0) and the implications of the rapid motorisation occurring in some of the most heavily populated nations in the world ([WHO, 2004\)](#page-6-0) appear to have driven an interest in identifying and documenting best practice. Several recent attempts have been made ([WHO, 2004; FHWA, 2006; Koornstra](#page-6-0) [et al., 2002; Wegman et al., 2006; OECD, 2002, 2008](#page-6-0)).
+
+### 2. Which are the best practice road safety management nations?
+
+The European research focussed initially on the ''SUN" countries – Sweden, United Kingdom and the Netherlands [\(Koornstra et al.,](#page-6-0) [2002; Wegman et al., 2006](#page-6-0)) – which, consistently over more than a decade now, have recorded the lowest population-based traffic crash fatality rates among motorised countries.While this is an eminently sensible starting point, the lowest rate does not necessarily imply the existence of best practice road safety thinking and management. [Trinca et al. \(1988\)](#page-6-0), and more recently [Johnston \(2006\),](#page-6-0) examined the difficulties with international comparisons, even among western motorised nations, highlighting the importance of relative level of motorisation, nation size, topography, road user mix and the nature of the prevailing land transport system. Within Australia, for example, the Australian Capital Territory has a markedly lower rate than the SUN countries [\(http://www.infrastruc](http://www.infrastructure.gov.au/roads/safety/road_fatality_statistics)[ture.gov.au/roads/safety/road\\_fatality\\_statistics\)](http://www.infrastructure.gov.au/roads/safety/road_fatality_statistics) but it is small, almost entirely urban, has a higher income level than other States (leading to amore modern vehicle fleet) and has a very high standard road system. These structural factors strongly influence the expected safety performance within that jurisdiction.
+
+The SUN countries have similar motorisation rates (between 540 and 580 vehicles per 1000 population in 2007) and these rates are at the low end of the scale among western motorised nations (Australia and New Zealand have rates over 700 and the USA rate is over 800). Higher motorisation results in higher exposure to risk for most classes of road user and this must partially explain the international differences in rates which, as [Table 2](#page-1-0) shows, have existed for a long time.
+
+In one sense it does not matter whether the lowest rates truly reflect the existence of best practice. The initial SUNflower study
+
+<sup>\*</sup> Tel.: +61 419 350 930. E-mail address: [irj111@bigpond.net.au.](mailto:irj111@bigpond.net.au)
+
+<span id="page-1-0"></span>Table 1 The absolute number of deaths and the population-based death rates from traffic crashes in the United States of America and Australia. Source: [www.infrastruc](http://www.infrastructure.gov.au/roads/safety/road_fatality_statistics)[ture.gov.au/roads/safety/road\\_fatality\\_statistics.](http://www.infrastructure.gov.au/roads/safety/road_fatality_statistics)
+
+|                          | Number of<br>deaths | Death rate per 100,000<br>population |
+|--------------------------|---------------------|--------------------------------------|
+| United States of America |                     |                                      |
+| 1995                     | 41,798              | 15.9                                 |
+| 2007                     | 41,059              | 13.6                                 |
+| Percentage improvement   | 1.8%                | 14.5%                                |
+| Australia                |                     |                                      |
+| 1995                     | 2013                | 11.2                                 |
+| 2007                     | 1603                | 7.6                                  |
+| Percentage improvement   | 20.3%               | 32.1%                                |
+
+Table 2 Road crash fatality rate improvements over time for four European and four ''new world" nations. Source: [www.infrastructure.gov.au/roads/safety/road\\_fatality\\_](http://www.infrastructure.gov.au/roads/safety/road_fatality_statistics) [statistics.](http://www.infrastructure.gov.au/roads/safety/road_fatality_statistics)
+
+|                   |      | Road crash fatality rates per 100,000 population |                         |  |
+|-------------------|------|--------------------------------------------------|-------------------------|--|
+|                   | 1995 | 2007                                             | % Improvement (rounded) |  |
+| European nations  |      |                                                  |                         |  |
+| France            | 15.3 | 7.5                                              | 51%                     |  |
+| Great Britain     | 6.4  | 5.0                                              | 22%                     |  |
+| Netherlands       | 8.6  | 4.3                                              | 50%                     |  |
+| Sweden            | 6.5  | 5.2                                              | 20%                     |  |
+| New world nations |      |                                                  |                         |  |
+| Australia         | 11.2 | 7.6                                              | 32%                     |  |
+| Canada            | 11.4 | 9.1                                              | 20%                     |  |
+| New Zealand       | 15.9 | 10.0                                             | 37%                     |  |
+| USA               | 15.9 | 13.6                                             | 15%                     |  |
+
+examined the interventions applied and the evidence of effectiveness, together with operational issues in application and conceptual methods of approach ([Koornstra et al., 2002](#page-6-0)). Seeking lessons from the safest nations is an excellent starting point.
+
+A supplementary method for identifying potential best practice nations is to identify those with the greatest gains in safety over the last decade or so, the assumption being that these nations have introduced new, effective management systems. Table 2 compares four European nations (including the three SUN countries) and four ''New World" nations. The Netherlands and France stand out among European nations as dramatic improvers along with New Zealand and Australia among the ''new world" nations. Of course, as [Wegman et al. \(2006\)](#page-6-0) point out, the greatest rates of improvement are more likely to be found in those nations with a poorer starting point. Nevertheless, useful road safety management lessons can be found in the experience of the rapid improvers.
+
+### 3. Scope and purpose
+
+This paper explores the lessons from the ''best practice" research and considers their implications in the context of (assumed) prevailing car use and safety cultures, seeking to promote systematic future research to fill an important current void. It is confined to western, motorised nations where the bulk of the relevant research has been done. It should be noted that little scientific work on safety culture in road use has been conducted ([Wiegmann and](#page-6-0) [von Thaden, 2007](#page-6-0)) so the descriptions of extant culture are speculative.
+
+#### 4. The broad culture around road safety
+
+In most western motorised nations from the mid 1950s through the early 1970s road safety strategic thinking and planning was the province of government-appointed coalitions (''councils") of concerned citizens; police forces; and road and traffic, particularly traffic, engineers. Crashes tended to be viewed simplistically as having one primary cause, blameworthy human behaviour, and the countermeasures were reactive, small-scale, isolated, directed largely at symptoms and with a clear tendency to be punitive.
+
+Gradually road crashes became the subject of scientific study. Ironically however, the early in-depth, crash research, while profoundly influencing the thinking of the researchers, appears to have helped perpetuate the simplistic view of the public and the bureaucratic and political decision makers and a generation of road and traffic engineers. The public catchcry was – and still is – that 95% of crashes were caused by error or misbehaviour, which is, of course, the principal finding of these pioneering studies. Not surprisingly in an open loop system where the human is, in effect, the only active element, it was indeed rare to investigate scientifically a crash in which no operator behaviours could be found among the multiple causal elements identified. However, the importance of the interactions between road, traffic and vehicle features and particular types of behaviour never featured in public discussion of the research and rarely appeared in briefings to policymakers. Further, the concept of latent conditions within the road transport system rendering certain types of error more likely had not found its way from the study of industrial safety ([Reason, 1997](#page-6-0)).
+
+Coupled with a widespread ''blame the victim" mindset within communities was a relative apathy towards the extent of death and serious injury on the road. [Williams and Haworth \(2007\)](#page-6-0) offer three explanations: the constant ''drip feed" of daily events ensures the ''road toll" becomes an accepted part of the road transport landscape; the accurate perception that the (personal) risk of a casualty crash on any given trip is remote; and the belief in one's personal ability to control risk, the problem being ''the other driver". While personal safety is valued, community safety is not ([Johnston, 2004](#page-6-0)).
+
+This historical perspective is important to understanding the social context in which attempts to advance strategic thinking have had to struggle with widespread, accepted views that if poor behaviour is the principal cause behaviour change must be the primary answer ([Howard and Sweatman, 2007](#page-6-0)). The prevailing community culture around road safety needs to be explicitly addressed. [Rose \(1992\)](#page-6-0) wrote a superb book over fifteen years ago outlining the strategy of preventive medicine and while road safety was not his primary focus it featured in his illustrative case studies. His thesis – that a relatively low level of individual risk spread relatively uniformly across an entire population requires populationwide measures to achieve an effect at the community level – is a perfect prescription for most road safety problems yet it is still uncommon as a strategy for behaviour change among road users.
+
+## 5. The transport setting and its cultural influence
+
+Road safety has traditionally been an issue for the transport system to address and there was, for a long time, no systematic engagement with either the field of public health or the field of occupational health and safety. Ironically, the World Health Organisation is now playing a lead role in promoting global action as road trauma assumes a more prominent position in the global burden of disease, yet it lacks the networks meaningfully to influence institutional change among the transport players [\(WHO, 2004,](#page-6-0) [2009\)](#page-6-0). Watkins has argued that, in emerging nations, governments ''... measure the success of their roads policies in kilometres of asphalt and the speed at which goods can be carried" and that ''road safety is the poor cousin of road building" [\(Watkins, in Ward 2009\)](#page-6-0).
+
+The efficient movement of people and freight is the over-arching goal of road transport with the level of death, injury and property damage being one of a number of by-products to be managed. This conception of death and injury as an undesirable but necessary by-product of road transport appears to have dominated thinking in most motorised nations in the early days of rapid motorisation, still dominates in some (notably the USA), and is a common characteristic of emerging nations ([Sleet et al., 2007\)](#page-6-0).
+
+Rarely were safety targets set and when they were it was in terms of the rate of death and serious injury (typically death alone) per unit of road use. As road use continues to grow, nations have continuously to improve their road and transport systems to cope with the need for more capacity and the need to minimise delay. Efficiency-inspired improvements in vehicles, roads and traffic management systems contribute to an ongoing improvement in safety per unit road use, over and above the effect of any specific safety measures; for example converting a two lane road to a four lane divided road to meet capacity demands dramatically improves safety by eliminating the risk of head-on crashes. However, if the safety gains do not outweigh the growth in road use then the absolute number of deaths must increase. That has been the case in the USA ([FHWA, 2006](#page-6-0) and [Table 1\)](#page-1-0) where the absolute number of citizens being killed has not meaningfully decreased.
+
+Public policy around road transport in the USA appears to accept that a commensurate price must be paid for what is, by a considerable margin in terms of vehicles per head of population, the highest level of personal mobility in the motorised world. Judgements of priority among competing goals are, of course, the daily business of governments on behalf of their communities. The US public policy around road transport where unfettered personal mobility appears the primary goal implies the absence of a widespread, public constituency for safety ([Kissinger, 2007\)](#page-6-0).
+
+Most western motorised nations have moved from using, as their main performance measure, deaths per unit road use to deaths per head of population; the common public health measure of the burden of disease, in this case the disease of mobility. The move appears to have had two drivers – the need for a measure that facilitated direct comparison with other public health issues and a desire to break from the mindset that assumes that a price for personal mobility in death and serious injury is inevitable. The USA remains one of the few nations to rely on deaths per unit road use (100 million miles driven) as its principal safety performance measure.
+
+[Elvik \(2003\)](#page-6-0) has shown that what matters most in creating commitment and action is the setting of ambitious, quantitative targets. The nations considered as being at the leading edge of safety thinking and performance set their safety performance targets in terms of reductions in the absolute number of deaths and serious injuries, not merely as reductions in population-based rates. This forces a shift in the institutional mindset from one of managing a by-product to one of viewing safety as a fundamental outcome of a sustainable road transport system.
+
+Many nations continue to rely heavily upon economic analyses of cost and benefit when choosing not only whether to introduce a particular safety measure but the optimum level of resource to be allocated. A monetary value is placed on travel time (particularly congestion and delay), on environmental impacts (fuel usage, emissions, noise) and on safety (death and injury) as a method for making economically rational trade-off decisions between mobility, safety and environmental impacts among competing road and traffic investment proposals.
+
+No distortion is introduced if the choices to be made are between competing safety projects – or congestion ameliorating projects – as such decisions are about best safety – or congestion reduction – value for money but complex socio-political issues arise when the decisions involve trading safety for mobility. Sweden is the only nation to date to have explicitly excluded the use of monetary values for death and serious injury from benefit-cost analyses of the impact of investment options. Its Vision Zero philosophy requires that there be no increase in adverse safety outcomes from any proposal ([Tingvall, 2004](#page-6-0)).
+
+Debates around the measurement of the so-called externalities within the road transport system, and what to include and exclude, are conducted within academe and the institutional players. No evidence of a public debate has been uncovered, yet if the societal culture around road safety is to become an important target for change then the debate needs to enter new arenas. The recent attempts by [Wegman et al. \(2008\)](#page-6-0) to develop macro measures of a nation's road safety progress have begun this journey.
+
+## 6. Best practice strategic thinking is not aligned with prevailing culture
+
+The term safe system has come to represent the current consensus of what constitutes best practice strategic thinking in road safety ([OECD, 2008](#page-6-0)). It builds upon the Swedish evolution of road safety strategies from their Vision Zero and the dutch development of the principles of sustainable safety. Its key elements, drawn largely from [Wegman and Aarts \(2006\)](#page-6-0), are:
+
+- - The vision is to prevent crashes and, if this is not possible, to reduce crash severity to a point where (severe) injury risk is eliminated (since the human body has limited tolerance to kinetic energy impacts system design should be such as to keep the anticipated crash forces below human tolerance limits).
+- - All members of the population are road users, the full range of human abilities and personalities exist within the population, humans are fallible operators and mistakes must be anticipated as commonplace in the open-loop road transport system. (Vehicles, roads and traffic systems must be designed and operated to be error-tolerant, particularly through increased predictability in system operation, and the performance levels of users must be raised).
+- - Latent errors in what is an inherently dangerous traffic system must be reduced.
+- - All of the elements in the road safety system are interrelated and responsibility for safety must be shared across all players and efforts must be coordinated.
+
+It is sometimes claimed, incorrectly, that adopting a safe system model transfers responsibility from the individual road user to the vehicle, road, and traffic system designers. In reality, it shares the responsibility in a way not previously achieved. The safe system seeks error-tolerance, it does not demand designs which seek to eliminate deliberate misbehaviour. The individual road user is expected to comply with prescribed (and proscribed) behaviours – to be well-trained and licensed, to be fit to take part in traffic (unaffected by alcohol, other drugs or fatigue) and to comply with the operational traffic rules such as speed limits and movement priorities throughout his journey.
+
+The adoption of this conceptual model requires a level of change in institutional mindset that cannot be understated. While there are standards – and warrants for their use – that take safety explicitly into account their origins are rarely based on a sound understanding of human behaviour and their application is based typically on traffic volumes. [Hauer \(1990\)](#page-6-0), for example, described the origin of the method for computing sight distance in geometric road design, a standard intended to assure adequate braking distance when an unexpected object appeared on the road ahead. The standard had to be regularly modified to increase the size of the design object as vehicle design changes regularly resulted in substantial lowering of driver eye height. The alternative of rebuilding large lengths of road was not acceptable. [Lie and Tingv](#page-6-0)[all \(2009\)](#page-6-0) have opened debate on the development of an ISO standard for traffic safety as part of a process of driving the implementation of safe system thinking through the ranks of system designers and operators.
+
+Many discussions of the safe system model commence with a discussion of an assumed underlying ethical platform in which human life is sacrosanct. The Swedish parliament formally adopted ''Vision Zero" in 1997 which, in effect, made the prevention of death and serious injury the over-arching policy objective in the management of the road transport system ([Tingvall, 1998; Tingvall](#page-6-0) [and Haworth, 1999](#page-6-0)). It has been argued that this formal public adoption of policy was fundamental in paving the way for the radical change in institutional mindsets necessary.
+
+While this may have been so in Sweden the question remains as to whether it is a necessary first step in all nations. While the ultimate objective is a system in which no-one behaving reasonably is killed or seriously injured in road traffic the socio-political realities prevailing in most countries are more likely to result in incremental steps along a safer system continuum rather than in short-term radical reform. The keys to starting the safe system journey appear to be securing top level political commitment, the setting of ambitious safety targets and the adoption of a new design and operational philosophy by the key institutions [\(Johnston, 2003](#page-6-0)).
+
+The acceptance of a safer (as opposed to a safe) system mindset may appear to run the risk of leading us back into the mire of trade-off decision making, explicitly balancing the various goals of road transport against each other. But, this is not necessarily so. In many nations, utilisation of the principles of the safe system in road safety strategy formulation and the application of a formal road safety management system will be a simpler socio-political task and, providing the targets set are ambitious and, ideally, in terms of absolute reductions in serious casualties, will lead to substantial gains. Current Australian road safety planning reflects this pragmatic approach; the communications focus with the politicians and with the professionals in the critical institutions revolves around the need for error-tolerance and for kinetic energy reduction. Through this focus, strategic planning for future road safety has taken a new shape ([Western Australian Road Safety Council, 2008](#page-6-0)).
+
+## 7. How far has safe system thinking, planning and action permeated institutions and society?
+
+There are three issues to consider – the extent to which the safe system model underpins a nation's strategic planning for improving road safety; the degree to which the affected institutions have adopted the operational implications of the safe system; and the level of political and community understanding and acceptance of the consequences of safe system adoption. It is not possible to examine these issues comparatively across national borders so this section is confined to observations about Australia with which the author is most familiar.
+
+In Australia one needs to examine planning at the state level because Australia is a federation. While there is a national road safety strategy it is more a consensus document than a model. The current national strategy (2001–2010) contains no mention of the safe system ([Australian Transport Council 2002](#page-6-0)). The most recent national Action Plan, however, does refer to the safe system, specifically the need to ''... include and apply safe system principles in infrastructure documents, safety audit processes and professional development programs..." [\(Australian Transport Council 2009\)](#page-6-0).
+
+The State of Western Australia has recently issued its strategy for the period 2008–2020. It is explicitly anchored in safe system principles and its target is a reduction in the absolute number of deaths and serious injuries. It is an excellent example of best practice strategic planning. It recognises that the State is not homogeneous but comprises a large metropolis and vast regional and remote areas, each requiring specific plans. It identifies the major problems in each; examines potential countermeasures for each major problem and evaluates the evidence of their effectiveness; models packages of evidence-based measures applied at different resource levels and selects a preferred package that optimises potential effectiveness in the context of resource opportunity and institutional readiness ([Western Australian Road Safety Council,](#page-6-0) [2008\)](#page-6-0). This approach is now being followed in the development of the new national strategy (Corben, 2009, personal communication).
+
+However, understanding and acceptance of the safe system principles seems not to be widespread among institutions such as road and traffic authorities, transport planning agencies or enforcement agencies. Using speed moderation as an example – because it is so fundamental to future progress ([Lie and Tingvall,](#page-6-0) [2009\)](#page-6-0) – road and traffic engineers seem generally not to accept the value of reducing speed limits, particularly urban speed limits. Little of the discussion around speed limit setting examines the speed distribution as a whole. Urban speed limits remain high in Australia, in comparison with most European nations. However, in terms of infrastructure design, there are promising signs of institutional progress, for example a national roundtable was held in March 2009 to share knowledge and experience in seeking to implement safe system innovations ([Turner et al., 2009\)](#page-6-0).
+
+Police officers are another group of players who seem not to accept the population-wide, preventive medicine approach to enforcement. Using speed moderation again as an example, there is little acceptance of the case for reducing enforcement tolerances when enforcing speed limits. The majority of traffic infringements issued through speed camera detection are for speeding less than 10 km/h over the limit; almost none of those issued by police officers are for less than 15 km/h over the limit. The author's personal experience in presenting in-house training to large numbers of police officers in Victoria, Australia and in New Zealand demonstrates that the vast majority of front-line officers and their immediate supervisors do not understand the importance of shifting the speed distribution to the left. They remain convinced that their principal task is to apprehend those speeding at very high levels.
+
+While in-house training can, and should, be used to increase the understanding of officers around the critical importance of shifting the speed distribution to the left, a more immediate process is available. Assigning line officers the target of changing the speed distribution on nominated high-risk sections of the network in their patch and holding them accountable for achieving this outcome will alter enforcement practice. It is rare at present for police to set targets for other than hours of speed enforcement or number of tickets issued, both of which sidestep the objective of shifting the speed distribution rather than just impacting the tail of the distribution. Changing the speed distribution operationalises the safe system kinetic energy management principle in a realistic way for police officers.
+
+It is clear that the spread of safe system thinking in key institutions is still embryonic in Australia. How critical and how urgent is its widespread transfer in changing institutional behaviour? It is extremely important in the longer term and will likely require the targeting of future generations of practitioners as an integral part of the curriculum for their qualifications training. In the short term, however, the key to changing institutional behaviour is the creation of appropriate targets, sound strategies, effective performance monitoring and clear lines of accountability; in short the adoption of an effective road safety management system.
+
+There have been significant changes in institutional behaviour with regard to institutional co-operation and coordination, which captures that element of the safe system model related to the interconnectedness of all elements in the road transport system. The traditional silo approach in which each institution did its own thing independently is disappearing. The synergies across the efforts of agencies are not only reasonably well understood but in many states are being overtly and explicitly managed. In Victoria, the chief executives of the road and traffic agency, the judicial agency, the police and the compensation insurer are jointly accountable to parliament for achieving the state's road safety targets ([FHWA,](#page-6-0) [2006](#page-6-0)). Again, it is the setting of targets, the clear strategies and the effectiveness of the lines of accountability that have proven crucial in driving institutional cooperation and coordination.
+
+## 8. Seeking culture change
+
+Road safety progress in the last decade appears to have stemmed from an increased emphasis on strategic planning – comprising the data-driven selection of the major problems to address, the setting of objective but ambitious targets and a focus on effective implementation of programs of measures through institutional co-operation and coordination [\(OECD, 2008](#page-6-0)). Nevertheless, as [Wegman et al. \(2006\)](#page-6-0) argue, the culturally-anchored unsafe behaviours of drink-driving and speeding continue to prove barriers to the achievement of sustainable safety. It is worth noting here that the success stories of Australia (FHWA) and France [\(FlA Founda](#page-6-0)[tion, 2006](#page-6-0)) stemmed primarily from massive investments in road policing – both equipment and enforcement intensity – to reduce the incidence of these unsafe behaviour by extreme levels of deterrence. It is an open question as to whether the underlying culture around these behaviours has changed fundamentally as a result.
+
+It is not possible to generalise across national boundaries and arguably of dubious validity to do so across even state boundaries in a federation like Australia. Within Australia, however, it seems clear that there is little understanding within the media or the public-at-large of the safe system principles. The media reporting of new state road safety strategies has focussed on the targets and on specific measures, particularly those that involve changes to regulation and enforcement intensity.
+
+How important is it that the public understand the thinking behind the strategies? It likely matters only when public support is needed for strategy implementation. In the early 2000s in the State of Victoria, the lowering of general urban speed limits, the intensification of speed limit enforcement, the reduction of enforcement tolerances, and the use of automated enforcement such as fixed and mobile speed cameras generated considerable public controversy. Accusations of revenue-raising dominated the newspapers and radio airwaves for many months. One political party in the lead up to an election promised to reduce the number of speed cameras and to restore enforcement tolerances, based on their interpretation of public sentiment. Interestingly, official surveys of public opinion reported a majority of citizens in favour of intense speed enforcement but this was not apparent in political reaction or media coverage ([FHWA, 2006](#page-6-0)).
+
+There is an underdeveloped constituency for safety within the Australian public. For example, both government and corporate fleet managers have resisted specifying, in their purchase specifications, vehicle safety features that are not required by regulation. Thus proven collision avoidance innovations such as electronic stability control and proven injury reduction features such as curtain side airbags are not specified. Employees have not pressed their employers to provide safety above the regulated minimum, despite duty of care provisions in occupational health and safety laws that could be brought into play.
+
+The situation appears more acute in the United States. The AAA Foundation for Traffic Safety has lamented that ''... one of our greatest challenges (isn't) a specific safety issue, but rather a way of thinking about all of the issues, or our ''safety culture". There are several compendia in the literature detailing known effective safety measures and providing the evidence for each [\(Elvik et al., 2009\)](#page-6-0) but the challenge is choosing measures that the community and hence the political and institutional players will accept.
+
+It is important to establish acceptance by both politicians and the public of the rationale for the road safety strategy to be implemented. Where folk lore and official strategy are at odds – as is the case with speed moderation across the entire speed distribution – public understanding is vital [\(Johnston, 2004](#page-6-0)). There can be no clearer example of a mismatch between strategic thinking and cultural mindset.
+
+There is a total lack of action by governments worldwide and by the automobile industry to limit the power, top-speed and acceleration capability of cars and trucks. Injury reduction is a matter of limiting the (potential) transfer of kinetic energy to the human body to a level which can be borne. Industry has certainly made great progress in reducing energy transfer through seat belts, air bags, collapsible steering columns, improved side door strength and so on across a long list of crashworthiness measures but impact speeds are frequently above human tolerance levels. The principal determinant of the amount of kinetic energy to be managed in the event of a crash is impact speed and speed control has never been on the vehicle design or regulatory agenda. There is not one vehicle safety design rule that pertains to top-speed capability or acceleration performance. Moreover, the speedometer through which the driver manages his speed of travel contains a scale that displays speeds of double or more that permitted on all but a handful of the world's roads. A manufacturer can get a five star safety rating for a model that can exceed 250 km/h!
+
+The automobile industry markets and sells on power and performance far more so than on any other feature; there is a vibrant global motor sports industry that links manufacturers (if tenuously in reality) to publicly available brands; there is a thriving aftermarket in speed-related accessories; and motoring journalists tend to promote speed and performance in their reviews of new models. The fact that governments have not intervened illustrates the global economic importance of the industry in all its forms as well as an apparent belief that the community (the market) does not believe that speed limiting is important at the vehicle design level.
+
+A wider issue is the relative roles of public versus political acceptance in the adoption of safe system principles and, more importantly, in their effective implementation. The chicken and egg argument of whether the politicians develop public policy that reflects public opinion or whether public policy is used to shape public opinion is often heard when safety is discussed.
+
+It is at least arguable that the politicians in most Australian states have not accepted the safe system as such but have merely accepted the package of measures that comprise the particular strategy put to them, after weeding out any specific measures deemed unacceptable politically. The new strategy adopted by the government in Western Australia, for example, excluded the recommended reductions in speed limits because of their perceived unpopularity ([Western Australian Road Safety Council,](#page-6-0) [2008](#page-6-0)). None of this is surprising, or even disappointing; it is simply part and parcel of the functioning of a democracy. What matters is that the politicians accept a strategy that contains challenging targets of reductions in the absolute numbers of serious casualties, a package of evidence-based measures that are likely to achieve those targets and transparent mechanisms for public accountability for achieving those targets. This requires leadership, especially for those critical measures in the package that may prove controversial. The safety strategists have a crucial role in providing politicians with an understanding of the evidence underpinning the strategy and its package of measures.
+
+The safe system model requires shared responsibility. Road users must comply with the prescribed and proscribed behaviours and norms of road use while system designers and operators must provide a system that is error-tolerant and manages potential kinetic energy for a compliant road user population. How well are we managing compliance with appropriate behavioural norms? The cornerstones are requiring self-protection behaviours, such as seat belt wearing; controlling entry to the road system (for other than pedestrians and bicyclists) by licensing; promoting fitness to drive through the prescription of blood alcohol and similar limits; and prescribing a set of road rules governing precedence at conflict points and maximum permitted speeds under various circumstances.
+
+It is with regard to behavioural norms that differences in cultural underpinnings across nations assume paramount importance. The author was requested to prepare, for the US Department of Transportation, an account of how Victoria achieved a halving of road crash fatalities in a 15 year period ([FHWA, 2006\)](#page-6-0). Victoria's success stemmed largely from a preparedness to legislate, regulate and enforce, supported by extensive public education specifically to underpin these measures. In France, the dramatic improvements noted in [Table 2](#page-1-0) followed the adoption at the end of 2002 of ''a comprehensive plan to automate the ''control and sanction" chain for driving offences and particularly for speeding offences" [\(FlA Foundation,](#page-6-0) [2006\)](#page-6-0). The USA has been unable, given its cultural norms, to sustain basic self protection legislation requiring helmets or seat belts to be worn. Only 21 states require all vehicle occupants to wear seat belts, one state has no belt wearing law at all. A large number of states do not require motorcyclists to wear helmets. Random breath testing, a proven effective measure, is not acceptable. Thus, the United States is severely handicapped in terms of its strategies for road user behaviour compliance under the shared responsibility principle. The need for individual nations to develop tailored approaches to culture change to reflect their unique patterns is crucial.
+
+What we have not done, at least not systematically, is address the cultural mores that underpin behaviour. Earlier, the conflicts surrounding the speeding issue were discussed as a prime example. A population-wide speed behaviour change (a general slowing down) in the interest of the common good is hardly likely without significant cultural change. Individuals perceive a personal benefit from setting their own travel speed, do not regard low level speeding as dangerous and seem not to accept that if everyone moderated their speed by a small margin there would be significant safety gains for the community.
+
+The Dutch have started examining the theoretical principles underpinning what they call 'social forgivingness" to begin to understand how to formulate a strategy for changing the prevailing cultural more of personal gain to one of community gain in dictating road user behaviour ([Houtenbos, 2009](#page-6-0)). The concept is a general one of how we might engender a culture around road safety that would facilitate a preventive medicine approach to behaviour change [\(Rose, 1992\)](#page-6-0). [Howard and Sweatman \(2007\)](#page-6-0) use the term ''social contract" to describe the bringing together of prevailing community views and strategic thinking around road safety. Future road safety gains demand increased scientific attention to the issues around changing the culture of road safety.
+
+Urban geographers concerned with social cohesion and community well-being [\(May et al., 2008\)](#page-6-0) have issued a call for a shift towards an ethic of care, a call which to date appears to have escaped the attention of mainstream road safety professionals. Cooperative research between public health, urban geography, environmental sustainability and road safety professionals may well be a promising way forward in developing a culture change research agenda.
+
+#### 9. Conclusions
+
+Having an appropriate strategy and action plan is one thing, achieving implementation is quite another. There are several excellent road safety management systems in the literature (for example [Bliss and Breen, 2009\)](#page-6-0) but they cannot be implemented effectively without a political and community willingness to tread down a safe (or even a safer) system path. The recent OECD report on achieving ambitious road safety targets [\(OECD, 2008](#page-6-0)) stresses the importance of political commitment and public support before going on to list the detailed requirements for institutions and governments, requirements which are necessary but not sufficient.
+
+The keys to current best practice can be summed up in the four Cs:
+
+- - Constituency – unless the public demands action, appropriate resource will not be applied. In Victoria, for example, each of the new programs introduced in 1970, in the period around the mid 1970s, in the mid 1980s and in the early 2000s followed public ''outrage" at apparent, dramatic increases in death totals [\(FHWA, 2006](#page-6-0)). However, ''outrage" generates only an immediate response. A constituency based upon a cultural underpinning of an ethic of care for community is likely to be required for sustained progress.
+- - Commitment – without political will from the top, success must be limited. The outstanding demonstration is in France which went from laggard to best practice within three years after the President personally intervened ([FlA Foundation, 2006\)](#page-6-0). Commitment and constituency are bedfellows, with political leaders tending to move with publicly held behavioural norms.
+- - Cooperation – road safety is institutionally very complex with the actions of numerous agencies impacting upon progress. Without meaningful cooperation integrated packages of measures cannot be successfully applied ([FHWA, 2006; Wegman](#page-6-0) [et al., 2008](#page-6-0)).
+- - Coordination – the necessary bedfellow of cooperation, coordination is vital to integration and synergy across institutional efforts [\(FHWA, 2006; Wegman et al., 2008](#page-6-0)).
+
+It is instructive that the latest work out of Europe on how to measure road safety progress recommends three indices – one based upon policy performance (e.g. the quality of the strategic and operational planning and the political will); one based upon implementation performance (e.g. process indicators like resource applied); and one based on more conventional safety outcome measures (e.g. deaths and injuries) with the three linked causally in a chain from policy to outcome [\(Wegman et al., 2008\)](#page-6-0).
+
+Ideal best practice is when the principles of the safe system have been adopted within all of the institutions impacting the design and operation of the road transport system and applied through an effective management system with transparent lines of institutional accountability for the achievement of tough targets for reductions in death and serious injury. Institutional acceptance is related to community understanding and acceptance.
+
+We must systematically address several critical socio-cultural issues around car use and road safety and explore ways of moving towards community acceptance of a duty of care. It may well be that a duty of care in road use can best be developed through public engagement in sustainability. This will require political leadership, duty of care role-modelling by governments and large companies, and the creation of partnerships with others concerned with the sustainability of urban society. A community constituency needs to be created and the beginning likely lies in vigorous and well-informed public debate [\(Howard and Sweatman, 2007\)](#page-6-0). The ethic of care (conservation) created in nations like Australia around water and waste needs extending to road transport.
+
+As [Wiegmann and von Thaden \(2007\)](#page-6-0) conclude: ''While the concept of safety culture has been fundamentally applied to organisations or groups, it has yet to be systematically applied to the population at large". Given the discord between some of the key <span id="page-6-0"></span>behavioural norms of road use and the principles of safe system thinking addressing the research void is likely to make a significant progress to ongoing safety gains.
+
+#### References
+
+- Australian Transport Council, 2002. National Road Safety Strategy 2001–2010. ATC, Canberra.
+- Australian Transport Council, 2009. National Road Safety Action Plan for 2009 and 2010. ATC, Canberra.
+- Bliss, T., Breen, J., 2009. Implementing the Recommendations of the World Report on Road Traffic Injury Prevention. Country Guidelines for the Conduct of Road Safety Capacity Reviews and the Related Specification of Lead Agency Reforms, Investment Strategies and Safety Projects. World Bank Global Road Safety Facility, Washington, DC.
+- Elvik, R., 2003. An overview of target setting in Europe. In: Best in Europe 2003: Targeted Road Safety Programmes in the EU. European Transport Safety Council, Brussels.
+- Elvik, R., Vaa, T., Erke, E., 2009. The Handbook of Road Safety Measures, second ed. Emerald Publishing.
+- FHWA, 2006. Halving Deaths From Road Traffic Crashes A Case Study From Victoria, Australia 1989–2004. US Department of Transportation, Washington, DC.
+- FIA Foundation, 2006. Road Safety in France Reflections on Three Decades of Road Safety Policy. FIA Foundation for the Automobile and Society, London.
+- Hauer, E., 1990. The engineering of safety and the safety of engineering. In: Rothe, J.P. (Ed.), Challenging The Old Order – New Directions in Traffic Safety Theory Transactions. New Brunswick.
+- Houtenbos, M., 2009. Social Forgivingness: A Theoretical Exploration SWOV Report R-2009-8 SWOV, Leidschendam (in Dutch with an English summary).
+- Howard, E., Sweatman, P., 2007. Road safety culture development for substantial road trauma reduction. In: AAA Foundation for Traffic Safety (2007) Improving Traffic Safety Culture in the United States – The Journey Forward. AAA Foundation for Traffic Safety, Washington, DC
+- Johnston, I.R., 2003. Improving road safety in the longer term finding the right buttons to push. In: Proceedings 2003 Road Safety Research, Policing and Education Conference. Monash University Accident Research Centre, Melbourne.
+- Johnston, I.R., 2004. Reducing injury from speed-related road crashes towards a population-based preventive strategy (guest editorial). Injury Prev. 10 (5), 257–
+- 259. Johnston, I.R., 2006. Highway safety. In: Fwa, T.F. (Ed.), The Handbook of Highway
+
+Engineering. Taylor and Francis, New York.
+
+- Kissinger, J.P., 2007. Preface. In: AAA Foundation for Traffic Safety (2007) Improving Traffic Safety Culture in the United States – The Journey Forward. AAA Foundation for Traffic Safety, Washington, DC.
+- Koornstra, M., Lynam, D., Nillson, G., Noordzij, P., Petterson, H.-E., Wegman, F., Wouters, P., 2002. SUNflower; A Comparative Study of the Development of Road Safety in Sweden, The United Kingdom and the Netherlands. SWOV, Leidschendam.
+
+- Lie, A., Tingvall, C., 2009. What Can an ISO-Standard for Traffic Safety Do. Swedish Road Administration, Borlange.
+- May, M., Tranter, P.J., Warn, J.R., 2008. Towards a holistic framework for road safety in Australia. J. Transport Geogr. 16, 395–405.
+- OECD, 2002. Road Safety: What's the Vision? OECD, Paris
+- OECD, 2008. Towards Zero: Ambitious Safety Targets and the Safe System Approach. OECD, Paris.
+- Reason, J., 1997. Managing the Risks of Organizational Accidents. Ashgate, UK. Rose, G., 1992. The Strategy of Preventive Medicine. OUP, Oxford.
+- Sleet, D.A., Dinh-Zarr, T.B., Dellinger, A.M., 2007. In: AAA Foundation for Traffic Safety Improving Traffic safety Culture in the United States – The Journey Forward. AAA Foundation for Traffic safety, Washington, DC.
+- Tingvall, C., Haworth, N., 1999. Vision Zero: an ethical approach to safety and mobility. In: 6th Institute of Transport Engineers International Conference on Road Safety and Traffic Enforcement, Melbourne, September 1999.
+- Tingvall, C., 1998. The Swedish Vision Zero and how parliamentary approval was obtained. In: Proceedings 1998 Road Safety Research. Policing and Education Conference Monash University Accident Research Centre, Melbourne.
+- Tingvall, C., 2004. The Swedish Vision Zero. In: WHO (2004) World Report on Road Traffic Injury Prevention WHO, Geneva.
+- Trinca, G.W., Johnston, I.R., Campbell, B.J., Haight, F.A., Knight, P.R., Mackay, G.M., McLean, A.J., Petrucelli, E., 1988. In: Reducing Traffic Injury – A Global Challenge. Royal Australasian College of Surgeons, Melbourne.
+- Turner, B, Tziotis, M, Cairney, P, Jurewicz, C., 2009. Safe System Infrastructure National Roundtable Report. Report ARR 370, ARRB Group Ltd., Melbourne.
+- Watkins, K., 2009. The world is turning a blind eye. In: Ward, D. (Ed.), Make Roads Safe: A Decade of Action for Road Safety. Commission for Global Road Safety, London.
+- Wegman, F., Aarts, L., (Eds.), 2006. Advancing Sustainable Safety National Road Safety Outlook for 2005–2020. SWOV, Leidschendam.
+- Wegman, F., Commandeur, J., Eksler, S., Gitelman, V., Hakkert, S., Lynam, D., Oppe, S., 2008. SUNflower Next – Towards a Composite Road Safety Performance Index. SWOV, Leidschendam.
+- Wegman, F., Eksler, S., Hayes, S., Lynam, D., Morsink, P., Oppe, S., 2006. SUNflower+6: A Comparative Study of the Development of Road Safety in European countries. SWOV, Leidschendam.
+- Western Australian Road Safety Council, 2008. Towards Zero: getting there together. In: Council's Recommendations to Government to Reduce Road Trauma in Western Australia 2008–2020. Road Safety Council, Perth.
+- WHO, 2004. World Report on Road Traffic Injury Prevention. WHO, Geneva.
+- WHO, 2009. Global Status Report on Road Safety Time for Action. WHO, Geneva. Wiegmann, D.A., von Thaden, T.L., 2007. A review of safety culture theory and its potential application to traffic safety. In: AAA Foundation for Traffic Safety (2007) Improving Traffic Safety Culture in the United States – The Journey Forward. AAA Foundation for Traffic Safety, Washington, DC.
+- Williams, A.F., Haworth, N. 2007. Overcoming Barriers to Creating a Well-Functioning Safety Culture: A Comparison of Australia and the United States in AAA Foundation for Traffic Safety (2007) Improving Traffic Safety Culture in the United States – The Journey Forward. AAA Foundation for Traffic Safety. Washington, D.C.

Zotero/001_artiklid/Road Safety Indicators/jovanovicapplicationreliabilitymodelstraffic2011.md

@@ -0,0 +1,289 @@
+---
+category: literaturenote
+citekey: jovanovicapplicationreliabilitymodelstraffic2011
+title: The application of reliability models in traffic accident frequency analysis
+authors: "Jovanović, Dragan; Bačkalić, Todor; Bašić, Svetlana"
+year: 2011
+date: 2011-00-00 2011
+doi: 10.1016/j.ssci.2011.04.008
+publication: Safety Science
+url: "https://doi.org/10.1016/j.ssci.2011.04.008"
+zotero_key: A5UDWPP9
+zotero_storage: 5W63N9RM
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Jovanović, Dragan"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+Contents lists available at [ScienceDirect](http://www.sciencedirect.com/science/journal/09257535)
+
+# Safety Science
+
+journal homepage: [www.elsevier.com/locate/ssci](http://www.elsevier.com/locate/ssci)
+
+![](_page_0_Picture_5.jpeg)
+
+## The application of reliability models in traffic accident frequency analysis
+
+Dragan Jovanovic´ ⇑ , Todor Bacˇkalic´, Svetlana Bašic´
+
+Faculty of Technical Sciences, Department of Transport, Trg Dositeja Obradovic´a 6, Novi Sad, Serbia
+
+#### article info
+
+Article history: Received 28 June 2010 Received in revised form 22 March 2011 Accepted 8 April 2011 Available online 4 May 2011
+
+Keywords: Traffic accident Frequency Reliability theory
+
+#### abstract
+
+Analysis of traffic accident frequency represents an important subject of research of many authors. From the aspect of temporal analysis of traffic accident occurrence, two approaches have been singled out in previous practice: the collective (analyzes traffic accidents over a longer period of time) and individual (analyzes traffic accidents in real time). The paper shows that the system reliability theory, with certain adjustments, can be largely used to analyze traffic accident frequency based on the individual approach. A certain similarity has been observed between the system reliability theory and the traffic safety theory, and conceptual adjustment of equivalent terms and states has been performed based on this. A model has been successfully tested on the basis of which, for the road and sections, we have determined the traffic accident frequency, the probability of the occurrence of a certain number of traffic accidents and the mean time between two consecutive traffic accidents.
+
+> -2011 Elsevier Ltd. All rights reserved.
+
+#### 1. Introduction
+
+The frequency of traffic accidents traditionally represents the subject of a great deal of research and until now, many different methodological approaches to the modeling of the occurrence of traffic accidents have been developed ([Lord and Mannering,](#page-5-0) [2010\)](#page-5-0). From the aspect of temporal analysis of traffic accident occurrence, two approaches were singled out: (1) collective ([Hauer, 1986; Persaud, 1991; Miaou and Lum, 1993; Milton and](#page-5-0) [Mannering, 1998; Golob and Recker, 2003\)](#page-5-0), which determines the frequency of traffic accidents over a longer period of time and (2) individual [\(Hughes and Council, 1999; Lee et al., 2003; Go](#page-5-0)[lob and Recker, 2004\)](#page-5-0), which determines the probability of traffic accident occurrence in real time [\(Abdel-Aty and Pande, 2007\)](#page-5-0).
+
+In this paper, we went from the assumption that the traffic accident frequency, based on the individual approach, can be analyzed through the application of a reliability model. The reliability theory deals with the issues of calculation, experimental evaluations, provision and optimization of technical systems reliability [\(Ushakov](#page-5-0) [and Harrison, 1994](#page-5-0)). The broad field of application of the reliability theory and technical servicing has had an impact on a number of approaches and levels of study of the reliability problem ([Ushakov](#page-5-0) [and Harrison, 1994; O'Connor et al., 2002; Rausand and Høyland,](#page-5-0) [2004; Dhillon, 2005\)](#page-5-0). The mathematical approach to reliability analysis includes general problems, regardless of the nature of the system. Therefore, in engineering, general mathematical models are adapted to real problems and serve towards their solution.
+
+At the state change analysis of the real system the basic assumption is that the time is uniform and absolute. In accordance with that hypothesis, the order of the analysis aspects is as follows: 1. temporal (when or how often), 2. spatial (where or which part). The reliability theory is based on the analysis of failure occurrence, or in other words the duration of the period without failure in the coordinate system time-state. The process of continuous road traffic, observed as a system, can be compared to the functioning without failure of a technical system. In other words, similar to models used for the analysis of traffic accident frequency, the theory of reliability of technical systems is used to determine the probability and possibility of predicting the occurrence of a failure. In this paper, the application of the reliability theory in the analysis of traffic accident frequency is limited to the temporal–spatial aspect. The occurrence of traffic accidents on road sections was analyzed from a temporal aspect, i.e. the occurrence of traffic accidents was analyzed in the coordinate system time-location.
+
+#### 2. Possibilities of associating the basic principles of the reliability theory with the analysis of traffic accident frequency
+
+In the reliability theory, technical systems represent the formal objects of study. They can be represented as complex systems, at any level of complexity, composed of components. The system components can be complex (subsystems) or simple (indivisible components or units). Units are constituent components of a system and they are simple systems for which, within the given research, there is no need for further parsing [\(Ushakov and Harri](#page-5-0)[son, 1994\)](#page-5-0). Roads are complex systems consisting of a number of sections and they can be identified with complex technical systems.
+
+<sup>⇑</sup> Corresponding author. Tel.: +381 21 485 2494; fax: +381 21 450 644. E-mail address: [draganj@uns.ac.rs](mailto:draganj@uns.ac.rs) (D. Jovanovic´).
+
+<span id="page-1-0"></span>In the field of traffic safety, as a complex and heterogeneous system, there have been no attempts to consider the process of occurrence of traffic accidents from the aspect of system reliability. Analogous to defining the reliability of technical systems [\(Ushakov](#page-5-0) [and Harrison, 1994; Dhillon, 2005](#page-5-0)), the term reliability of the traffic safety system can be viewed as the ability of the system to fulfil the given functions within a certain time interval and at the same time maintain determined projected characteristics within given limits, in appropriate conditions of exploitation and level of traffic regulation.
+
+By looking at and comparing characteristic processes in traffic and technical systems, many similarities and dependencies studied by theory of reliability of technical systems have been noticed (e.g. traffic accident failure). A traffic accident as an unplanned and unwanted event causes an interruption in road traffic flow and requires an emergency response in order to ease and neutralize the congestion as quickly as possible.
+
+Apart from the attempt to analyze the process of road traffic flow from the aspect of technical system reliability, the second task was conceptual definition. This procedure required an analysis of terms and conditions of the reliability theory, followed by finding potential similar occurrences and states in the traffic safety theory, i.e. the model for the analysis of traffic accident frequency. Thus the terms of the reliability theory received their equivalent terms in the model for traffic accident frequency analysis. Traffic safety covers a much wider area and contains a significantly greater number of disciplines and factors than the reliability of technical systems. In a technical system a failure causes an interruption in the functioning of the system and requires repair or replacement, i.e. recovery of the system, followed by putting the system back into operation. In the process of road traffic flow, the occurrence of a traffic accident causes a delay or significant slowing down of traffic, but following the investigation (departure of the police and other services from the scene of the accident) traffic flow spontaneously and relatively quickly returns to its standard regime. A traffic accident has the character of a sudden failure, or in other words it instantly switches the system from an operational (properly working) state to a state of failure. In accordance with this, basic concept, terminological mapping has been done from the technical system reliability theory to the theory of road traffic safety. Table 1 provides an overview of the terminological mapping of the basic terms.
+
+#### 3. Defining the model
+
+#### 3.1. Determining road reliability
+
+In determining the reliability of a road a preliminary analysis is necessary that would determine its structure and level of complexity, or the defining of the system units – the sections. Each road as a complex system consists of n units – sections (Fig. 1), where the traffic flow interruption in any unit causes the traffic flow interruption in the whole road.
+
+Table 1 Terminological mapping from the technical systems reliability theory into the theory of road traffic safety.
+
+| Reliability of technical systems | Road safety                         |
+|----------------------------------|-------------------------------------|
+| System                           | Road                                |
+| System unit                      | Road section                        |
+| Failure                          | Traffic accident                    |
+| Failure rate                     | Traffic accident frequency          |
+| Reliability                      | Reliability                         |
+| Unreliability                    | Unreliability                       |
+| Mean time between failures       | Mean time between traffic accidents |
+| Work without failure             | Period without accident             |
+| Renewal                          | Traffic recovery                    |
+
+![](_page_1_Picture_10.jpeg)
+
+Fig. 1. A road observed as a series system.
+
+In order to achieve an adequate mathematical approach with the real system, reliability theory applies different statistical distributions with the help of which real processes are approximated, e.g. distributions of time between two events. Traffic flow on the section of road begins at the time t = 0 and the time period without traffic accidents is the interval [0, s1], where s<sup>1</sup> is the moment of occurrence of the first traffic accident. The next period without traffic accidents is the interval [s1, s2], where s<sup>2</sup> is the moment of occurrence of the second traffic accident.
+
+Poisson's distribution of events in stochastic traffic processes has a significant role and represents a starting distribution for most of the analyses ([Jovanis and Chang, 1986; Elvik and Vaa, 2004;](#page-5-0) [Shankar et al., 2008](#page-5-0)). The basic form of Poisson process is a continuous-time counting process that possesses the following properties: independent increments, stationary increments and no counted occurrences are simultaneous. A homogeneous Poisson process is characterized by its rate parameter k, which is the expected number of ''events'' or ''arrivals'' that occur per unit time. If the stream of events has Poisson's distribution, then the time between the occurrences of two events (T) can be described with an exponential distribution with parameter k. Therefore, it has been adopted that all the empirical distributions of the duration of the period without accidents can be replaced with the corresponding exponential distributions.
+
+The parameters of road reliability for Poisson' process are calculated from a temporal aspect according to the forms of the technical systems reliability theory (Eqs. (1)–(13)) [\(Ushakov and](#page-5-0) [Harrison, 1994; O'Connor et al., 2002; Rausand and Høyland,](#page-5-0) [2004; Dhillon, 2005](#page-5-0)).
+
+The density function of the distribution of the time between two accidents f(t) (fS(t) – road, fi(t) – ith section) represents a measure of the speed of the accident's occurrence. The function of accident distribution density for the ith section is:
+
+$$f_i(t) = \lambda_i e^{-\lambda_i t}$$
+, where is  $\lambda_i > 0, t \ge 0$  (1)
+
+Distribution function F(t) (FS(t) – road, Fi(t) – ith section) of the random variable T (the time between two accidents), is equal to the probability that an accident will occur before the moment t. This function is also called the function of unreliability. The distribution function of the time between two accidents on the ith section – unreliability function of the ith section is:
+
+$$F_i(t) = \int_0^\infty f_i(t) dt = \int_0^\infty \lambda_i e^{-\lambda_i t} dt = 1 - e^{-\lambda_i t}$$
+ (2)
+
+Using the unreliability function F(t) we introduce the reliability function R(t) (RS(t) – road, Ri(t) – ith section), as the probability of a time period without accidents until the moment t. The reliability function of the ith section is:
+
+$$R_i(t) = 1 - F_i(t) = e^{-\lambda_i t} \tag{3}$$
+
+The mean time between two consecutive accidents T0 (T0S(t) – road, T0i(t) – ith section) is obtained as a mathematical expectation of the random variable T. The mean time between two consecutive accidents on the ith section is:
+
+$$T_{0i} = \int_0^\infty R(t) \ dt = \int_0^\infty e^{-\lambda_i t} \ dt = \frac{1}{\lambda_i}$$
+ (4)
+
+The accident frequency a(t) (aS(t) – road, ai(t) – ith section) also represents an important and widely used reliability characteristic, and represents the measure of the current speed of the accident's occurrence. The a(t) can be expressed in various time units and depends on the time unit which is used for calculation of T<sup>0</sup> (minute<sup>1</sup> , hour<sup>1</sup> , ..., year<sup>1</sup> ). For Poisson's distribution, the accident frequency as the current speed of the accident's occurrence is constant at any moment in the observed time period. The accident frequency function on the ith section is:
+
+$$a_i(t) = \lambda_i = const$$
+ (5)
+
+The expected number of accidents into the certain period A(t) (AS(t) – road, Ai(t) – ith section) is calculated as:
+
+$$A_i(t) = \lambda_i \cdot t \tag{6}$$
+
+where is t – duration of the certain time period expressed in basic time unit as k.
+
+If the time unit is the same as duration of the observed period, the accident frequency a(t) and the expected number of accidents A(t) will be equal.
+
+The road reliability is conditioned by the reliability of the sections that it consists of. The observed road represents a simple system, because of the serial connection of the sections. Since the sections are independent in terms of reliability, in the given case the reliability function of the road is equal to:
+
+$$R_{S}(t) = \prod_{i=1}^{n} R_{i}(t) = e^{-(\lambda_{1} + \lambda_{2} + \dots + \lambda_{n})t}$$
+(7)
+
+If the reliability functions are expressed using the accident frequency, the reliability function of the road is equal to:
+
+$$\lambda_S = \sum_{i=0}^{n} \lambda_i = \lambda_1 + \lambda_2 + \dots + \lambda_n$$
+(8)
+
+Since it is a serial connection in the case of exponential distribution, the parameter k<sup>S</sup> of the road is equal to the sum of the parameters k<sup>i</sup> of the sections. The parameters of road reliability as a system, for Poisson' process, are calculated according to equations from [\(1\)–\(5\).](#page-1-0)
+
+## 3.2. The basic characteristics of the traffic recovery process
+
+In the previous section we determined the basic parameters of the reliability of the road and sections until the first traffic accident. Traffic flow on the road and sections represents a process that is being recovered. The duration of traffic interruption on the section is negligibly small compared to the time between two consecutive traffic accidents, so the calculation of road and section reliability is performed with the assumption that the traffic recovery is executed instantaneously. It is also assumed that the moments of accident occurrences on the sections are independent, and that traffic flow, accidents, and the traffic recovery on a section do not affect the reliability of the other sections.
+
+The moments of the occurrence of accidents on the section, i.e. the moments of recovery of traffic flow on the sections, s1, s2, ... , s<sup>n</sup> form the traffic recovery process (Fig. 2).
+
+The process of traffic recovery can be considered simple, since the random variables T1, T2, ... , Tn are independent and with the same probability distribution. It is assumed that the traffic recovery processes are Poisson's, from which it follows that the time intervals between two recoveries, or two consecutive accidents, have an exponential distribution.
+
+From this it follows that the density of time until the nth traffic recovery is:
+
+![](_page_2_Figure_18.jpeg)
+
+$$r_{ni}(t) = \frac{\lambda_i (\lambda_i t)^{n-1}}{(n-1)!} e^{-\lambda_i t}$$
+ (9)
+
+Then the average number of traffic recoveries H, as a linear time function, and the traffic recovery intensity h have the following forms:
+
+$$H_i(t) = \lambda_i t \tag{10}$$
+
+$$h_i(t) = a_i(t) = \lambda_i = const$$
+ (11)
+
+The probability of the occurrence of n accidents or traffic flow recoveries, during a time interval of a length of t, does not depend on the position of that interval and is expressed with the following formula:
+
+$$P_{ni}(t) = \frac{(h_i t)^n}{n!} e^{-h_i t} = \frac{(\lambda_i t)^n}{n!} e^{-\lambda_i t}$$
+ (12)
+
+The probability that there will not be an accident, during the time interval of a length of t, is equal to:
+
+$$P_{0i}(t) = e^{-h_i t} = e^{-\lambda_i t} \tag{13}$$
+
+#### 4. Testing the model
+
+To test the model we selected a main road (the road M-22), which is made up of 11 sections of a total length of 115 km [\(Table](#page-3-0) [2](#page-3-0)). The sections represent parts of the road network between two consecutive traffic nodes and are used to provide for continuous and unobstructed traffic flows ([Hauer et al., 2002](#page-5-0)). Each section is specific by the structure and volume of the traffic, road environment, units and road equipment.
+
+During the analyzed period, from 2001 to 2008, 796 traffic accidents occurred on the observed road. All the traffic accidents have been allocated based on the time of occurrence (year/month/hour) and location on the road (kilometer/meter), which provides us with the temporal–spatial distribution of traffic accidents per section. Testing whether the empirical time distributions between successive accidents on the sections match the theoretical distributions was performed using the v<sup>2</sup> test.
+
+Due to a small sample, distributions of traffic accidents in sections 3 and 6 were approximated using Poisson process, i.e. it was assumed that in those sections as well, time between two traffic accidents has exponential distribution.
+
+With probability of 1.0 we can point out that, on the road analyzed, in the period of 365 days, the number of accidents will be between 43 and 170, and with probability of 0.0397537246, that in the same time period, there will be exactly 100 accidents on the road.
+
+By comparing the reliability, or the mean time between two consecutive accidents on the sections, we may conclude that section 10 is the least reliable or it has the shortest T0, followed by sections 11, 7, 9, 8, 1, 4, 2 and 5, while the most reliable sections are 3 and 6. The reliability of a road during 1 week is 0.144869487, within one year 0, or unreliability is 0.855130513 and 1, respectively. On the observed road, on average, 1.932 accidents happen within 1 week, that is 100.736 accidents per year, i.e. a traffic accident happens every 86.96 h.
+
+When analyzing the probability and the occurrence of accidents on the sections, and on the road as a whole, according to Eqs. (12) and (13), we also observed the periods of t = 7 (days) = 168 (h) and t = 365 (days) = 8760 (h). [Figs. 3 and 4](#page-3-0) show the probabilities of accident occurrence (''equal or more than a certain number'', ''a certain number'' and ''less or equal than a certain number'') on the road as a whole, or in the [Figs. 5 and 6](#page-4-0) on the road sections.
+
+In pictures 3 and 4, probability curve points to the probability of a certain number of accidents, which is greater, smaller or equal to Fig. 2. The traffic recovery process.
+
+<span id="page-3-0"></span>Table 2 Basic characteristics of observed road and appropriate main results of analysis.
+
+| Road section | Length (km) | Observed number of accidents | T0i (h) | ki (h1<br>) | t = 7 (days) = 168 (h) |             | t = 365 (days) = 8 760 (h) |             |
+|--------------|-------------|------------------------------|---------|-------------|------------------------|-------------|----------------------------|-------------|
+|              |             |                              |         |             | Ai(t)                  | Ri(t)       | Ai(t)                      | Ri(t)       |
+| 1            | 4.823       | 59                           | 1171.47 | 0.000853625 | 0.143409002            | 0.866399637 | 7.477755111                | 0.000565526 |
+| 2            | 12.600      | 55                           | 1257.27 | 0.000795372 | 0.133622560            | 0.874920232 | 6.967462039                | 0.000942041 |
+| 3            | 6.293       | 12                           | 5487.50 | 0.000182232 | 0.030615034            | 0.969848860 | 1.596355353                | 0.202633702 |
+| 4            | 14.541      | 57                           | 1229.74 | 0.000813182 | 0.136614594            | 0.872306353 | 7.123475284                | 0.000805961 |
+| 5            | 12.403      | 52                           | 1346.42 | 0.000742709 | 0.124775045            | 0.882695447 | 6.506127346                | 0.001494255 |
+| 6            | 4.236       | 31                           | 2110.65 | 0.000473789 | 0.079596515            | 0.923488885 | 4.150389729                | 0.015758274 |
+| 7            | 16.486      | 102                          | 684.17  | 0.001461632 | 0.245554202            | 0.782270882 | 12.803897686               | 0.000002750 |
+| 8            | 10.094      | 65                           | 1062.03 | 0.000941592 | 0.158187507            | 0.853689694 | 8.248348592                | 0.000261690 |
+| 9            | 12.735      | 85                           | 817.64  | 0.001223039 | 0.205470582            | 0.814264048 | 10.713823220               | 0.000022235 |
+| 10           | 18.434      | 165                          | 420.04  | 0.002380712 | 0.399959600            | 0.670347127 | 20.855036288               | 0.000000001 |
+| 11           | 2.486       | 113                          | 612.88  | 0.001631651 | 0.274117392            | 0.760242823 | 14.293264024               | 0.000000620 |
+| Road         | 115.131     | 796                          | 86.96   | 0.011499536 | 1.931922035            | 0.144869487 | 100.735934672              | 0.000000000 |
+
+![](_page_3_Figure_4.jpeg)
+
+Fig. 3. The probabilities of accident occurrence (''equal or more than a certain number'', ''a certain number'' and ''less or equal than a certain number'') on the road as a whole for the period t = 7 (days) = 168 (h).
+
+n, in other words, it points to the probability of certain speed of traffic accident occurrence. Picture 3 shows that with probability of around 0.15 we can claim that there will not be any accidents on the road within a period of 7 days, i.e. the speed of traffic accident occurrence will be 0 in the observed period,. With probability of about 0.90 it can be asserted that the speed of traffic accident occurrence will be less than three accidents per week. Also, with probability of 1.0 it can be asserted that the speed of traffic accident occurrence, i.e. number of traffic accidents within a week, will be less than six accidents per week. It is important to point out, which is confirmed with probability of about 0.85, that the speed of traffic accident occurrence will be greater than 0 accidents per week. With probability of 0.28 it can be asserted that the speed of traffic accident occurrence will be 1.1 accidents per week. For the period of 365 days situation is different (picture 4). With probability of 1.0 it can be asserted that the speed of traffic accident occurrence will be greater than 75 and less than 130, i.e. with probability of about 0.9 it can be asserted that the speed of traffic accident occurrence will be between 85 and 115 traffic accidents in 365 days.
+
+Taking into account conclusions of the analysis of pictures 3 and 4, during the analysis of pictures 5 and 6 there are obvious differences between shapes of probability curves. In picture 4 some curves are narrower and higher, which means that cumulants, too, are close to each other and they are steep. In other words, that points to narrow interval of expected speeds of accident occurrence, with high probability of happening. Besides that, section 11 corresponds to flattened probability curve of a certain speed of accident occurrence, which means that the interval of possible speeds of accident occurrence is wider. On the other hand, all curves have prominent peak, i.e. the speed of accident occurrence with greatest probability.
+
+#### 5. Conclusions and discussion
+
+In recent years, traffic accident analysis models based on the individual approach are becoming more and more important. Defining and testing of the model confirmed the initial assumption that the traffic accident frequency, based on the individual approach, can be analyzed by applying the reliability theory model. To this end, an analysis of application possibilities has been performed as well as conceptual adaptation.
+
+In spatial analysis, the starting point is the total number of accidents in a certain period and causes and solutions are sought. In the temporal analysis, from the point of reliability theory, the start-
+
+<span id="page-4-0"></span>![](_page_4_Figure_2.jpeg)
+
+Fig. 4. The probabilities of accident occurrence (''equal or more than a certain number'', ''a certain number'' and ''less or equal than a certain number'') on the road as a whole for the period t = 365 (days) = 8760 (h).
+
+![](_page_4_Figure_4.jpeg)
+
+Fig. 5. The probabilities of the occurrence of a certain number of accidents on the sections as units for the period t = 7 (days) = 168 (h).
+
+ing point is the time between two accidents on road sections, i.e. the speed of traffic accident occurrence. Determining the laws of traffic accident occurrence with the goal of reduction the total number of accidents is the ultimate goal of both analyses. In the spatial analysis, the only indicator is the number of traffic accidents. In the temporal analysis, the main indicator is average time between two accidents and the number of traffic accidents is the consequence of that. The temporal analysis from the point of view of reliability theory does not exclude the spatial analysis; on the contrary, it represents the starting analysis. Therefore, the spatial analysis should not exclude the temporal either.
+
+The ranking of safety of two independent roads based on a simple comparison of number of accidents requires that the accidents occurred during the same, exactly defined period. The problem arises when it is necessary to compare the reliability of more roads, and to thereby vary the data collection period. In this case, the use of the proposed model which is based on the system approach will help to determine the mean time between accidents, i.e. the accident frequency. In that way, a better comparison will be allowed without reduction, extrapolation or averaging of heterogeneous data. In addition, reliability analysis based on the number of accidents does not contain a dynamic component and it is possible only after the data collection period. On the other hand, the proposed analysis of accidents frequency, which is based on the temporal aspect, allows monitoring in the real time and does not require defining the duration of the periods of observation and data collection. This possibility gives the represented model better application flexibility, i.e. much lower dependence on the length of observation
+
+<span id="page-5-0"></span>![](_page_5_Figure_2.jpeg)
+
+Fig. 6. The probabilities of the occurrence of a certain number of accidents on the sections as units for the period t = 365 (days) = 8760 (h).
+
+period. While most models demand observation over a relatively long period of time (3–5 years), this model can be applied on relatively short time periods (a week, a month, a year). In other words, instead of passively registering the number of accidents in the defined period, the suggested model promotes permanent and active monitoring and measuring of time between two accidents.
+
+Traffic safety analysis, i.e. traffic accident occurrence frequency in this model, as the output result gives reliability of the road and its sections. It can be seen that the model considers all recorded traffic accidents, regardless of the type of consequence, which contributes to greater reliability of the model.
+
+The more unreliable section is, i.e. the shorter the time between two accidents, the harder it is to determine the probability of the accurate number of accidents. Because of accidents happen often, i.e. high speed of accident occurrence diminishes the possibility of precise prediction of the number of accidents. Therefore, the proposed model also points to the importance of directing attention to unreliable sections, i.e. sections with higher speed of traffic accident occurrence. Since the suggested model is based on system approach to monitoring of traffic accident occurence, it also enables determination of occurrence probability of a certain number of accidents on the sections, as well as the road as a whole. Also, it is possible to determine probability that there would not be any accidents in a certain period, i.e. reliability analyses of both the sections and the road by following the periods of functioning without accidents.
+
+Through application of the given model, for the road and sections over a defined period of time, the following parameters are determined:
+
+- the mean time between two consecutive traffic accidents;
+- the traffic accident frequency;
+- the probability of the occurrence of a certain number of traffic accidents on any section or whole road.
+
+The proposed model may serve the road authority as a significant tool in decision making on the application of traffic safety measures on the roads. Namely, the road authority tends to make the time between two consecutive traffic accidents on a road to be as long as possible, or that the accident occurrence rate is as low as possible. The reliability of the road is increased in this way, as well as the traffic flow system. On the other hand, the proposed model has great potential for expansion and development.
+
+## Acknowledgment
+
+This paper is based on research within projects financed by the Ministry of Science and Technological Development of the Republic of Serbia.
+
+### References
+
+Abdel-Aty, M., Pande, A., 2007. Crash data analysis: collective vs. individual crash level approach. Journal of Safety Research 38 (5), 581–587.
+
+Dhillon, B.S., 2005. Reliability, Quality, and Safety for Engineers. Boca Raton, Florida, USA, ISBN 0-8493-3068-8.
+
+Elvik, R., Vaa, T., 2004. The Handbook of Road Safety Measures. Elsevier.
+
+Golob, T.F., Recker, W.W., 2003. Relationships among urban freeway accidents, traffic flow, weather, and lighting conditions. Journal of Transport 129 (4), 342– 353.
+
+Golob, T., Recker, W., 2004. A method for relating type of crash to traffic flow characteristics on urban freeways. Transportation Research Part A 38 (1), 53– 80.
+
+Hauer, E., 1986. On the estimation of the expected number of accidents. Accident Analysis and Prevention 18 (1), 1–12.
+
+Hauer, E. et al., 2002. Screening the road network for sites with promise. Transportation Research Record 1784, 27–32.
+
+Hughes, R., Council, F., 1999. On establishing relationship(s) between freeway safety and peak period operations: performance measurement and methodological considerations. In: Presented at the 78th Annual Meeting of Transportation Research Board, Washington, DC.
+
+Jovanis, P.P., Chang, H.L., 1986. Modeling the relationship of accidents to miles traveled. Transportation Research Record 1068, 42–51.
+
+Lee, C., Hellinga, B., Saccomanno, F., 2003. Real-time crash prediction model for the application to crash prevention in freeway traffic. Transportation Research Record 1840, 67–78.
+
+Lord, D., Mannering, F., 2010. The statistical analysis of crash-frequency data: a review and assessment of methodological alternatives. Transportation Research Part A 44 (5), 291–305.
+
+Miaou, S.-P., Lum, H., 1993. Modeling vehicle accidents and highway geometric design relationships. Accident Analysis and Prevention 25 (6), 689–709.
+
+Milton, J., Mannering, F., 1998. The relationship among highway geometrics, traffic related units and motor vehicle accident frequencies. Transportation 25 (4), 395–413.
+
+O'Connor, P.D.T., Newton, D.W., Bromley, R.C., 2002. Practical Reliability Engineering. John Wiley & Sons Ltd., Chichester, England, ISBN 0-470-84462-0. Persaud, B.N., 1991. Estimating accident potential of Ontario road sections.
+
+Transportation Research Record 1327, 47–53. Rausand, M., Høyland, A., 2004. System Reliability Theory – Models, Statistical
+
+Methods and Applications. John Wiley & Sons Inc., Hoboken, New Jersey, USA, ISBN 0-471-47133-X.
+
+Shankar, V., Jovanis, P., Aguerde, J., Gross, F., 2008. Analysis of naturalistic driving data: prospective view on methodological paradigms. Transportation Research Record 2061, 1–9.
+
+Ushakov, I.A., Harrison, R.A., 1994. Handbook of Reliability Engineering. John Wiley & Sons, Inc., New York, USA, ISBN 0-471-57173-3.

Zotero/001_artiklid/Road Safety Indicators/khandevelopmentroundaboutcrashspecific2025.md

@@ -0,0 +1,400 @@
+---
+category: literaturenote
+citekey: khandevelopmentroundaboutcrashspecific2025
+title: Development of roundabout crash-specific safety performance functions in Kentucky using different techniques
+authors: "Khan, Tathagatha; Pathivada, Bharat Kumar; Banerjee, Arunabha; Haleem, Kirolos"
+year: 2025
+date: 2025-00-00 2025
+doi: 10.1016/j.jsr.2025.03.001
+publication: Journal of Safety Research
+url: "https://doi.org/10.1016/j.jsr.2025.03.001"
+zotero_key: PI7FTB7G
+zotero_storage: WP8TXGP8
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Khan, Tathagatha"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+Contents lists available at [ScienceDirect](www.sciencedirect.com/science/journal/00224375)
+
+# Journal of Safety Research
+
+journal homepage: [www.elsevier.com/locate/jsr](https://www.elsevier.com/locate/jsr)
+
+![](_page_0_Picture_6.jpeg)
+
+# Development of roundabout crash-specific safety performance functions in Kentucky using different techniques
+
+Tathagatha Khan , Bharat Kumar Pathivada [,](https://orcid.org/0000-0003-2485-701X) Arunabha Banerjee [,](https://orcid.org/0000-0002-6314-295X) Kirolos Haleem \*
+
+*Center for Transportation Research (CTR), School of Engineering & Applied Sciences, Western Kentucky University, USA*
+
+ARTICLE INFO
+
+*Keywords:* Roundabout Safety performance functions Conway-Maxwell-Poisson Negative binomial Unobserved heterogeneity Varying dispersion parameter Kentucky
+
+#### ABSTRACT
+
+*Introduction*: Roundabouts are one of the innovative intersection designs that can reduce the number of fatal and serious injury crashes compared to traditional intersections. Nevertheless, relatively limited studies are available that have developed roundabout crash-specific safety performance functions (SPFs) using different modeling techniques. This study develops roundabout crash-specific SPFs using crashes in the state of Kentucky over a four-year period (2019–2022). *Method:* Crash data exhibited over-dispersion; hence, negative binomial (NB) and heterogenous negative binomial (HTNB), as well as Conway-Maxwell-Poisson (CMP) modeling approaches with fixed and varying dispersion parameters (while accounting for unobserved heterogeneity) were investigated. The performance of the fitted models was evaluated using different goodness-of-fit (GOF) statistics, e.g., Akaike information criterion "AIC," Bayesian information criterion "BIC," and McFadden pseudo R2 . *Results:* The GOF measures showed that the heterogeneous CMP (or HTCMP) model was the best-fit model for developing roundabout SPFs. The results of the HTCMP model showed that annual average daily traffic (AADT) on major and minor approaches to the roundabout and inscribed circle diameter (ICD) were significantly associated with increased crash frequency at roundabouts. Conversely, single circulatory lanes, presence of two lanes along the major roundabout approach, and wider circulatory roadway width (greater than 16 feet) were significantly associated with reduced crashes at roundabouts. *Practical Applications:* Based on the study results, safety countermeasures were proposed to enhance safety at roundabouts. One example is ensuring roundabout circulatory roadway widths greater than 16 feet to provide adequate room for driving error rectification while navigating roundabouts.
+
+## **1. Introduction and study objective**
+
+Intersections are one of the risky traffic facilities and present substantial safety concerns due to the presence of several conflicting points ([Al-Omari et al. 2020; Khan](#page-8-0) & Mohapatra, 2021). According to the Federal Highway Administration (FHWA), around 2.5 million crashes occurred at intersections, consisting of 23% fatal crashes and 50% serious injury crashes [\(FHWA, 2022\)](#page-8-0). In the state of Kentucky in 2022, more than 30% of traffic crashes occurred at intersections, consisting of ~ 18% fatal crashes (Kentucky Traffic Collision [Facts, 2023\)](#page-8-0). The reduction in crashes and their severities can be achieved through significant geometric design improvements at intersections, and one innovative improvement is the conversion to roundabouts.
+
+Roundabouts are at-grade intersections having circular or elliptical
+
+islands at the center, around which vehicular traffic moves in a single direction and traffic in the roundabout is given the right of way. Roundabouts are believed to be one of the safest intersection types, where drivers are required to reduce their speed, yield, and slowly merge with circulating traffic. Roundabouts have fewer conflicting points compared to traditional intersections. A typical two-way intersection has 32 vehicle-to-vehicle conflicts (16 crossing, 8 merging, and 8 diverging conflicts), whereas a single-lane roundabout has only eight conflicts (4 merging, 4 diverging, and no crossing conflicts).
+
+Several researchers advocated the conversions of intersections into roundabouts based on their findings, due to significant reduction in traffic fatalities and injuries [\(Isebrands, 2009\)](#page-8-0). Researchers also reported that the crash rates at roundabouts were comparatively less than other types of at-grade intersections [\(Daniels et al., 2011; Li et al., 2024](#page-8-0)).
+
+<sup>\*</sup> Corresponding author at: Center for Transportation Research (CTR), School of Engineering & Applied Sciences, Western Kentucky University, 1906 College Heights Blvd, EBS 2122, Bowling Green, KY 42101, USA.
+
+*E-mail addresses:* [tathagatha.khan@wku.edu](mailto:tathagatha.khan@wku.edu) (T. Khan), [bharatkumar.pathivada@wku.edu](mailto:bharatkumar.pathivada@wku.edu) (B.K. Pathivada), [arunabha.banerjee@wku.edu](mailto:arunabha.banerjee@wku.edu) (A. Banerjee), [kirolos.](mailto:kirolos.haleem@wku.edu)  [haleem@wku.edu](mailto:kirolos.haleem@wku.edu) (K. Haleem).
+
+Nevertheless, relatively few studies have developed roundabout crashspecific safety performance functions (SPFs) or crash prediction models at roundabouts. SPFs are developed using different statistical techniques by considering a hypothesis that the crash count data follows the NB distribution ([Hauer et al., 2002](#page-8-0)). The Highway Safety Manual (HSM) considered the NB model primarily for SPF development. However, the NB model is restricted to over-dispersed count data and unable to handle small sample size [\(AASHTO, 2010; Lord](#page-8-0) & Mannering, 2010; [Hosseinpour et al. 2022](#page-8-0)). The Conway-Maxwell-Poisson (CMP) model is a flexible approach which could appropriately accommodate both overdispersed and under-dispersed count data and low sample size (Geedipally & Lord, 2008; Lord & [Park, 2008; Hosseinpour et al. 2022](#page-8-0)).
+
+It should be noted that both NB and CMP models restrict the dispersion parameter to be fixed across sites, but the dispersion parameter could vary across the sites, in what so-called unobserved heterogeneity, where the influence of different independent variables on crash frequency may differ across different sites. Several researchers have reported that the dispersion parameters are often site-specific and vary from one site to another ([Wang et al., 2020; Geedipally et al., 2009;](#page-9-0)  Lord & [Park, 2008\)](#page-9-0). For this, this study applied two promising extensions of the NB and CMP models, the heterogeneous negative binomial (HTNB) and heterogeneous CMP (HTCMP) models (that exhibit varying dispersion parameter as a function of site-specific features), along with the random parameter negative binomial (RPNB) modeling approach. The RPNB is an extension of the NB model that has fixed dispersion parameter; however, accounts for unobserved heterogeneity. This study used roundabout-related crashes in the state of Kentucky over a fouryear period (2019–2022) and a comprehensive list of variables, as will be shown next.
+
+#### **2. Prior research**
+
+#### *2.1. Roundabout-Related safety studies*
+
+Several researchers have evaluated the safety effectiveness of roundabouts across the globe ([Persaud et al., 2001; Elvik, 2003; Daniels](#page-8-0)  [et al., 2010\)](#page-8-0). In another study, [Retting et al. \(2001\)](#page-9-0) reported that the conversion of intersections into roundabouts was safety-beneficial, where the overall crash severities and injury crashes were reduced by 38% and 76%, respectively. [Isebrands and Hallmark \(2012\)](#page-8-0) investigated the safety effectiveness of roundabouts in rural areas with high-speed approaches by collecting data from 19 locations in the United States. They reported a reduction of 63% in total crashes and 88% in injury crashes, when intersections were converted to roundabouts.
+
+The safety effectiveness of converting signalized intersections to roundabouts was carried out by [Gross et al. \(2013\)](#page-8-0), while considering 28 sites in the United States and using the empirical Bayes (EB) method. The EB results suggested a roundabout safety benefit, with a reduction in both total and injury crashes. [Elvik \(2017\)](#page-8-0) carried out a study in Europe and reported that the conversion of intersections to roundabouts reduced fatal crashes and injury crashes by 65% and 40%, respectively. [Mamlouk and Souliman \(2019\)](#page-8-0) evaluated the safety effectiveness of single-lane and double-lane roundabouts on crash rates in Arizona. The authors reported that the roundabout conversion resulted in a reduction in crash severity. Furthermore, single-lane roundabouts reduced the crash rates, while double-lane roundabouts increased them.
+
+## *2.2. Roundabout-Related crash frequency studies*
+
+Past studies have largely focused on investigating the safety effectiveness of roundabouts; however, relatively very few studies have developed roundabout crash-specific SPFs. For instance, [Qin et al.](#page-8-0)  [\(2011\)](#page-8-0) evaluated the crash patterns at 41 roundabouts in Wisconsin and reported a reduction in crash severity. They developed nine models using the NB model by considering several variables such as annual average daily traffic (AADT), central island diameter, and number of
+
+circulatory lanes. They reported AADT as the only significant variable in all nine models.
+
+[Kim and Choi \(2012\)](#page-8-0) explored crash data from 14 roundabouts to investigate the factors influencing roundabout crashes in South Korea. The authors developed models to associate the relationship between geometric design elements and occurrence of roundabout crashes. They reported that the NB model had a better ability to explain the occurrence of vehicle crashes at roundabouts compared to the Poisson model. [Kamla](#page-8-0)  [et al. \(2016\)](#page-8-0) investigated the influence of geometric and traffic characteristics on the number of crashes at roundabouts by collecting data from 70 sites in the U.K. The authors developed two count data models (fixed parameter NB and RPNB) and reported RPNB to be the better crash prediction model for roundabouts. They also found that geometric features (e.g., entry width, inscribed circle diameter, number of lanes, and presence of traffic signals) and traffic characteristics (e.g., total approach traffic and truck percentage) significantly influenced the crash frequency at roundabouts.
+
+Recently, [Vinayaraj and Perumal \(2023\)](#page-9-0) employed the NB approach for developing SPFs and estimated the crash modification factors (CMFs) for roundabouts using historical crash data collected over five years (2015–2019) from 21 sites in India. The result showed that AADT, central island diameter, entry angle, posted speed limit, circulating roadway width, number of circulatory lanes, and central island height influenced the occurrence of crashes in the vicinity of roundabouts.
+
+### *2.3. Safety-Related studies using NB-Extension and CMP-Extension models*
+
+The NB model has been extensively used for developing SPFs for different traffic facilities. However, the NB model is restricted to overdispersed count data only and is incapable of handling a small sample size (Lord & [Mannering, 2010; Hosseinpour et al. 2022](#page-8-0)). On the other hand, the CMP model addresses the aforementioned issue; however, both NB and CMP models still restrict the dispersion parameter to be fixed across sites. Therefore, several extensions of NB and CMP model have been used in safety studies across the globe. For example, El-Basyouny and Sayed (2006) developed and compared NB and HTNB models using crash data for 392 arterial segments located in British Columbia, Canada between 1994 and 1996. The authors reported that the HTNB model outperformed the NB model. [Anastasopoulos and](#page-8-0)  [Mannering \(2009\)](#page-8-0) carried out a study to investigate the use of random parameter count models as an alternative methodology for analyzing crash frequencies using five-year crash data (1995 to 1999). The authors reported that random parameter count models could provide a detailed understanding of the variables that influence the crash frequencies. They also highlighted that disregarding the possibility of random parameters could result in substantially different marginal effects and consequent interpretations relating to the magnitude of the effect of factors affecting crash frequencies.
+
+[Khattak et al. \(2021\)](#page-8-0) developed SPFs for crash severity for urban signalized and unsignalized intersections using six-year crash data, geometric design data, and traffic flow data by utilizing three different functional forms of NB models (i.e., NB- type 1, NB-type 2, NB-type P) and a generalized Poisson (GP) model. They compared all developed SPFs based on the goodness of fit and predictive performance measures. The authors reported that the NB-type P model outperformed the other models for signalized intersections and the GP model outperformed the other models for unsignalized intersections. Recently, [Haleem and](#page-8-0)  [Hosseinpour \(2023\)](#page-8-0) developed SPFs for predicting commercial motor vehicle (CMV) crashes on interchange ramp segments using five-year (2015 to 2019) CMV-related crashes while employing four modeling approaches (NB, HTNB, CMP, and HTCMP). The study results indicated that the HTCMP model significantly outperformed the other three comparative models in terms of crash prediction accuracy and goodnessof-fit statistics. In another study, [Haleem and Hosseinpour \(2024\)](#page-8-0) developed SPFs for CMVs at signalized and unsignalized intersections in
+
+<span id="page-2-0"></span>the state of Kentucky using five-year crash data (2015–2019) and employing different modeling approaches (NB, CMP, HTCMP, zeroinflated CMP "ZI-CMP," and ZI-HTCMP). The authors reported that the ZI-HTCMP model outperformed the other models for both unsignalized and signalized intersections.
+
+## *2.4. Summary of literature review and study contribution*
+
+In summary, the majority of the research has been carried out to evaluate the safety effectiveness of roundabouts; however, relatively limited research has been carried out to develop roundabout crashspecific SPFs using different modeling approaches. Furthermore, most previous studies have applied the conventional NB model for developing roundabout specifics SPFs, which could restrict the dispersion parameter to be fixed across the sites and neglect unobserved heterogeneity. Therefore, this study aims to address these gaps by:
+
+- 1. Developing comprehensive roundabout crash-specific SPFs using multiple statistical approaches, including the NB and CMP models with fixed and varying dispersion parameters, along with a myriad list of variables, as shown in [Section 4](#page-3-0) "Data Collection and Extraction".
+- 2. Identifying the key operational and geometrical attributes influencing crashes at roundabouts.
+- 3. Recommending countermeasures to enhance safety at roundabouts.
+
+#### **3. Methodological approaches**
+
+#### *3.1. NB, HTNB, and RPNB modeling approaches*
+
+The NB model has been extensively used for developing SPFs and for predicting the number of crashes (Mannering & [Bhat, 2014; Vinayaraj](#page-8-0) & [Perumal, 2023\)](#page-8-0). The NB model accounts for over-dispersion data. The NB model incorporates an error term into Poisson means and allows the variance to differ from the mean. For example, let *λi*be the predicted number of crashes at roundabout *i,* then the NB model can be expressed as follows:
+
+$$\lambda_i = \exp(\beta X_i + \varepsilon_i) \tag{1}$$
+
+where: *Xi* is a vector of explanatory variables at roundabout *i*, *β* is a vector of estimable parameters, and exp (*ε*i) denotes the Gammadistributed error term with variance equal to *α* (i.e., the dispersion parameter) and mean equal to 1. The inclusion of *εi* permits the variance to differ from the mean. For more information about the NB model, readers are referred to [Washington et al. \(2003\)](#page-9-0) and [Anastasopoulos and](#page-8-0)  [Mannering \(2009\)](#page-8-0).
+
+The NB model assumes the dispersion parameter (*α*) to be fixed across all observations, though this may not be true as the geometry of different facilities vary significantly from each other ([Geedipally](#page-8-0) & Lord, 2008; Anastasopoulos & [Mannering, 2009; Miranda-Moreno et al.,](#page-8-0)  [2011\)](#page-8-0). Therefore, two prominent extensions of the NB model, namely the HTNB and RPNB models, were used in this study. The HTNB model is a flexible approach that addresses the issue of unobserved heterogeneity through allowing the dispersion parameters to vary across crash sites as a function of site-specific characteristics. The HTNB model uses the same probability density function as the NB model and the varying dispersion parameter is expressed as a function of site-specific attributes, as follows:
+
+$$\alpha_i = \frac{1}{\theta_i} = \exp(\gamma Z_i) \tag{2}$$
+
+where: *Z*i denotes the vector of site-specific variables and γ denotes the vector of regression coefficients corresponding to covariates *Z*i. If none of the variables are found significant in the dispersion parameter section, then the HTNB model is reduced to the traditional NB model.
+
+The RPNB is an extension of the NB model, which accounts for the
+
+unobserved heterogeneity issue. The estimable parameters in the count data model can be expressed as follows:
+
+$$\beta_i = \beta + \varphi_i \tag{3}$$
+
+where: *φ*i denotes the randomly distributed term (e.g., a normally distributed term with mean equals to zero and variance equals to σ<sup>2</sup> ), which lets the effects of the parameter to vary across the observations. A parameter is considered random when the standard deviation is greater than zero or else it is considered as a fixed parameter. This suggests that if the parameter is considered as fixed, then *φ*i is zero.
+
+#### *3.2. CMP and HTCMP modeling approaches*
+
+The CMP distribution is a generalization of the Poisson distribution introduced by Conway and Maxwell (1962) to model queue and service rates. The CMP distribution is one of the most popular techniques used by researchers for modeling count data [\(Lord et al., 2008; Haleem](#page-8-0) & [Hosseinpour, 2024\)](#page-8-0). The statistical properties of the CMP distribution were further explained by Shmueli et al. (2005) and the conjugate distributions for the parameters of the CMP distribution were developed by [Kadane et al. \(2006\).](#page-8-0) The probability mass function of the CMP distribution can be expressed as follows:
+
+$$P(y) = \frac{1}{Z(\lambda, \nu)} \frac{\lambda^{y}}{(y!)^{\nu}} \tag{4}$$
+
+$$Z(\lambda, \nu) = \sum_{n=0}^{\infty} \frac{\lambda^n}{(n!)^{\nu}}$$
+ (5)
+
+where: *y* is a discrete count; *λ* is a centering parameter that is approximately the mean of the observations in many cases; and *ν* is defined as the shape parameter or dispersion parameter of the CMP distribution. The important attribute of the CMP distribution is that it can handle both under-dispersed (*ν >* 1) and over-dispersed (*υ <* 1) data. However, when (*υ* = 1), then the Poisson distribution is generated. Using an asymptotic approximation for Z (*λ, υ*), the approximation can be obtained as follows [\(Lord et al., 2008; Huang, 2017](#page-8-0)).
+
+$$E[Y] \approx \lambda^{1/\nu} - \frac{\nu - 1}{2\nu} \tag{6}$$
+
+The HTCMP is also a promising generalization of the standard CMP model that allows the dispersion parameter (*ν*) to vary across observations depending on site-specific characteristics [\(Geedipally](#page-8-0) & Lord, [2011\)](#page-8-0). The (*υ*) of the HTCMP model can be expressed as follows:
+
+$$\ln(v) = \gamma \psi_i \tag{7}$$
+
+where: *ψi*denotes a vector of explanatory variables associated with *υ*  and *γ* denotes a vector of regression coefficients corresponding to covariates *ψ<sup>i</sup>* . If none of the variables are observed to be significant in the dispersion part, then the HTCMP model is reduced to the CMP model.
+
+#### *3.3. Goodness-of-Fit (GOF) statistics*
+
+# *3.3.1. Deviance (D)*
+
+The deviance statistic can be calculated as a Chi-square (*x*<sup>2</sup> ) distribution as follows:
+
+$$D = -2(LL_0 - LL_\beta) \cong \chi^2_{(df,=P)}$$
+(8)
+
+where: *LLβ* denotes the log-likelihood at convergence, *LL0* denotes the log-likelihood of the null model (i.e., constant only model), and *P*  denotes the number of predictors in the model.
+
+## *3.3.2. McFadden pseudo R2*
+
+The McFadden pseudo *R*2 statistic is calculated as follows:
+
+<span id="page-3-0"></span>
+$$McFadden Pseudo R^2 = 1 - (LL_{\beta}/LL_0)$$
+ (9)
+
+where: *LLβ* and *LL0* denote the same as in Equation [\(8\)](#page-2-0). Note that a higher value of McFadden pseudo R2 suggests a better fit model.
+
+## *3.3.3. Akaike information criterion (AIC) and Bayesian information criterion (BIC)*
+
+The AIC and BIC statistics are calculated as follows:
+
+$$AIC = -2*LL_{\beta} + 2*P \tag{10}$$
+
+$$BIC = -2*LL_{\beta} + P*ln(n)$$
+(11)
+
+where: *LLβ*, *LL0*, and *P* denote the same as in Equation [\(8\),](#page-2-0) and *n*  denotes the number of site observations. Note that a lower value of AIC and BIC statistics signifies a better fit model.
+
+# *3.3.4. Likelihood ratio test (LRT)*
+
+The likelihood ratio test (LRT) follows a Chi-square distribution with degrees of freedom equal to the difference in the number of estimated parameters between the two models (best-fit model and comparable model). The LRT statistic is calculated as follows:
+
+$$LRT = 2*\left(LL_{bestmodel} - LL_{compared\ model}\right) \ \cong \ \chi^2_{(df. = P_{best\ model} - P_{compared\ model})} \eqno(12)$$
+
+For more clarity, the adopted research methodology for this study is presented in Fig. 1. As shown in the figure, the flowchart displays the different techniques used in this study.
+
+## **4. Data Collection and Extraction**
+
+Crash data from 33 roundabouts were collected over a four-year period (2019–2022) in the state of Kentucky. Initially, the locations of the roundabouts were identified from Kittelson & [Associates \(2023\)](#page-8-0) and later confirmed from the Kentucky Transportation Cabinet (KYTC) database. Several geometrical and operational data were collected from all the roundabouts. For example, speed limits on major and minor roads were collected from Highway Information View and Extract Interface (HIVEi) KYTC-Query ([2024\)](#page-8-0). Area type and AADT were collected from the KYTC Interactive Statewide Traffic Counts Map 2024 as depicted in [Fig. 2-](#page-4-0)a and 2-b.
+
+It should be noted that AADT was collected at each roundabout during each of the four years (2019 through 2022), then the average AADT (for each of the major and minor approaches) was calculated and used to represent the specific major approach and minor approach
+
+![](_page_3_Figure_16.jpeg)
+
+**Fig. 1.** Adopted Research Methodology.
+
+<span id="page-4-0"></span>![](_page_4_Figure_2.jpeg)
+
+**Fig. 2.** Sample Data Collected Manually at Roundabout Using Google [Maps \(2024\)](#page-8-0) and KYTC Interactive Map 2024: (a) AADT, (b) Area Type, (c) Central Island Diameter (CID), and (d) Inscribed Circle Diameter (ICD).
+
+AADTs. This ensures that the AADT represents an average daily traffic estimate before and during the COVID-19 pandemic. Regarding roundabouts with varying traffic volumes observed on different road directions around them, the average approach AADT in each of the four years (2019–2022) was calculated, then the overall average (for each of the major and minor approaches) was determined and used in the SPF development.
+
+The geometric roundabout attributes, such as inscribed circle diameter (ICD), central island diameter (CID), number of lanes on major and minor approaches, entry width, number of circulatory lanes, circulatory width, and presence or crosswalk and yield markings, were collected manually using Google [Maps \(2024\)](#page-8-0) (refer to Fig. 2-c and 2-d). Furthermore, a 250-foot radius from each roundabout's centroid was drawn using [ArcGIS \(2023\)](#page-8-0) and all crashes within this buffer zone were considered as roundabout-related crashes (refer to [Fig. 3](#page-5-0)). Note that this radius has been deemed sufficient for accurately capturing roundabout crashes (Hu & [Cicchino, 2019](#page-8-0)).
+
+The crash frequency at each roundabout was extracted from [ArcGIS](#page-8-0)  [\(2023\)](#page-8-0) and further used as the dependent variable for developing the roundabout crash-specific SPFs. A total of 621 crashes occurred at 33 roundabouts in Kentucky. The composition of the roundabout crashes was as follows, 5.15% were either involved in suspected serious injuries (A), suspected minor injuries (B), or possible injury (C), whereas 94.85% were involved in property damage only (PDO) crashes. This illustrates that roundabouts have the potential to reduce the severity of crashes as advocated by several researchers across the globe (see for examples: Isebrands & [Hallmark, 2012; Gross et al., 2013](#page-8-0)). Full descriptive statistics of the explored variables are presented in [Table 1](#page-5-0). Note that the mean and variance of roundabout crash data were found to be 18.81 and 1147.60, respectively, which signifies that the crash data were overdispersed. Furthermore, the crash distribution at the 33 roundabouts is depicted in [Fig. 4](#page-5-0).
+
+# **5. Results and Discussion**
+
+## *5.1. Multicollinearity analysis*
+
+The two most used methods for determining data multicollinearity are the Pearson correlation matrix of predictor variables and the variance inflation factor (VIF) [\(Alin, 2010](#page-8-0); Ali et al., 2021; [Cheng et al.,](#page-8-0)  [2022; Pathivada et al., 2024](#page-8-0)). One major drawback of the Pearson correlation matrix is that it could potentially establish a correlation between only two independent variables at once. However, VIF is an effective methodology for determining multicollinearity, which overcomes the drawback of the Pearson correlation matrix. Note that higher VIF values indicate higher collinearity between the independent variables [\(Cheng et al., 2022](#page-8-0)). In this study, both multicollinearity detection techniques between the independent variables were used.
+
+<span id="page-5-0"></span>![](_page_5_Figure_2.jpeg)
+
+![](_page_5_Figure_3.jpeg)
+
+**Fig. 3.** Roundabout-Specific Crash Data Extraction Process: (*a*) Creating Buffer Zones (250 ft) around Roundabout Centroids and (*b*) Sample Crashes within a Buffer Zone.
+
+The results of the Pearson correlation matrix of all the variables explored in the development of roundabout crash-specific SPF are shown in [Fig. 5](#page-6-0). It is suggested that any variable, having a correlation (≥ |0.7|) with other variables, should be removed before moving forward with model development. As shown in [Fig. 5](#page-6-0), no variables were found to have a correlation factor of 0.7 or greater. The VIF values of all the variables explored in the development of roundabout crash-specific SPFs are presented in [Fig. 6](#page-6-0). [Cheng et al. \(2022\)](#page-8-0) reported that a VIF value of less than 10 indicates that there is no multicollinearity between the independent variables. As shown in [Fig. 6,](#page-6-0) there existed no multicollinearity between the independent variables used for developing roundabout crash-specific SPFs in this study.
+
+## *5.2. Roundabout-Specific SPF modeling results*
+
+The results of the five fitted SPF models (NB, HTNB, RPNB, CMP, and HTCMP) are presented in [Table 2](#page-7-0) with a 10% significance level threshold. The bottom portion of [Table 2](#page-7-0) represents the GOF measures for all five models. The deviance statistic for all the developed models was found to be statistically significant (P-value *<* 0.001), indicating rejection of the null hypothesis (i.e., the models had different prediction performances than their corresponding constant-only models).
+
+The GOF measures presented in [Table 2](#page-7-0) indicated that the HTCMP model had the highest McFadden pseudo R2 (0.246) and lowest AIC (215.209) values compared to the other models. Overall, the HTCMP model outperformed the other comparative models for development of roundabout-specific SPFs in Kentucky. The difference in the degrees of freedom between (HTCMP vs. HTNB) was found to be zero; therefore, the LRT test was only conducted between (HTCMP vs. NB), (HTCMP vs. RPNB), and (HTCMP vs. CMP). As shown in [Table 2](#page-7-0), the HTCMP
+
+**Table 1**  Descriptive Statistics of Explanatory Variables Used in Developing Roundabout Crash-Specific SPFs in Kentucky.
+
+| Variable                                               | Variable<br>Description                                    | No. of Sites<br>Level<br>Level<br>1<br>0 |        | Crash Frequency<br>Level 1 | Level 0 |  |
+|--------------------------------------------------------|------------------------------------------------------------|------------------------------------------|--------|----------------------------|---------|--|
+| Indicator Dummy Variables                              |                                                            |                                          |        |                            |         |  |
+| Area type                                              | If urban = 1,                                              | 27                                       | 6      | 545                        | 76      |  |
+| Number of<br>circulatory                               | otherwise = 0<br>If single = 1,<br>otherwise = 0           | 25                                       | 8      | 195                        | 426     |  |
+| lanes "cln"<br>Circulatory road                        | If > 16 feet = 1,                                          | 20                                       | 13     | 326                        | 295     |  |
+| width "crw"<br>Number of lanes<br>on major<br>approach | otherwise = 0<br>If two lanes = 1,<br>otherwise = 0        | 25                                       | 8      | 363                        | 258     |  |
+| Number of lanes<br>on minor<br>approach                | If two lanes = 1,<br>otherwise = 0                         | 27                                       | 6      | 416                        | 205     |  |
+| Crosswalk                                              | If present = 1,<br>otherwise = 0                           | 19                                       | 14     | 480                        | 141     |  |
+| Yield markings                                         | If present = 1,<br>otherwise = 0                           | 27                                       | 6      | 605                        | 16      |  |
+| Number of<br>approaches                                | If four = 1,<br>otherwise = 0                              | 28                                       | 5      | '578                       | 43      |  |
+| Variable                                               | Variable<br>Description                                    | Min.*                                    | Max.*  | Mean                       | SD*     |  |
+| Continuous Variables                                   |                                                            |                                          |        |                            |         |  |
+| Major approach<br>AADT                                 | Major<br>roundabout<br>approach AADT                       | 496                                      | 19,181 | 6692.74                    | 4530.73 |  |
+| Minor approach<br>AADT                                 | Minor<br>roundabout<br>approach AADT                       | 496                                      | 18,550 | 3896.87                    | 3914.42 |  |
+| Major approach<br>posted speed<br>limit<br>"majorpsl"  | Speed limit on<br>major approach<br>(in miles/hour)        | 25                                       | 55     | 36.36                      | 10.10   |  |
+| Minor approach<br>posted speed<br>limit<br>"minorpsl"  | Speed limit on<br>minor approach<br>(in miles/hour)        | 15                                       | 55     | 32.58                      | 12.32   |  |
+| Entry width<br>"ew" (ft)                               | Roundabout<br>entry width (in<br>feet)                     | 10.50                                    | 29.07  | 16.51                      | 3.40    |  |
+| Central island<br>diameter<br>"CID" (ft)               | Diameter of<br>roundabout<br>central island (in<br>feet)   | 31.18                                    | 181.46 | 107.79                     | 32.18   |  |
+| Inscribed circle<br>diameter<br>"ICD" (ft)             | Diameter of<br>roundabout<br>inscribed circle<br>(in feet) | 71.10                                    | 237.88 | 147.48                     | 39.34   |  |
+
+<sup>\*</sup>Min. = minimum; Max. = maximum; and SD = standard deviation.
+
+![](_page_5_Figure_12.jpeg)
+
+**Fig. 4.** Crash Distribution at Roundabouts in Kentucky (2019–2022).
+
+<span id="page-6-0"></span>![](_page_6_Figure_2.jpeg)
+
+**Fig. 5.** Pearson Correlation Matrix of Explored Variables.
+
+![](_page_6_Figure_4.jpeg)
+
+**Fig. 6.** Variance Inflation Factor (VIF) Values of Explored Variables.
+
+outperformed the NB, RPNB, and CMP models at the 90% confidence level (P-value ≤ 0.10).
+
+The SPF model results in [Table 2](#page-7-0) indicated that higher AADT on major and minor roundabout approaches, larger central island diameter, and larger inscribed circle diameter were associated with increased crash frequency at roundabouts. On the other hand, single roundabout circulatory lanes, higher minor approach posted speed limit, circulatory roadway width (*>*16 feet), and two lanes on the major approach reduced the number of crashes at roundabouts.
+
+For the HTNB model, the major and minor approaches' posted speed limits were found to be significantly associated with the dispersion parameter. Similarly, for the HTCMP model, presence of two lanes on minor roundabout approach road was found to be significantly associated with the dispersion parameter. This indicates that the HTCMP dispersion parameter significantly varied by the number of lanes on the minor roundabout approach.
+
+The findings drawn from the best-fit model in [Table 2,](#page-7-0) the HTCMP model, were intuitive. For example, AADT on major and minor roundabout approaches was significantly associated with increased crash frequency at roundabouts. This outcome is expected as AADT reflects the exposure and with higher exposure, the risk of crashes at roundabouts increases. This finding is consistent with previous studies (e.g., [Kamla](#page-8-0)  [et al., 2016; Vinayaraj](#page-8-0) & Perumal, 2023). The ICD was observed to be associated with increased crash frequency at roundabouts. This could be since larger ICD provides a false sense of security for vehicles to increase their circulating speed, which in turn increases crash risk at roundabouts. This finding is consistent with prior studies by [Rodegerdts et al.](#page-9-0)  [\(2007\)](#page-9-0) and [Ferguson et al. \(2018\)](#page-8-0). The presence of single circulatory lanes was found to have a negative association with crash frequency. This implies that single-lane roundabouts tend to have lesser crash frequency compared to multi-lane roundabouts. This could be since multilane roundabouts are inherently more complex, with vehicles operating in multiple lanes simultaneously; hence, increasing the risk of collisions. Furthermore, this risk is further compounded by drivers' lack of awareness of multilane roundabout maneuvers. This finding is consistent with the study by [Mamlouk and Souliman \(2019\)](#page-8-0).
+
+Also, from the HTCMP model results, roundabouts with two lanes on the major approach were associated with lesser crash frequency. This is mainly since major roundabout approaches with two lanes are less prone to conflicting traffic movements compared to approaches with more
+
+<span id="page-7-0"></span>**Table 2**  Parameter Estimates for Roundabout Crash-Specific SPFs in Kentucky.
+
+| Variable                                             | NB               |                   | HTNB      |                                    | RPNB       |                   | CMP           |                   | HTCMP     |         |
+|------------------------------------------------------|------------------|-------------------|-----------|------------------------------------|------------|-------------------|---------------|-------------------|-----------|---------|
+|                                                      | Coef.            | P-Value           | Coef.     | P-Value                            | Coef.      | P-Value           | Coef.         | P-Value           | Coef.     | P-Value |
+| Count part                                           |                  |                   |           |                                    |            |                   |               |                   |           |         |
+| Constant                                             | − 5.782          | 0.000             | − 4.735   | 0.002                              | − 7.449    | 0.000             | − 1.599       | 0.009             | − 1.580   | 0.003   |
+| Natural logarithm of major approach AADT             | 0.714            | 0.000             | 0.905     | 0.000                              | 0.475      | 0.002             | 0.160         | 0.026             | 0.164     | 0.017   |
+| Natural logarithm of minor approach AADT             | 0.345            | 0.023             | N/S       | N/S                                | 0.755      | 0.000             | 0.067         | 0.089             | 0.064     | 0.096   |
+| Single circulatory lanes                             | − 0.958          | 0.000             | − 1.282   | 0.000                              | N/S        | N/S               | − 0.205       | 0.018             | − 0.186   | 0.026   |
+| Central island diameter (CID) (in 100 feet)          | N/S*             | N/S               | 1.283     | 0.009                              | N/S        | N/S               | 0.126         | 0.097             | N/S       | N/S     |
+| Minor approach posted speed limit (mph)              | N/S              | N/S               | − 0.019   | 0.094                              | N/S        | N/S               | N/S           | N/S               | N/S       | N/S     |
+| Circulatory roadway width (>16 feet)                 | N/S              | N/S               | − 0.742   | 0.026                              | N/S        | N/S               | − 1.997       | 0.000             | − 0.214   | 0.019   |
+| Two lanes on major approach                          | N/S              | N/S               | N/S       | N/S                                | N/S        | N/S               | N/S           | N/S               | − 0.177   | 0.029   |
+| Inscribed circle diameter (ICD) (in 100 feet)        | N/S              | N/S               | N/S       | N/S                                | N/S        | N/S               | N/S           | N/S               | 0.198     | 0.030   |
+| Random parameters                                    |                  |                   |           |                                    |            |                   |               |                   |           |         |
+| Two lanes on minor approach                          | na**             | na                | na        | na                                 | − 0.549    | 0.077             | na            | na                | na        | na      |
+| St. deviation of "Two lanes on minor approach"       | na               | na                | na        | na                                 | 0.421      | 0.012             | na            | na                | na        | na      |
+| Dispersion part                                      |                  |                   |           |                                    |            |                   |               |                   |           |         |
+| Constant                                             | 1.044***         | 0.001             | − 3.444   | 0.037                              | 2.866      | 0.013             | − 1.997***    | 0.000             | − 1.710   | 0.000   |
+| Major approach posted speed limit (mph)              | na               | na                | 0.129     | 0.023                              | na         | na                | na            | na                | N/S       | N/S     |
+| Minor approach posted speed limit (mph)              | na               | na                | − 0.084   | 0.055                              | na         | na                | na            | na                | N/S       | N/S     |
+| Two lanes on minor approach                          | na               | na                | N/S       | N/S                                | na         | na                | na            | na                | − 0.298   | 0.017   |
+| Goodness-of-fit (GOF) statistics                     | NB               |                   | HTNB      |                                    | RPNB       |                   | CMP           |                   | HTCMP     |         |
+| Number of observations                               | 33               |                   | 33        |                                    | 33         |                   | 33            |                   | 33        |         |
+| Number of significant parameters                     | 5                |                   | 9         |                                    | 6          |                   | 7             |                   | 9         |         |
+| Log-likelihood at convergence (LLβ)<br>− 103.797     |                  |                   | − 100.944 |                                    | − 107.010  |                   | − 101.35      |                   | − 98.604  |         |
+| Log-likelihood at zero (LL0)                         | − 128.463        |                   | − 128.463 |                                    | − 128.463  |                   | − 130.750     |                   | − 130.750 |         |
+| Deviance Chi-sq. statistic (2*(LLβ − LL0))           | 49.33            |                   | 55.04     |                                    | 42.906     |                   | 58.80         |                   | 64.29     |         |
+| (P-Value < 0.001)<br>(P-Value)                       |                  | (P-Value < 0.001) |           | (P-Value < 0.001)                  |            | (P-Value < 0.001) |               | (P-Value < 0.001) |           |         |
+| McFadden pseudo R2                                   | 0.192            |                   | 0.214     |                                    | 0.167      |                   | 0.225         |                   | 0.246     |         |
+| Akaike Information Criterion (AIC)<br>217.594        |                  | 219.887           |           | 226.000                            |            | 216.710           |               | 215.209           |           |         |
+| Bayesian Information Criterion (BIC)<br>225.077      |                  | 233.357           |           | 234.999                            |            | 227.176           |               | 228.677           |           |         |
+| Likelihood Ratio Test (LRT): Chi-sq. (d.f., P-value) | HTCMP vs. NB     |                   | na        |                                    |            |                   | HTCMP vs. CMP |                   | na        |         |
+|                                                      | Chi-sq. = 10.386 |                   |           | HTCMP vs. RPNB<br>Chi-sq. = 16.812 |            | Chi-sq. = 5.492   |               |                   |           |         |
+|                                                      | (4, 0.034)       |                   |           |                                    | (3, 0.000) |                   | (2, 0.064)    |                   |           |         |
+
+<sup>\*</sup> N/S = not significant. \*\* na = not applicable.
+
+than two lanes. The circulatory roadway (*>* 16 feet) was observed to have a significant negative association with crash frequency at roundabouts. This implies that roundabouts with circulatory roadways greater than 16 feet are safer than roundabouts with circulatory roadways lesser than 16 feet. This is mainly since wider circulatory roadway width provides adequate room for rectification of any errors committed by drivers while navigating roundabouts.
+
+## **6. Conclusions and study recommendations**
+
+#### *6.1. Main study Conclusions*
+
+Numerous studies have advocated the conversion of intersections into roundabouts due to their safety benefits. However, relatively limited research has been conducted to develop roundabout crashspecific SPFs (roundabout crash prediction models) using different modeling approaches. To fill this gap, this study investigated crash frequency at roundabouts in Kentucky and developed roundabout crashspecific SPFs using various geometrical and operational attributes of roundabouts. Crash data from 33 roundabouts over four years (2019–2022) in the state of Kentucky were retrieved from KYTC and used in the analysis. Furthermore, additional effort was made to manually collect the geometric attributes (e.g., CID and ICD) at each roundabout for developing the SPFs. The crash data exhibited overdispersion; therefore, the NB and CMP modeling approaches, with fixed and varying dispersion parameters, were applied in the study while accounting for unobserved heterogeneity across the roundabout locations.
+
+Five modeling techniques (NB, HTNB, RPNB, CMP, and HTCMP) were employed for the development of roundabout crash-specific SPFs. The performance of all the models were evaluated using GOF measures, such as McFadden pseudo R<sup>2</sup> , AIC, BIC, and LRT statistics. The results of the GOF measures indicated the HTCMP was the best-fit model for developing roundabout crash-specific SPFs in the state of Kentucky. Overall, the study findings reveal that it is imperative, when modeling roundabout crashes, to use a count data model that accounts for unobserved heterogeneity (i.e., a model with heterogeneous dispersion parameter, e.g., the HTCMP model) than using a model that merely accounts for random parameters (i.e., the RPNB model).
+
+The results of the HTCMP model highlighted that AADT of both major and minor roundabout approaches and ICD had a significant association with increased crash frequency at roundabouts. Conversely, single circulatory roundabout lanes, presence of two lanes on major roundabout approach, and circulatory roadway width (*>* 16 feet) were significantly associated with reduced crash frequency at roundabouts.
+
+## *6.2. Roundabout safety recommendations*
+
+The following recommendations are suggested based on the study findings to help improve roundabout safety:
+
+- 1. Consider adding single-lane roundabouts, wherever possible, in those areas where traffic is not generally high, as they are safer and less confusing than those multi-lane roundabouts.
+- 2. Provide roundabout circulatory roadway widths greater than 16 feet to provide adequate room for rectification of any errors committed by drivers who are navigating roundabouts.
+- 3. The study results could be amalgamated with driver assistance and vehicle-to-vehicle communication technology as part of Intelligent Transportation Systems (ITS) to help drivers navigate roundabouts safely and efficiently with a diminished risk of crashes.
+
+<sup>\*\*\*</sup> Constant equivalent to "ln (dispersion parameter)" for NB and CMP models (i.e., models with fixed dispersion parameter).
+
+### <span id="page-8-0"></span>*6.3. Limitations and future scope*
+
+It should be noted that the findings and safety recommendations of this study are specific to roundabouts in Kentucky. However, the findings and safety recommendations can be transferable to other states in the United States or to other regions having similar roundabout configurations, roundabout crashes, and traffic characteristics as those in Kentucky. For future research, it is recommended to develop roundabout SPFs in other states/regions and compare the results to those SPFs developed in Kentucky. This will help compare the influence of the significant variables from the models and how they differ or concur.
+
+#### **CRediT authorship contribution statement**
+
+**Tathagatha Khan:** Writing – original draft, Software, Methodology, Investigation, Formal analysis. **Bharat Kumar Pathivada:** Writing – review & editing, Methodology, Formal analysis. **Arunabha Banerjee:**  Writing – review & editing, Methodology. **Kirolos Haleem:** Writing – review & editing, Supervision, Conceptualization.
+
+#### **Declaration of competing interest**
+
+The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
+
+#### **Acknowledgment**
+
+The authors would like to acknowledge the Kentucky Transportation Cabinet (KYTC) for the grant provided to conduct this research and for providing the necessary crash and roadway data used in this research. The opinions, findings, and conclusions in this paper are those of the authors and not necessarily those of KYTC.
+
+## **References**
+
+- Alin, A. (2010). Multicollinearity. *[Wiley Interdisciplinary Reviews: Computational Statistics,](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0010)  2*[\(3\), 370](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0010)–374.
+- [Al-Omari, M. E., Abdel-Aty, M. A., Lee, J., Yue, L., & Abdelrahman, A. \(2020\). Safety](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0015) [evaluation of median U-turn crossover-based intersections.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0015) *Transportation Research [Record: Journal of the Transportation Research Board, 2674](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0015)*(7), 206–218.
+- American Association of State Highway and Transportation Officials (AASHTO). (2010). Highway Safety Manual (First edition). Washington, D.C.
+- [Anastasopoulos, P. C., & Mannering, F. L. \(2009\). A note on modeling vehicle accident](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0025)  [frequencies with random parameters count models.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0025) *Accident Analysis & Prevention, 41*[\(1\), 153](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0025)–159.
+- ArcGIS Version 10.8.1. ESRI. Environmental Systems Research Institute, Redlands, CA, 2023.
+- [Cheng, J., Sun, J., Yao, K., Xu, M., & Cao, Y. \(2022\). A variable selection method based](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0035)  [on mutual information and variance inflation factor.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0035) *Spectrochimica Acta Part A: [Molecular and Biomolecular Spectroscopy, 268](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0035)*, Article 120652.
+- [Daniels, S., Brijs, T., Nuyts, E., & Wets, G. \(2010\). Explaining variation in safety](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0045)  [performance of roundabouts.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0045) *Accident Analysis & Prevention, 42*(2), 393–402.
+- [Daniels, S., Brijs, T., Nuyts, E., & Wets, G. \(2011\). Extended prediction models for crashes](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0050)  at roundabouts. *[Safety Science, 49](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0050)*(2), 198–207.
+- [Elvik, R. \(2003\). Effects on road safety of converting intersections to roundabouts:](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0055) [Review of evidence from non-US studies.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0055) *Transportation Research Record: Journal of [the Transportation Research Board, 1847](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0055)*(1), 1–10.
+- [Elvik, R. \(2017\). Road safety effects of roundabouts: A meta-analysis.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0060) *Accident Analysis & [Prevention, 99](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0060)*, 364–371.
+- Federal Highway Administration (FHWA). About Intersection safety, FHWA (dot.gov). https://highways.dot.gov/safety/intersection-safety/about. Accessed January 7, 2024.
+- Ferguson, E., Bonneson, J., Rodegerdts, L., Foster, N., Persaud, B., Lyon, C., and Rhoades, D. (2018). Development of roundabout crash prediction models and methods. National Cooperative Highway Research Program (NCHRP) Project 17-70, Transportation Research Board, Washington, D.C.
+- [Geedipally, S. R., & Lord, D. \(2008\). Effects of Varying Dispersion Parameter of Poisson](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0075) [Gamma Models on Estimation of Confidence Intervals of Crash Prediction Models.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0075) *[Transportation Research Record: Journal of the Transportation Research Board, 2061](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0075)*(1), 46–[54](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0075).
+- [Geedipally, S., & Lord, D. \(2011\). Examination of Crash Variances Estimated by](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0080) Poisson–Gamma and Conway–Maxwell– Poisson Models. *[Transportation Research](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0080) [Record: Journal of the Transportation Research Board, 2241](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0080)*, 59–67.
+
+- [Geedipally, S., Lord, D., & Park, B. J. \(2009\). Analyzing Different Parameterizations of](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0085)  [the Varying Dispersion Parameter as a Function of Segment Length.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0085) *Transportation [Research Record: Journal of the Transportation Research Board, 2103](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0085)*, 108–118.
+- Google Maps. https://google.maps.com. Accessed March 9, 2024.
+- [Gross, F., Lyon, C., Persaud, B., & Srinivasan, R. \(2013\). Safety effectiveness of](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0095) [converting signalized intersections to roundabouts.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0095) *Accident Analysis & Prevention, 50*[, 234](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0095)–241.
+- [Haleem, K., & Hosseinpour, M. \(2023\). Developing safety performance functions for](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0100)  [commercial motor vehicle crashes at interchange ramp segments in Kentucky.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0100) *[Transportation Research Record: Journal of the Transportation Research Board, 2677](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0100)*(8),
+- [Haleem, K., & Hosseinpour, M. \(2024\). Comprehensive investigation of commercial](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0105)  [motor vehicle crashes at signalized and unsignalized intersections in Kentucky.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0105) *[Transportation Research Record: Journal of the Transportation Research Board, 2678](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0105)*(1), 350–[365](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0105).
+- [Hauer, E., Harwood, D. W., Council, F. M., & Griffith, M. S. \(2002\). Estimating safety by](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0110)  the empirical Bayes method. A tutorial. *[Transportation Research Record: Journal of the](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0110)  [Transportation Research Board, 1784](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0110)*, 126–131.
+- [Hosseinpour, M., Love, R., Williams, B., & Haleem, K. \(2022\). Comprehensive](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0115)  [investigation of commercial motor vehicle crashes along roadway segments in](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0115)  Kentucky. *[Transportation Research Record: Journal of the Transportation Research](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0115) [Board, 2676](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0115)*(7), 655–671.
+- [Hu, W., & Cicchino, J. \(2019\). Long-term crash trends at single-and double-lane](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0120) [roundabouts in Washington State.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0120) *Journal of Safety Research, 70*, 207–212.
+- [Huang, A. \(2017\). Mean-parametrized Conway](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0125)–Maxwell–Poisson regression models for dispersed counts. *[Statistical Modelling, 17](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0125)*(6), 359–380.
+- [Isebrands, H., & Hallmark, S. \(2012\). Statistical analysis and development of crash](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0130)  [prediction model for roundabouts on high-speed rural roadways.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0130) *Transportation [Research Record: Journal of the Transportation Research Board, 2312](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0130)*(1), 3–13.
+- [Isebrands, H. \(2009\). Crash analysis of roundabouts at high-speed rural intersections.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0135) *[Transportation Research Record: Journal of the Transportation Research Board, 2096](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0135)*(1), 1–[7.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0135)
+- [Kadane, J. B., Shmueli, G., Minka, T. P., Borle, S., & Boatwright, P. \(2006\). Conjugate](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0140) [analysis of the Conway](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0140)–Maxwell–Poisson distribution. *Bayesian Analysis, 1*, 363–[374](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0140).
+- [Kamla, J., Parry, T., & Dawson, A. \(2016\). Roundabout accident prediction model:](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0145)  [Random-parameter negative binomial approach.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0145) *Transportation Research Record: [Journal of the Transportation Research Board, 2585](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0145)*(1), 11–19.
+- Kentucky Traffic Collision Facts. (2023). Report number: KTC-24-05 doi: 10.13023.ktc. rr.2024.05. Accessed April 20, 2024.
+- Kentucky Transportation Cabinet (KYTC) Interactive Statewide Traffic Counts Map. https://maps.kytc.ky.gov/trafficcounts/. Accessed March 9, 2024. Kentucky Transportation Cabinet. Highway Information View and Extract Interface (HIVEi) KYTC – Query. https://datamart.kytc.ky.gov/edsb\_solutions/hisextracts/. Accessed January 7, 2024.
+- [Khan, T., & Mohapatra, S. S. \(2021\). Effect of operational attributes on lateral merging](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0165) [position characteristics at mid-block median opening.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0165) *Transportation Letters, 13*(2), 83–[96](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0165).
+- [Khattak, M. W., Pirdavani, A., De Winne, P., Brijs, T., & De Backer, H. \(2021\). Estimation](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0170)  [of safety performance functions for urban intersections using various functional](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0170)  [forms of the negative binomial regression model and a generalized Poisson](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0170)  [regression model.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0170) *Accident Analysis & Prevention, 151*, Article 105964.
+- [Kim, S., & Choi, J. \(2013\). Safety analysis of roundabout designs based on geometric and](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0175)  speed characteristics. KSCE. *[Journal of Civil Engineering, 17](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0175)*, 1446–1454.
+- Kittelson & Associates, Inc. Roundabout database. https://roundabout.kittleson.com/. Accessed April 25, 2023.
+- [Li, L., Zhang, Z., Xu, Z. G., Yang, W. C., & Lu, Q. C. \(2024\). The role of traffic conflicts in](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0185)  [roundabout safety evaluation: A review.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0185) *Accident Analysis & Prevention, 196*, Article [107430](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0185).
+- [Lord, D., & Mannering, F. L. \(2010\). The Statistical Analysis of Crash-Frequency Data: A](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0190)  [Review and Assessment of Methodological Alternatives.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0190) *Transportation Research Part [A: Policy and Practice, 44](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0190)*(5), 291–305.
+- [Lord, D., & Park, P. Y. J. \(2008\). Investigating the Effects of the Fixed and Varying](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0195) [Dispersion Parameters of Poisson- Gamma Models on Empirical Bayes Estimates.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0195)  *[Accident Analysis](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0195) & Prevention, 40*(4), 1441–1457.
+- [Lord, D., Guikema, S. D., & Geedipally, S. R. \(2008\). Application of the](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0200) Conway–Maxwell–[Poisson Generalized Linear Model for Analyzing Motor Vehicle](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0200) Crashes. *[Accident Analysis](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0200) & Prevention, 40*(3), 1123–1134.
+- [Mamlouk, M., & Souliman, B. \(2019\). Effect of traffic roundabouts on accident rate and](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0205)  severity in Arizona. *[Journal of Transportation Safety](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0205) & Security, 11*(4), 430–442.
+- [Mannering, F. L., & Bhat, C. R. \(2014\). Analytic Methods in Accident Research:](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0210) [Methodological Frontier and Future Directions.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0210) *Analytic Methods in Accident [Research, 1](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0210)*, 1–22.
+- [Miranda-Moreno, L. F., Morency, P., & El-Geneidy, A. M. \(2011\). The Link Between Built](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0215)  [Environment, Pedestrian Activity and Pedestrian](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0215)–Vehicle Collision Occurrence at [Signalized Intersections.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0215) *Accident Analysis & Prevention, 43*(5), 1624–1634.
+- [Pathivada, B. K., Banerjee, A., & Haleem, K. \(2024\). Impact of real-time weather](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0220) [conditions on crash injury severity in Kentucky using the correlated random](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0220) [parameters logit model with heterogeneity in means.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0220) *Accident Analysis & Prevention, 196*[, Article 107453.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0220)
+- [Persaud, B. N., Retting, R. A., Garder, P. E., & Lord, D. \(2001\). Safety effect of](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0225) [roundabout conversions in the United States: Empirical bayes observational before](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0225)after study. *[Transportation Research Record: Journal of the Transportation Research](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0225) [Board, 1751](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0225)*(1), 1–8.
+- [Qin, X., Khan, G., Bill, A., & Noyce, D. A. \(2011\). Comprehensive safety evaluation of](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0230)  roundabouts in Wisconsin. *[Journal of Transportation Safety](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0230) & Security, 3*(4), 289–303.
+
+- <span id="page-9-0"></span>[Retting, R. A., Persaud, B. N., Garder, P. E., & Lord, D. \(2001\). Crash and injury reduction](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0235)  [following installation of roundabouts in the United States.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0235) *American Journal of Public [Health, 91](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0235)*(4), 628–631.
+- [Rodegerdts, L., Blogg, M., Wemple, E., Myers, E., Kyte, M., Dixon, M., & Carter, D.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0240) (2007). *[Roundabouts in the United States \(National Cooperative Highway Research](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0240) Program Report, No. 572)*[. Washington, DC: Transportation Research Board](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0240).
+- [Vinayaraj, V. S., & Perumal, V. \(2023\). Developing safety performance functions and](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0245) [crash modification factors for urban roundabouts in heterogeneous non-lane-based](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0245) traffic conditions. *[Transportation Research Record: Journal of the Transportation](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0245) [Research Board, 2677](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0245)*(8), 644–661.
+- [Wang, K., Zhao, S., & Jackson, E. \(2020\). Investigating Exposure Measures and](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0250) [Functional Forms in Urban and Suburban Intersection Safety Performance Functions](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0250)  [Using Generalized Negative Binomial-P Model.](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0250) *Accident Analysis & Prevention, 148*, [Article 105838](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0250).
+- [Washington, S. P., Karlaftis, M. G., & Mannering, F. L. \(2003\).](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0255) *Statistical and Econometric [Methods for Transportation Data Analysis](http://refhub.elsevier.com/S0022-4375(25)00049-0/h0255)*. New York, NY: Chapman & Hall/CRC.
+
+**Tathagatha Khan**, is a Postdoctoral Research Associate at the Center for Transportation Research (CTR) in the School of Engineering and Applied Sciences at Western Kentucky University. His main interests are traffic safety analysis, driver behavior, human factors, and traffic flow modeling. He also holds a Doctorate Degree in transportation engineering.
+
+**Bharat Pathivada**, is a Post-Doctoral Research Associate at the Center for Transportation Research (CTR) in the School of Engineering and Applied Sciences at Western Kentucky University (WKU). He has over two years of research experience in the field of transportation engineering with a focus on traffic safety and human factors. His research interests include Traffic Safety Analysis, Crash Modeling, Road User Behavior, and Intelligent Transportation Systems (ITS). He has a Doctorate Degree in transportation engineering and has participated in accomplishing several externally funded projects.
+
+**Arunabha Banerjee**, is a Postdoctoral Research Associate at the Center for Transportation Research (CTR) in the School of Engineering and Applied Sciences at Western Kentucky University. His main interests are traffic safety analysis, crash modeling, and pedestrian walking behavior modeling. He also holds a Doctorate Degree in transportation engineering.
+
+**Kirolos Haleem**, is an associate professor of civil (transportation) engineering at Western Kentucky University (WKU). Dr. Haleem is the founder & director of the Center for Transportation Research (CTR) at WKU. Kirolos received his master's and Ph.D. degrees in civil (transportation) engineering from the University of Central Florida (UCF) in 2007 and 2009, respectively. He has more than 15-year experience in transportation engineering. His areas of expertise include highway traffic safety analysis, application of statistical models and data mining techniques in transportation data analysis, driving simulator studies, and intelligent transportation systems (ITS). Kirolos led and conducted numerous transportation research projects sponsored by the Kentucky Department of Transportation (Kentucky DOT), Alabama Department of Transportation (ALDOT), Florida DOT, and the U.S. DOT.

Zotero/001_artiklid/Road Safety Indicators/khazraeebayesianpoissonhierarchicalmodels2018.md

@@ -0,0 +1,538 @@
+---
+category: literaturenote
+citekey: khazraeebayesianpoissonhierarchicalmodels2018
+title: "Bayesian Poisson hierarchical models for crash data analysis: Investigating the impact of model choice on site-specific predictions"
+authors: "Khazraee, S. Hadi; Johnson, Valen; Lord, Dominique"
+year: 2018
+date: 2018-00-00 2018
+doi: 10.1016/j.aap.2018.04.016
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2018.04.016"
+zotero_key: EC8U6TDT
+zotero_storage: 3FAJSKU6
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Khazraee, S. Hadi"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+## Contents lists available at [ScienceDirect](http://www.sciencedirect.com/science/journal/00014575)
+
+# Accident Analysis and Prevention
+
+journal homepage: [www.elsevier.com/locate/aap](https://www.elsevier.com/locate/aap)
+
+![](_page_0_Picture_5.jpeg)
+
+# Bayesian Poisson hierarchical models for crash data analysis: Investigating the impact of model choice on site-specific predictions
+
+![](_page_0_Picture_7.jpeg)
+
+S. Hadi Khazraee[a](#page-0-0),[⁎](#page-0-1) , Valen Johnson[b](#page-0-2) , Dominique Lord[c](#page-0-3)
+
+- <span id="page-0-0"></span><sup>a</sup> Uber Technologies, Inc., San Francisco, CA, 94103, United States
+- <span id="page-0-2"></span><sup>b</sup>Department of Statistics, Texas A&M University, College Station, TX, 77843-3143, United States
+- <span id="page-0-3"></span>Zachry Department of Civil Engineering, Texas A&M University, College Station, TX, 77843-3136, United States
+
+### ARTICLE INFO
+
+Keywords: Poisson hierarchical Bayesian model Site-specific prediction Model choice Poisson-inverse gamma
+
+# ABSTRACT
+
+The Poisson-gamma (PG) and Poisson-lognormal (PLN) regression models are among the most popular means for motor vehicle crash data analysis. Both models belong to the Poisson-hierarchical family of models. While numerous studies have compared the overall performance of alternative Bayesian Poisson-hierarchical models, little research has addressed the impact of model choice on the expected crash frequency prediction at individual sites. This paper sought to examine whether there are any trends among candidate models predictions e.g., that an alternative model's prediction for sites with certain conditions tends to be higher (or lower) than that from another model. In addition to the PG and PLN models, this research formulated a new member of the Poissonhierarchical family of models: the Poisson-inverse gamma (PIGam). Three field datasets (from Texas, Michigan and Indiana) covering a wide range of over-dispersion characteristics were selected for analysis.
+
+This study demonstrated that the model choice can be critical when the calibrated models are used for prediction at new sites, especially when the data are highly over-dispersed. For all three datasets, the PIGam model would predict higher expected crash frequencies than would the PLN and PG models, in order, indicating a clear link between the models predictions and the shape of their mixing distributions (i.e., gamma, lognormal, and inverse gamma, respectively). The thicker tail of the PIGam and PLN models (in order) may provide an advantage when the data are highly over-dispersed. The analysis results also illustrated a major deficiency of the Deviance Information Criterion (DIC) in comparing the goodness-of-fit of hierarchical models; models with drastically different set of coefficients (and thus predictions for new sites) may yield similar DIC values, because the DIC only accounts for the parameters in the lowest (observation) level of the hierarchy and ignores the higher levels (regression coefficients).
+
+# 1. Introduction
+
+Over the past three decades, highway safety researchers have been concerned with developing statistical models to predict motor vehicle crashes. The purpose of such models is to relate the frequency of crashes to the traffic, geometrical, and/or environmental characteristics of highway entities. A mass of published work in the field of crash data modeling has focused on evaluating the application of countless statistical models by comparing their goodness-of-fit (GOF) to field data sets. These studies seek to answer the question of "which model performs better?" Measuring the GOF of statistical models is not a straightforward task; researchers use many different methods for assessing GOF. These methods can yield contradictory results in determining which model performs best.
+
+Even if there was a consensus among researchers in employing a
+
+unique method for GOF assessment, the performance of different models would depend on the characteristics of the data set to which they are fitted. Based on the model structure and application to a limited number of datasets, some studies have suggested that certain models are expected to perform better than other specified models for data with certain characteristics. For example, [Geedipally et al. \(2012\)](#page-13-0) noted that the heavy-tailed negative binomial-Lindley (NB-L) distribution is expected to outperform the negative binomial model for datasets with abundant zero crash observations and a long/heavy tail. While rules of thumb like the aforementioned may be valid, it is difficult to predict with certainty which model performs better before the models are actually fitted to the data. Therefore, one must be very cautious when making conclusions of the type "one model is better than another" based on a comparison of the models' GOF to a few data sets.
+
+GOF analyses in the literature of crash data modeling have been
+
+E-mail addresses: [hadi@uber.edu](mailto:hadi@uber.edu) (S.H. Khazraee), [vjohnson@stat.tamu.edu](mailto:vjohnson@stat.tamu.edu) (V. Johnson), [d-lord@tamu.edu](mailto:d-lord@tamu.edu) (D. Lord).
+
+<span id="page-0-1"></span><sup>⁎</sup> Corresponding author.
+
+focused on the overall fit of a model to an entire dataset. An important question that has often been overlooked is "how are the site-specific predicted crash frequencies affected by model choice?" Can we find a trend in the difference between the predicted crash frequencies of different models; e.g., that a model's prediction for sites with certain conditions is higher (or lower) than that from a different model? This paper conducts such a microscopic analysis for three hierarchical-Poisson regression models, including the two most popular models in crash data analysis, the Poisson-gamma and Poisson-lognormal models. We also examine a new model, the Poisson-Inverse Gamma (PIGam) model; note: the PIGam should not be confused with the Poisson-Inverse Gaussian (PIG) model, as detailed in [Zha et al. \(2016\)](#page-14-0). The PIGam model was formulated and added to this study to investigate the potential benefit of having a long/heavy-tailed mixing distribution (i.e., inverse Gamma) in handling data sets with unusually high crash count observations. Three crash data sets with a range of dispersion characteristics were used to compare the performance of the candidate models.
+
+This research is concerned with the Bayesian estimation of Poissonhierarchical models. The Bayesian structure of the models and the practical interpretation of the model parameters are described in detail and distinguished from the traditional frequentist models. The models' predictions are compared at two levels: a) for sites in the dataset where an observed crash count is available, and b) for hypothetical new sites without an observed crash count. Rather than considering several candidate models and employing the best one (based on GOF comparisons) for prediction, this study aims to add depth to our understanding of how the magnitude of predictions is influenced if each of the three considered models is selected.
+
+#### 2. Background
+
+Because crashes are random events and typically independent of one another, the Poisson distribution is the intuitive means to describe the randomness in crash counts. In practice, however, the Poisson is rarely an appropriate distribution because the crash data are often over-dispersed ([Poch and Mannering, 1996](#page-14-1); [Hauer, 2001;](#page-13-1) [Mitra and](#page-13-2) [Washington, 2006](#page-13-2)), meaning that the conditional variance of observed crash counts is greater than the mean, whereas the Poisson distribution is equi-dispersed (i.e., the mean equals the variance).
+
+The over-dispersion in crash count data is mainly attributed to the heterogeneity among the different sites (highway segments, intersections, etc.) where crash data is collected [\(Hauer, 2001;](#page-13-1) [Washington](#page-14-2) [et al., 2010;](#page-14-2) [Lord and Mannering, 2010](#page-13-3); [Mannering and Bhat, 2014](#page-13-4); [Mannering et al., 2016\)](#page-13-5). Although the effect of some important factors (such as traffic volume) on the expected number (mean) of crashes is usually accounted for by a regression model, some degree of heterogeneity is always believed to remain unobserved due to factors either unknown or known but hard to collect and include in the model. In addition, [Lord et al. \(2005\)](#page-13-6) showed that the theory behind the fundamental crash process itself gives rise to over-dispersion. They argued that a crash count is the sum of a series of Bernoulli random variables, with each trial representing a vehicle/driver going through a site with an un-equal probability of success (crash). These assumptions lead to an over-dispersed distribution of crash counts.
+
+To accommodate for possible over-dispersion in crash data, researchers have extensively used mixed-Poisson models, where crash counts are assumed to have a Poisson distribution with a variable mean that follows an underlying distribution often referred to as the mixing distribution. Mixed-Poisson models are indeed hierarchical, where at the lowest level of hierarchy, conditional on the mean, the observed crash counts are mutually independent and Poisson-distributed while at the second level, the unobservable mean of crash counts varies across sites with an assumed probability distribution. Importantly, hierarchical Poisson models particularly suit the theoretical nature of crash frequency data. Such conceptual suitability, often referred to as "goodness-of-logic", has been sacrificed for a better statistical goodnessof-fit (GOF) in a number of regression models proposed for crash data analysis [\(Miaou and Lord, 2003\)](#page-13-7).
+
+Two types of mixed-Poisson models have gained extensive attention among highway safety researchers: Poisson-gamma, and Poisson-lognormal:
+
+### 1) Poisson-Gamma (PG)
+
+When the Poisson parameter is assumed to have a gamma probability distribution, the mixed-Poisson distribution will have a closed form probability density function (pdf) which turns out to be that of the Negative Binomial (NB). The over-dispersion parameter (α) of the NB distribution captures the unmodeled heterogeneity ([Miaou and Lord,](#page-13-7) [2003\)](#page-13-7). The value of this parameter defines the relationship between the distribution mean and variance as Var(yi) = E[yi] + αE[yi] . The simplicity of model fitting and parameter interpretation has made the Poisson-gamma/NB the most popular model in crash data analysis. In its classical applications, modelers estimated the parameters using the Maximum Likelihood Estimate (MLE) method (e.g. [Maycock and Hall,](#page-13-8) [1984;](#page-13-8) [Hauer et al., 1989](#page-13-9); [Bonneson and McCoy, 1993](#page-13-10); [Vogt and Bared,](#page-14-3) [1998\)](#page-14-3). Recently, the Poisson-gamma models have also been estimated using Bayesian methods ([Schluter et al., 1997](#page-14-4); [Miaou and Lord, 2003](#page-13-7); [Miaou and Song, 2005](#page-13-11); [Lord and Miranda-Moreno, 2008\)](#page-13-12).
+
+### 2) Poisson-Lognormal (PLN)
+
+The Poisson-lognormal model results when the Poisson parameter is assumed to follow a lognormal distribution. Unlike the Poisson-gamma model, the marginal distribution of the Poisson-lognormal model does not have a closed form and the model parameters cannot be estimated analytically using the Maximum Likelihood Estimates (MLE) method. Despite the availability of Hinde'[s \(1982\)](#page-13-13) numerical integration method to approximate the MLE parameters, the Poisson-lognormal safety performance models in the published studies have all been estimated using the Bayesian approach. The Poisson-lognormal model is potentially more flexible than the Poisson-gamma [\(Lord and Mannering,](#page-13-3) [2010\)](#page-13-3) and the model has become increasingly popular in crash data analysis over the past few years ([Miaou et al., 2003](#page-13-14); [Aguero-Valverde](#page-13-15) [and Jovanis, 2008;](#page-13-15) [Aguero-Valverde, 2013](#page-13-16)). Highway safety researchers have also used multivariate Poisson-lognormal models to jointly model crash frequency by severity while accounting for the correlation among different severity levels [\(Park and Lord, 2007](#page-14-5); [Ma](#page-13-17) [et al., 2008](#page-13-17); [El-Basyouny and Sayed, 2009\)](#page-13-18).
+
+The MLE method has traditionally been used to estimate safety performance model parameters. Nonetheless, the MLE method cannot be used straightforwardly when the marginal likelihood function is difficult to characterize, as in the case of the Poisson-lognormal model. Nonetheless, the development of such models and others with complex functional forms was greatly facilitated after the rediscovery of Markov Chain Monte Carlo (MCMC) simulation methods ([Tanner and Wong,](#page-14-6) [1987;](#page-14-6) [Gelfand and Smith, 1990](#page-13-19)) for model estimation from the Bayesian perspective. Unlike the traditional frequentist approach, the Bayesian approach to statistics aims to estimate the probability distribution of model parameters using the information in the observed data as well as the prior knowledge about model parameters. The drastic growth in the processing speed of personal computers and availability of Bayesian software programs such as WinBUGS ([Spiegelhalter et al., 2003](#page-14-7)) and MLwiN [\(Yang et al., 1999\)](#page-14-8) has helped significantly in the increasing popularity of Bayesian models.
+
+The fully (hierarchical) Bayesian (FB) estimation of safety performance models has been explored only in the last two decades ([Schluter](#page-14-4) [et al., 1997;](#page-14-4) [Davis and Yang, 2001](#page-13-20); [Miaou and Lord, 2003](#page-13-7); [Carriquiry](#page-13-21) and [Pawlovich, 2004;](#page-13-21) [Miaou and Song, 2005](#page-13-11); [Park and Lord, 2007](#page-14-5); [Daziano et al., 2013](#page-13-22)). Prior to FB models, however, the empirical Bayes (EB) method was introduced into the highway safety literature [\(Hauer](#page-13-23) [and Persaud, 1983;](#page-13-23) [Hauer, 1986,](#page-13-24) [1992\)](#page-13-25). The EB method basically uses the Bayes rule to combine the information from some reference population (or results of a regression model) with the observed crash counts at a certain site to estimate the expected (long-term) mean of crashes. There is extensive documentation and application of the EB method in highway safety, especially in before-after studies to estimate the effect of safety countermeasures [\(Hauer, 1997;](#page-13-26) [Persaud, 1988;](#page-14-9) [Harwood et al.,](#page-13-27) [2002;](#page-13-27) [Persaud and Lyon, 2007;](#page-14-10) [Fitzpatrick and Park, 2009\)](#page-13-28). EB methods have also been used for identification of hotspot locations ([Persaud et al., 1999](#page-14-11); [Heydecker and Wu, 2001](#page-13-29); [Miranda-Moreno et al.,](#page-13-30) [2005;](#page-13-30) [Lord and Park, 2008\)](#page-13-31).
+
+The FB approach has a fundamental advantage over the MLE and EB methods; it takes into account the uncertainty associated with model parameters and provides exact measures of uncertainty [\(Miaou and](#page-13-7) [Lord, 2003](#page-13-7)). In the MCMC method, this is carried out by sampling from the posterior distribution of model parameters. The MLE and EB methods, on the other hand, ignore this uncertainty and thus overestimate the model precision [\(Carriquiry and Pawlovich, 2004](#page-13-21); [Goldstein, 2010;](#page-13-32) [Park et al., 2010](#page-14-12)). This advantage of the FB approach is especially important when the sample size is relatively small [\(Miaou](#page-13-7) [and Lord, 2003](#page-13-7)).
+
+The performance of safety prediction models is often evaluated and compared using statistical GOF measures. In the Bayesian paradigm, measures such as the Bayes factor, Bayesian information criterion (BIC), Watanabe-Akaike information criterion (WAIC), and others are used to assess and compare the performance of fitted models [\(Gelman et al.](#page-13-33) [2013\)](#page-13-33). However, the GOF of Bayesian hierarchical regression models is most commonly compared using the deviance information criterion (DIC). Proposed by [Spiegelhalter et al. \(2002\),](#page-14-13) the DIC is a Bayesian generalization of the Akaike Information Criterion (AIC), and is defined as:
+
+$$DIC = \overline{D} + P_D \tag{1}$$
+
+where *D yθ* = − E( 2 log(Pr( | )) is the expectation of the model deviance under the posterior distribution of the model parameters (collectively denoted as θ), and PD is the effective number of parameters, defined as:
+
+$$P_D = \overline{D} - D(\overline{\theta}) \qquad (2)$$
+
+where *D θ*( ) is the deviance under the posterior expectation of parameters.
+
+*D* is a classical estimate of fit; a smaller *D* indicates a better fit as it corresponds to a greater log-likelihood. PD is indeed a penalty for model complexity and ensures a fair comparison between competing models with different degrees of complexity. DIC is particularly useful when the posterior distributions of model parameters are obtained via MCMC simulation, which is the case for the models in this study.
+
+It should be pointed out that the DIC can be very sensitive to the structure of the hierarchical Bayesian model. [Geedipally et al. \(2013\)](#page-13-34) noted that when different hierarchical Bayesian models are compared, the DIC can be used as a model selection criterion only if the likelihood structure remains the same across all the models under consideration. This requirement was met for the model comparisons performed in this work.
+
+# 3. Model specification
+
+This section reviews the characteristcs of the models evaluated in this paper, which all belong to the Bayesian Poisson hierarchical family of models. First, the common structure of these models is described and compared with the traditional frequentist models (such as the negative binomial). Next, the alternative Bayesian Poisson hierarchical models selected for this study are listed and their mathematical formulation is presented. Finally, the fundamental properties of the models' mixing distribution (i.e., the distinctive element of the alternative models) are characterized.
+
+## 3.1. Common model structure
+
+Let yi denote the number of crashes observed at the i'th site (road segment, intersection, etc.) during the study period. In Poisson hierarchical models, yi's, when conditional on their mean λi, are assumed to be Poisson distributed:
+
+$$y_i | \lambda_i = \text{Poisson}(\lambda_i) \quad i = 1, 2, ..., n$$
+ (3)
+
+In the second level of hierarchy, the mean of the Poisson distribution is variable according to an underlying mixing distribution: gamma, lognormal, or inverse gamma. In the most common type of crash prediction regression models, the site-specfic mean of the mixing distribution, μi, is modeled as a log-linear function of the prevailing traffic, geometric, and/or environmental variables:
+
+$$E(\lambda_i | \beta) = \mu_i = \exp(X_i \beta) \ i = 1, 2, ..., n$$
+ (4)
+
+where Xi is the vector of covariates for site i, and β is the vector of unknown coefficients. This functional formulation is adopted in this paper.
+
+At this point, it is critical to emphasize the distinction between the frequestist and Bayesian approaches to hierarchical regression modeling. The frequentist approach is based on the assumption that the model parameters are fixed and the observed data is a repeatable random sample. Therefore, λi's are constant values from a mixing distribution with mean μi. Once the coefficients are estimated based on the data, the expected crash frequency at site i is simply calculated as:
+
+$$E(y_i) = E(\lambda_i | \beta) = \exp(X_i \beta) \quad i = 1, 2, ..., n$$
+ (5)
+
+In contrast, the Bayesian paradigm views the parameters probabilistically. Every parameter is assumed to be drawn from a specified prior distribution (representing the modeler's prior knowledge), which is combined with the information in the data to produce posterior distributions using Bayes rule. In the hierarchical framework under study here, the λi's are assumed a priori to follow an undelying mixing distribution p(λi) with mean exp(Xiβ), where the hyper-parameter β itself follows a prior distribution (a.k.a. hyper-prior). However, the posterior distribution of λ<sup>i</sup> (for every i) neither follows the specified mixing distribution (gamma, lognormal, or inverse gamma) nor has an expectation necessarily equal to μi.
+
+In the Bayesian model formulation, it is critical to understand the difference between μ<sup>i</sup> and λi. μ<sup>i</sup> is the unconditional mean that depends only on the covariates. This parameter captures the effects of site characteristics on the expected crash frequency through the regression model and is therefore referred to in this paper as the "regression mean." The regression mean can be interpreted as the expected crash frequency for sites with the same covariate values (i.e., traffic, geometric, and/or environmental conditions) as those of site i but without an observed crash count. λi, on the other hand, is the site-specific expected crash frequency after μ<sup>i</sup> is adjusted by the observed crash frequency through the Bayesian model.
+
+In the Bayesian hierarchical models in this paper, each λ<sup>i</sup> represents a model parameter. Therefore, for a dataset with n sites, a Bayesian Poisson hierarchical model will have n parameters more than its frequentist counterpart. However, these parameters are not regarded as being independent, but are instead assumed to be drawn from a common distribution. As the variance of this common distribution approaches 0, the frequentist, fixed-effect model is obtained as a limiting case. As the case is made in this paper, the greater inherent complexity of the Bayesian approach demands greater attention when the performance of alternative Bayesian models are evaluated and compared.
+
+# 3.2. Alternative models
+
+This research thoroughly evaluates and compares the crash frequency predictions of the following hierarchical Poisson regression
+
+<span id="page-3-0"></span>![](_page_3_Figure_2.jpeg)
+
+Fig. 1. Probability distribution function of the gamma, lognormal, and inverse gamma distributions for different mean-variance combinations.
+
+models using a full Bayesian approach:
+
+- 1) Poisson-Gamma (PG)
+- 2) Poisson-Lognormal (PLN)
+- 3) Poisson-Inverse Gamma (PIGam)
+
+The first two models are commonly used in highway safety analyses, whereas the latter is new and its appropriateness for crash data modeling is to be examined. As mentioned before, the Poisson-Inverse Gamma model is abbreviated as "PIGam" to avoid confusion with the Poisson-Inverse Gaussian (PIG) model, whose application to crash data has been investigated by [Zha et al. \(2016\).](#page-14-0)
+
+Every mixing distribution selected for this study has two parameters. Below, these distributions are reparametrized in terms of their mean (μi) and a remaining hyper-parameter (shape or scale) to structure the regression models:
+
+1) Poisson-Gamma (PG): the Poisson parameter (mean) follows a gamma distribution with shape parameter φ and scale parameter θi:
+
+$$\Pr(\lambda_i | \varphi, \theta_i) = \frac{1}{\Gamma(\varphi)\theta_i^{\varphi}} \lambda_i^{\varphi - 1} \exp(\frac{-\lambda_i}{\theta_i}) \quad \varphi, \theta_i > 0$$
+(6)
+
+$$E(\lambda_i | \beta) = \mu_i = \varphi \theta_i = \exp(X_i \beta)$$
+ (7)
+
+$$\Pr(\lambda_i | \varphi, X_i, \beta) = \frac{\varphi^{\varphi}}{\Gamma(\varphi) \exp(\varphi. (X_i \beta))} \lambda_i^{\varphi-1} \exp(\frac{-\lambda_i \varphi}{\exp(X_i \beta)})$$
+(8)
+
+<span id="page-4-0"></span>Table 1 Summary statistics of the data sets in this study.
+
+| Variable                                                                      | Min.  | Max.   | Average | (std. dev) | Total  |
+|-------------------------------------------------------------------------------|-------|--------|---------|------------|--------|
+| Texas Data                                                                    |       |        |         |            |        |
+| Crash count                                                                   | 0     | 41     | 2.83    | (5.06)     | 1415   |
+| Segment length (mi)                                                           | 0.10  | 4.54   | 0.54    | (0.61)     | 270.87 |
+| AADTa (veh/day)                                                               | 420   | 23800  | 6548.5  | (41.1)     | –      |
+| Lane width (ft) (LW)                                                          | 10    | 16.5   | 12.63   | (1.57)     | –      |
+| Shoulder widthb (ft) (SW)                                                     | 0     | 32     | 10.15   | (7.80)     | –      |
+| Curve density (curve/mi)<br>(CD)                                              | 0     | 16.95  | 1.37    | (2.30)     | –      |
+| Michigan Data                                                                 |       |        |         |            |        |
+| Crash count                                                                   | 0     | 30     | 0.77    | (2.16)     | 383    |
+| Segment length (mi)                                                           | 0.001 | 4.21   | 0.19    | (0.35)     | 93.61  |
+| AADT(veh/day)                                                                 | 270   | 19990  | 4699.0  | (3487.2)   | –      |
+| Right shoulder width (ft)<br>(RSW)                                            | 0     | 12     | 8.50    | (2.80)     | –      |
+| Lane width (ft)                                                               | 9     | 14     | 11.23   | (0.74)     | –      |
+| Speed limit (mph) (SL)                                                        | 25    | 55     | 53.01   | (5.71)     | –      |
+| Left shoulder width (ft)<br>(LSW)                                             | 0     | 12     | 8.46    | (2.87)     | –      |
+| Indiana Data                                                                  |       |        |         |            |        |
+| Crash count                                                                   | 0     | 329    | 16.97   | (36.30)    | 5737   |
+| Segment length (mi)                                                           | 0.009 | 11.53  | 0.89    | (1.48)     | 300.09 |
+| AADT(veh/day)                                                                 | 9,442 | 143422 | 30237.6 | (28776.4)  | –      |
+| Minimum friction reading<br>in the road segment<br>(FRICTION)                 | 15.9  | 48.2   | 30.51   | (6.67)     | –      |
+| Median width (ft) (MW)                                                        | 16    | 194.7  | 66.98   | (34.17)    | –      |
+| Pavement surface type<br>(PAVEMENT) (1 if asphalt,<br>0 if concrete)          | 0     | 1      | 0.77    | (0.42)     | –      |
+| Presence of median barrier<br>(BARRIER) (1 if present, 0<br>if absent)        | 0     | 1      | 0.16    | (0.37)     | –      |
+| Presence of interior rumble<br>strips (RUMBLE) (1 if<br>present, 0 if absent) | 0     | 1      | 0.72    | (0.45)     | –      |
+
+<span id="page-4-1"></span><sup>a</sup> Annual average daily traffic.
+
+2) Poisson-Lognormal (PLN): the Poisson parameter follows a lognormal distribution with location parameter υ<sup>i</sup> and shape parameter σ2
+
+$$Pr(\lambda_i | \nu_i, \sigma^2) = \frac{1}{\lambda_i \sigma \sqrt{2\pi}} exp(-\frac{(\ln(\lambda_i) - \nu_i)^2}{2\sigma^2}) \quad \sigma^2 > 0$$
+(9)
+
+$$E(\lambda_i | \beta) = \mu_i = \exp(\nu_i + \sigma^2/2) = \exp(X_i \beta)$$
+(10)
+
+$$\Pr(\lambda_i \mid \sigma^2, X_i, \beta) = \frac{1}{\lambda_i \sigma \sqrt{2\pi}} \exp\left(-\frac{(\ln(\lambda_i) - X_i \beta + \sigma^2/2)^2}{2\sigma^2}\right)$$
+(11)
+
+3) Poisson-Inverse Gamma (PIGam): The Poisson parameter follows an inverse gamma distribution with shape parameter φ and scale parameter θi:
+
+$$\Pr(\lambda_i \mid \varphi, \theta_i) = \frac{\theta_i^{\varphi}}{\Gamma(\varphi)} \lambda_i^{-\varphi - 1} \exp(\frac{-\theta_i}{\lambda_i}) \quad \varphi, \theta_i > 0$$
+ (12)
+
+$$E(\lambda_i | \beta) = \mu_i = \frac{\theta_i}{\varphi - 1} = \exp(X_i \beta) \text{ for } \varphi > 1$$
+(13)
+
+$$\Pr(\lambda_i | \varphi, X_i, \beta) = \frac{|(\varphi - 1)\exp(X_i\beta)|^{\varphi}}{\Gamma(\varphi)} \lambda_i^{-\varphi - 1} \exp(\frac{-(\varphi - 1)\exp(X_i\beta)}{\lambda_i}) \text{ for } \varphi > 1$$
+
+To distinguish between the parameters of the PG and PIGam models, the shape parameters are augmented by subscripts denoting the model name: φPG for the Poisson-gamma and φPIGam for the Poissoninverse gamma.
+
+The similar structure and number of parameters provides for a reasonable comparison between the predictions of the three models. No model can be presumed to take advantage of a higher number of parameters for its flexibility to fit the data. Site-specific λi's and μi's under each model can be directly and meaningfully compared, as carried out later in this paper.
+
+### 3.3. Mixing distribution properties
+
+Given the common structure of the alternative Poisson hierarchical models, the distinctive performance of the models is attributable to the fundamental characteristics of their mixing distributions (i.e., prior distribution for λi's).
+
+[Fig. 1](#page-3-0) compares the shape of the gamma, lognormal, and inverse gamma probability distribution functions (pdf's) for four different combinations of mean and variance. The inverse gamma distribution is the distribution of the reciprocal of a variable that follows a gamma distribution. Since the gamma distribution is light-tailed, the inverse of a gamma-distributed variable has a very low probability in the vicinity of zero (similar to the NB-L, see [Shirazi et al., 2016\)](#page-14-14). It is interesting to note that the lognormal distribution lies between the gamma and inverse gamma distributions almost in every important aspect including the probability density in vicinity of zero, probability density for very large values (i.e., tail thickness), mode, probability at mode, skewness, etc. As we move from the gamma distribution to lognormal and then to inverse gamma, near-zero values for expected crash frequencies (λi's) become less likely, and extremely high values for the expected crash frequency become more likely.
+
+Unlike the gamma distribution, the lognormal and inverse gamma distributions are classified as heavy-tailed because their right tails are not exponentially bounded. Moreover, the inverse gamma distribution has a thicker tail than the lognormal distribution and, therefore, the lognormal distribution lies between the gamma and inverse-gamma distribution in terms of the tail thickness. The thicker tail, which is representative of greater probabilities for higher expected crash frequencies, can theoretically be beneficial when modeling data with occasional unusually high crash count observations. Please note that [Fig. 1](#page-3-0) cannot demonstrate the aforementioned relation between the tail thickness of the distributions; as x → ∞ the tails become too thin and their different thicknesses are unrecognizable.
+
+This notion is evident in [Fig. 1](#page-3-0), which compares the shape of the gamma, lognormal, and inverse gamma probability distribution functions (pdf's) for four different combinations of mean and variance. It is interesting to note that the lognormal distribution lies between the gamma and inverse gamma distributions almost in every important aspect including the probability density in vicinity of zero, probability density for very large values (i.e., tail thickness), mode, probability at mode, skewness, etc. As we move from the gamma distribution to lognormal and then to inverse gamma, near-zero values for expected crash frequencies (λi's) become less likely, and extremely high values for the expected crash frequency become more likely.
+
+# 4. Data description
+
+Three relatively large crash data sets from highway segments in Texas, Michigan, and Indiana are used to compare the predictions of the selected models. [Table 1](#page-4-0) presents the summary statistics of the variables in these datasets. The source and main characteristics of the datasets are described below.
+
+The Texas dataset is a randomly selected sample of 500 rural fourlane undivided highway segments in Texas. The crash counts indicate the total number of crashes observed during a five-year period from 1997 to 2001. The original dataset included a total of 1499 segments and was used by [Lord et al. \(2008\)](#page-13-35) to develop safety prediction models for inclusion in the first edition of the Highway Safety Manual (HSM) ([AASHTO, 2010\)](#page-13-36).
+
+(14)
+
+<span id="page-4-2"></span><sup>b</sup> Right shoulder width + left shoulder width.
+
+<span id="page-5-0"></span>Table 2 Posterior Estimates of Model Parameters for Texas Data.
+
+| Parameter<br>Respective Variable |           | PG      |            | PLN     |            | PIGam   |            |
+|----------------------------------|-----------|---------|------------|---------|------------|---------|------------|
+|                                  |           | mean    | (std dev.) | mean    | (std dev.) | mean    | (std dev.) |
+| ln(β0)                           | Intercept | −7.3986 | (0.6683)   | −7.4276 | (0.7069)   | −7.4725 | (0.7413)   |
+| β1                               | AADT      | 0.9636  | (0.0692)   | 0.9752  | (0.0737)   | 0.9812  | (0.0753)   |
+| β2                               | LW        | −0.0869 | (0.0290)   | −0.0913 | (0.0300)   | −0.0909 | (0.0296)   |
+| β3                               | SW        | −0.0122 | (0.0057)   | −0.0131 | (0.0058)   | −0.0132 | (0.0056)   |
+| β4                               | CD        | 0.1040  | (0.0207)   | 0.1038  | (0.0209)   | 0.1022  | (0.0203)   |
+| φPG                              |           | 2.8169  | (0.4746)   |         |            |         |            |
+| σ                                |           |         |            | 0.6060  | (0.0502)   |         |            |
+| φPIGam                           |           |         |            |         |            | 3.9323  | (0.5741)   |
+
+The Michigan dataset contains the single-vehicle crashes that occurred on rural two-lane highways in Michigan in 2006. Similar to the Texas dataset, a randomly selected sample of 500 highway segments was taken from the original data and used for analysis. The data were collected for the Federal Highway Administrations's (FHWA) Highway Safety Informations System (HSIS) and originally included a total of 33,970 segments. In this dataset, about 67% of segments experienced no crashes in 2006. The selection of this dataset provides an opportunity to examine the performance of different models with data characterized by a large number of zeros, which is specifically common in road safety analyses.
+
+The Indiana dataset contains data collected for a five-year period (1995–1999) from 338 rural interstate freeway segments in Indiana. The Indiana data is the most highly over-dispersed dataset considered in this study, which includes a relatively large proportion of zero crash counts (about 36%) along with very high crash counts observed at some other sites.
+
+# 4.1. Functional form
+
+The following functional form is adopted for all models and datasets in this study:
+
+$$\mu_i = \beta_0 L F_i^{\beta_1} \exp\left(\sum_{j=2}^p \beta_j x_j\right) \tag{15}$$
+
+where Fi is the AADT through the highway segment, L is the segment length, and xj's are the additional covariates listed in [Table 1](#page-4-0). In this functional form, the segment length is treated as an offset (i.e., its power is kept fixed at 1) because theoretically, the crash risk is expected to have a linear relationship with the segment length.
+
+Given the selected functional form, the datasets cover a relatively wide range of dispersion characteristics. The overdispersion parameter (α = 1/φ) of the traditional negative binomial (NB) model (estimated using the MLE method) is considered as a rough measure of conditional dispersion in the data (i.e., dispersion of the crash counts conditional on the modeled mean). The NB model overdispersion parameter (α NB) is 0.361, 0.496, and 0.888 for the Texas, Michigan, and Indiana data, respectively.
+
+#### 5. Model estimation
+
+This section describes the assumptions used to fit the models to the datasets and presents the results of parameter estimation.
+
+#### 5.1. Choice of hyper-priors
+
+A critical step in Bayesian modeling is the selection of hyper-prior distributions [\(Gelman et al. 2013](#page-13-33); [Lee 2012\)](#page-13-37). In order to control for the potential influence of prior information on the performance of the alternative models, non-informative uniform distributions are considered for the models' hyperparameters. However, for large data sets such as the ones selected here, the choice of hyper-priors will not make a significant change in the parameter estimates. The following prior distributions are used:
+
+- Regression coefficients (all models): <sup>β</sup><sup>j</sup> ∼ Uniform (−∞,+∞) for j = 0,1, 2, …, p
+- <sup>φ</sup>PG (Poisson-gamma): <sup>φ</sup>PG <sup>∼</sup> Uniform (0,+∞)
+- <sup>φ</sup>PIGam (Poisson-inverse gamma): <sup>φ</sup>PIGam <sup>∼</sup> Uniform (1,+∞)
+- <sup>σ</sup> (Poisson-lognormal):<sup>σ</sup> <sup>∼</sup> Uniform (0,+∞)
+
+## 5.2. MCMC simulation for posterior inference
+
+Posterior distribution of model parameters were estimated using basic Markov Chain Monte Carlo (MCMC) simulation methods. Sampling from all parameters was carried out using the Metropolis-Hastings algorithm. The first 20,000 samples were discarded to diminish the influence of the starting values, and the posterior distribution of each parameter was constructed using the next 1,000,000 MCMC iterations. Gelman's *R*ˆ factor ([Gelman et al., 2013\)](#page-13-33) and visual inspection were utilized to ensure sufficient convergence.
+
+Table 3 Posterior Estimates of Model Parameters for Michigan Data.
+
+| Parameter<br>Respective Variable | PG        |            | PLN      |            | PIGam    |            |          |
+|----------------------------------|-----------|------------|----------|------------|----------|------------|----------|
+|                                  | mean      | (std dev.) | mean     | (std dev.) | mean     | (std dev.) |          |
+| ln(β0)                           | Intercept | −7.3166    | (1.6729) | −7.5750    | (1.6709) | −7.3671    | (1.7026) |
+| β1                               | AADT      | 0.5521     | (0.0949) | 0.5477     | (0.0982) | 0.5430     | (0.0903) |
+| β2                               | RSW       | 0.0733     | (0.0695) | 0.0710     | (0.0755) | 0.0654     | (0.0731) |
+| β3                               | LW        | 0.2366     | (0.1009) | 0.2569     | (0.1068) | 0.2553     | (0.1024) |
+| β4                               | SL        | 0.0193     | (0.0192) | 0.0204     | (0.0201) | 0.0165     | (0.0211) |
+| β5                               | LSW       | −0.0285    | (0.0631) | −0.0246    | (0.0680) | −0.0109    | (0.0745) |
+| φPG                              |           | 2.4660     | (1.1162) |            |          |            |          |
+| σ                                |           |            |          | 0.6898     | (0.1001) |            |          |
+| φPIGam                           |           |            |          |            |          | 3.4994     | (1.1192) |
+
+Table 4 Posterior Estimates of Model Parameters for Indiana Data.
+
+| Parameter<br>Respective Variable |           | PG         |          | PLN        |          | PIGam      |          |
+|----------------------------------|-----------|------------|----------|------------|----------|------------|----------|
+|                                  | mean      | (std dev.) | mean     | (std dev.) | mean     | (std dev.) |          |
+| ln(β0)                           | Intercept | −4.5245    | (1.4117) | −3.5477    | (1.4432) | −3.6814    | (1.4259) |
+| β1                               | AADT      | 0.6974     | (0.1314) | 0.6231     | (0.1335) | 0.6459     | (0.1223) |
+| β2                               | FRICTION  | −0.0269    | (0.0106) | −0.0301    | (0.0109) | −0.0283    | (0.0102) |
+| β3                               | PAVEMENT  | 0.4222     | (0.1886) | 0.4902     | (0.2106) | 0.4818     | (0.2048) |
+| β4                               | MW        | −0.0052    | (0.0019) | −0.0073    | (0.0022) | −0.0077    | (0.0021) |
+| β5                               | BARRIER   | −3.0591    | (0.3028) | −4.1406    | (0.4952) | −5.4245    | (0.5866) |
+| β6                               | RUMBLE    | −0.4050    | (0.1825) | −0.3138    | (0.2038) | −0.1574    | (0.1924) |
+| φPG                              |           | 1.1143     | (0.1421) |            |          |            |          |
+| σ                                |           |            |          | 0.9665     | (0.0661) |            |          |
+| φPIGam                           |           |            |          |            |          | 1.8844     | (0.2104) |
+
+### 5.3. Parameter estimation results
+
+[Table 2](#page-5-0)–4 summarize the posterior inference with the mean and standard deviation of the parameters. The results indicate that the difference between the models' coefficients intensifies as the over-dispersion in the data increases (from the Texas data to the Indiana data). The number of model covariates (increasing from the Texas models to the Indiana models) may also be a contributing factor. Variation in coefficient estimates across alternative models (especially for the Indiana dataset) may result in distinctive predictions by the alternative models, as demonstrated in the next section.
+
+### 6. Comparison of model predictions
+
+This section compares the predictions of the alternative models on two levels: 1) crash frequency expectation at sites in the dataset (i.e., posterior λi's), and 2) regression mean (μ) function across the models.
+
+It is crucial to understand the practical importance of each level of comparison. λi's, the actual site-specific crash frequency expectations, are used in before-after studies (to represent the before conditions) and to rank sites based on their crash-proneness.
+
+The μ function, on the other hand, not only affects the λi's estimates, but also has additional applications by itself. For instance, it is common for safety analysts to use calibrated crash prediction models to predict the expected crash frequency at planned facilities that have not yet been constructed. Here, the main objective is to estimate the safety impact of each design feature (e.g., lane width, type of median barrier, signal phasing, etc.) and help decision making in the design/planning stage. Also, the observed crash frequency may not be available at existing sites for reasons such as limited resources for data collection, etc. For such applications, the models' prediction is not affected by the sitespecific observed crash frequency and, as shown later, the pattern of the difference between the models' predictions is quite different from that for the sites with observed crash counts. Therefore, the two comparison cases are presented in separate subsections.
+
+In the comparisons hereafter, λ and μ variables are superscripted with the abbreviated name of the model with which they are associated; for example, λ<sup>i</sup> PLN is the λ of the i'th site when the Poisson-lognormal model is used.
+
+# 6.1. Crash frequecy expectation (λ<sup>i</sup> | y)
+
+At this level of comparison, we focus on the posterior mean of λi's (i.e., E(λ<sup>i</sup> | y) for i = 1, 2, …, n) and how they are affected by the model choice. Since the variations of λi's across the alternative models are relatively small (compared to the magnitude of λi's), a traditional scatter plot may conceal possible trends. Therefore,
+
+[Fig. 2](#page-7-0)–4 are constructed to magnify the differences between the models' predictions and observe their relationship. For each dataset, E (λ<sup>i</sup> | y)'s from every pair of models are compared in a separate scatter plot. In all these plots, the horizontal axis represents the value of E(λ<sup>i</sup> | y) predicted by one of the two models being compared, whereas the vertical axis indicates the difference between the two models' predicted E(λ<sup>i</sup> | y).
+
+The plots do not reveal a strong relationship between the alternative models' predictions for the site-specific expected crash frequencies. The only noticeable trend is that the most severe differences between the predictions are typically related to the sites where E(λ<sup>i</sup> PIGam| y)>E (λ<sup>i</sup> PLN| y) > E(λ<sup>i</sup> PG | y). The relationship between the E(λ<sup>i</sup> | y)'s of alternative models was further investigated and documented by [Khazraee](#page-13-38) [\(2016\);](#page-13-38) more complicated trends were identified when the regression mean and observed crash frequency were also considered in the analysis. However, from a practical standpoint, the main takeaway of the analysis is that the positive and negative differences between the predictions of different models balance and none of the models predicts an overall higher or lower crash frequency compared to the others. As [Table 5](#page-9-0) indicates, the total expected crash frequency (over all sites combined) is virtually equal across the three models and very close to the total observed crash frequency, indicating that none of the models predicts significantly biased E(λ<sup>i</sup> | y)'s for any of the datasets.
+
+# 6.2. Regression mean (μ)
+
+Crash prediction models are calibrated with historic crash data from a sample of roadway sites, and applied to sites without observed crash data to predict the expected crash frequency. Let j denote the index for a new roadway site to be constructed. In the absence of an observed crash count at site j, the expected crash frequency will follow a model-specific distribution (gamma, lognormal, or inverse gamma) with mean μ<sup>j</sup> = exp(Xjβ). Differences in posterior β of the alternative models, and therefore their respective μ's, lead to different predictions for expected crash frequency at new sites.
+
+To illustrate the implication of the model choice, we should compare μ<sup>j</sup> across the alternative models for new sites with a range of characteristics. In the interest of convenience, we consider a hypothetical set of new sites with exactly silimar features (covariate vector values) as those in the dataset used for fitting the models. The expected crash frequency at each new site will be equal to E(μi| y) of the site in the calibration data set that it is replicating. With this hypothestical new set of sites, comparison between expected crash frequencies will be equivalent to simply comparing E(μi| y)'s across the alternative models. This comparison is carried out in [Fig. 5](#page-10-0)–7 using the same scatter plots as those used earlier for comparing E(λ<sup>i</sup> | y)'s.
+
+One might expect that the differences between the values of each models' E(μi| y) predictions would increase with the increase in the magnitude of E(μi| y). That is, the difference between the alternative models' coefficients for traffic volume causes a larger difference between the models' E(μi| y)'s for sites with greater traffic volume and thus greater E(μi| y)'s. However, [Fig. 5](#page-10-0)–7 clearly indicate that the PLN model tends to predict larger μi's compared to the PG model, and the
+
+<span id="page-7-0"></span>![](_page_7_Figure_2.jpeg)
+
+Fig. 2. Difference between Texas dataset E(λ<sup>i</sup> | y)'s estimated using any two of the candidate models as a function of the E(λ<sup>i</sup> | y) from one of the models considered.
+
+PIGam model tends to predict larger μi's compared to PLN model. The figures also indicate that the E(μ<sup>i</sup> PG | y) < E(μ<sup>i</sup> PLN | y) < E(μ<sup>i</sup> PIGam | y) relationship becomes more pronounced for larger E(μi| y)'s. The aforementioned relationship is an important finding because, unlike the λ<sup>i</sup> predictions for sites with observed crash frequency (see [Table 5\)](#page-9-0), it results in each model over- or under-predicting μi's (collectively, over all sites combined) relative to the other two models (as indicated in [Table 6](#page-12-0)).
+
+It is evident from [Table 6](#page-12-0) that the PIGam model predicts higher posterior regression means (for all sites combined) than the PLN model and the PG model, in order. It is interesting to note that most E(μi| y)'s from the PLN model are in between those from the PG and PIGam models, indicating an apparent link between the shape properties of the three mixing distributions (as explained earlier) and the predictions of their respective Poisson-hierarchical models.
+
+Another important observation is that the degrees of variation between models' μ<sup>i</sup> predictions increases as the data become more overdispersed. As implied by [Fig. 5](#page-10-0)–7 and [Table 6](#page-12-0), the proportional (rather than absolute) difference between predicted E(μi| y)'s become more variant across the models in the following order of the datasets: Texas, Michigan, and Indiana, the same ranking of the datasets with respect to their NB model over-dispersion parameter. For the Indiana dataset, which is much more over-dispersed than the other two datasets, the alternative models result in drastically different regression means for sites with high traffic volume (AADT). For such datasets, the model choice is critical for any safety analysis.
+
+# 7. Goodness of fit analysis
+
+As explained earlier, the GOF of alternative Bayesian hierarchical models are most commonly compared using the DIC. In the context of Poisson-hierarchical models, the deviance (D) is typically calculated using the parameters in the lowest level of the hierarchy (λi's):
+
+$$\overline{D} = \sum_{i=1}^{T} -2\log\left(\frac{\lambda_i^{y_i} e^{\lambda_i}}{y_i!}\right)$$
+(16)
+
+where T is the total number of MCMC samples used for posterior inference. While the model coefficients (and regression means, in turn)
+
+![](_page_8_Figure_2.jpeg)
+
+Fig. 3. Difference between Michigan dataset E(λ<sup>i</sup> | y)'s estimated using any two of the candidate models as a function of the E(λ<sup>i</sup> | y) from one of the models considered.
+
+affect the λ<sup>i</sup> estimates, they do not directly enter the DIC calculation formula. [Table 7](#page-13-39) presents the DIC values obtained for each model-dataset.
+
+According to the DIC's in [Table 7,](#page-13-39) for the two moderately overdispersed datasets in this study (Texas and Michigan), the PG model fits the data better than the PLN model, which itself fits better than the PIGam model (the PLN model again falling in between the PG and PIGam models). For the highly over-dispersed Indiana dataset, however, the PLN and PIGam models provide a better fit than the PG model. Please note that the small difference between the DIC's of the PLN and PIGam models does not indicate a meaningful difference between the two models quality of fit to the Indiana dataset; as a rule of thumb, only a DIC difference of more than 10 constitutes a disparity between the models' fit to the data [\(MRC Biostatistics Unit, 2016\)](#page-13-40).
+
+The better performace of the PG model (compared to the PLN and PIGam models, in order) for the less over-dispersed datasets and its worse performance for a highly-overdispersed dataset may be attributable to the greater density near zero and thinner tail of the gamma distribution (resulting from higher skewness) compared to that of the lognormal and inverse gamma distributions, in order. The pattern hints to the possibility that the newly-proposed PIGam model might be the best-fitting model for data—even for more skewed and over-dispersed data than the Indiana dataset. The advantages of long/heavy distribution tails for fitting highly over-dispersed data have previously been demonstrated for the NB-L ([Geedipally et al., 2012](#page-13-0); [Shirazi et al., 2016\)](#page-14-14) and Sichel [\(Zou et al., 2012](#page-14-15)) regression models. Furthermore, the results shown here support the recent work on using heuristic methods for selecting models based on the characteristics of the data [\(Shirazi et al.,](#page-14-16) [2017\)](#page-14-16). However, based on the experience of the authors in fitting the three models to several different datasets (only three of which are included in this paper), the relative GOF of the models (judged based on DIC) is far too complicated to be reliably predicted based on a simple measure of over-dispersion (such as α NB) for the given dataset. In other words, one should not simply assume that a heavy-tailed model will perform better than a light-tailed model as over-dispersion in data increases; models have to be fitted before their GOF characteristics can be
+
+![](_page_9_Figure_2.jpeg)
+
+Fig. 4. Difference between Indiana dataset E(λ<sup>i</sup> | y)'s estimated using any two of the candidate models as a function of the E(λ<sup>i</sup> | y) from one of the models considered.
+
+<span id="page-9-0"></span>Table 5 Total expected crash frequency (E(λ<sup>i</sup> | y)) over all sites predicted by each model for each dataset.
+
+| Dataset  | n<br>∑ = E(<br>) λ y i<br>i 1 | Total Expected Crash Frequency, |                | Total Observed Crash<br>n<br>Frequency, ∑ = Y i<br>i 1 |
+|----------|-------------------------------|---------------------------------|----------------|--------------------------------------------------------|
+|          | PG Model                      | PLN Model                       | PIGam<br>Model |                                                        |
+| Texas    | 1413.5                        | 1413.5                          | 1414.7         | 1415                                                   |
+| Michigan | 384.0                         | 383.5                           | 383.3          | 383                                                    |
+| Indiana  | 5737.3                        | 5736.4                          | 5737.2         | 5737                                                   |
+
+reliably compared.
+
+The modeling results for the Indiana dataset reveal a very important limitation of the DIC for model selection. Virtually similar DIC's from the PLN and PIGam models implies that either model may be used for predicive applications without considerable difference. However, when the calibrated PIGam model is used to predict the expected crash frequency for new sites, for the majority of site conditions (i.e, covariate vector) it will predict a higher expectation than would the PLN model. As [Fig. 7](#page-12-1) illustrated, for sites with high traffic flow (and thus great crash frequency) the practical difference between the two models will be drastic. Therefore, it is by no means reasonable to conclude (based on the DIC's) that the two models are performing equally well.
+
+The aforementioned limitation of the DIC arises from the notion that it is solely calculated based on the conditional likelihood of the parameters in the lowest (i.e., observation) level of the hierarchy. As previously described in detail, in these models each λ<sup>i</sup> is a parameter whose posterior expectation is not necessarily equal to that of μi. Therefore, alternative models may have similar (or slightly different) λi's while having considerably different regression mean functions; for the Indiana dataset, the difference between λ<sup>i</sup> PLN's and λ<sup>i</sup> PIGam's is far smaller than that between μ<sup>i</sup> PLN's and μ<sup>i</sup> PIGam's.
+
+This weakness of the DIC has extensively been documented by statisticians [\(Spiegelhalter et al. 2002;](#page-14-13) [Celeux et al. 2006](#page-13-41); [Carlin et al.,](#page-13-42)
+
+<span id="page-10-0"></span>![](_page_10_Figure_2.jpeg)
+
+Fig. 5. Difference between Texas dataset E(μ<sup>i</sup> | y)'s estimated using any two of the candidate models as a function of the E(μ<sup>i</sup> | y) from one of the models considered.
+
+[2006;](#page-13-42) [Meng and Vaida et al., 2006;](#page-13-43) [Plummer et al., 2006\)](#page-14-17). In the context of hierarchcial bayesian models for over-dispersed count data (the typical case for crash count data), [Miller \(2009\)](#page-13-44) warned against the naïve use of the "condtional DIC" and reported major inconsistenies between the DIC and more robust measures of fit such as the Bayes factor. The analysis in this paper illustrates the shortcoming of the DIC by presenting an exmple (Indiana data) where the DIC fails to differentiate between alternative models that result in drastically different predictions (for μi's).
+
+Alternatives to the DIC are abundant (see e.g., [Celeux et al., 2006](#page-13-41)). The most familiar and theoretically robust is the Bayes factor, which is defined as the ratio of the marginal densities of the data under two alternative models:
+
+$$K = \frac{\Pr(D \mid M_1)}{\Pr(D \mid M_2)} = \frac{\int \Pr(\theta_1 \mid M_1) \Pr(D \mid \theta_1, M_1) d\theta_1}{\int \Pr(\theta_2 \mid M_2) \Pr(D \mid \theta_2, M_2) d\theta_2}$$
+(17)
+
+where D denotes the observed data, M1/M2 the alternative Model 1 and 2, and θ1/θ<sup>2</sup> the collective set of parameters for Model 1 and 2, respectively. Unlike DIC, the Bayes factor accounts for the marginal probability structure of the competing models. In addition, the value of the Bayes factor is easily interpretable. For example, a Bayes factor of 0.2 means that model 2 is 5 times more likely than model 1. These likelihood ratios can be used to mix the alternative models and obtain more accurate mixed models.
+
+The major drawback of the Bayes factor is the computational intensiveness due to the high-dimension integral over the parameter space. Researchers have proposed a plothera of methods for approximating the marginal density (e.g., [Gelman and Meng, 1998](#page-13-45); [Han and](#page-13-46) [Carlin, 2001](#page-13-46); [Sinharay and Stern, 2003\)](#page-14-18). Determination of the marginal density for the PG model is straightforward due to the closed form of the marginal likelihood function. [Miller \(2009\)](#page-13-44) used a complicated method developed by [Chib \(1995\)](#page-13-47) to approximate the marginal density for the PLN model. The marginal density for the PIGam model is expectedly more complicated and difficult to estimate.
+
+Given the complexity of estimating Bayes factors for the models in this paper, reviewing the existing methods, adjusting and implementing
+
+![](_page_11_Figure_2.jpeg)
+
+Fig. 6. Difference between Michigan dataset E(μ<sup>i</sup> | y)'s estimated using any two of the candidate models as a function of the E(μ<sup>i</sup> | y) from one of the models considered.
+
+them can be the subject of another research srtudy. The initial goal of this paper was to evaluate and compare Poisson-hierarchical models in terms of the relative magnitude of their predictions (rather than GOF) and the resulting practical implications of using one model instead of another. The methodology used for analysis of predictions revealed a known (but often neglected) weakness of evaluating the GOF using the DIC.
+
+### 8. Summary and conclusions
+
+This research investigated the model selection dilemma for crash prediction models from a new perspective. The methodology deviated from the conventional model selection studies in the following sense: rather than ranking the alternative models based on their overall GOF and predictive performance, the research focused on the magnitude of models predictions for the expected crash frequency at individual sites. Focusing on an important family of count regression models, Bayesian Poisson-hierarchical, the study found important trends among predictions of three alternative models: Poisson-gamma, Poisson-lognormal, and Poisson-inverse gamma.
+
+The common structure of the selected candidate models facilitated comparison between their predictions and the similar number of parameters provided for a fair comparison. The distinction between the regression mean (μi) and the site-specific crash count expectation (λi) was well explained; posterior λ<sup>i</sup> is the actual expected crash frequency at site i after the regression mean (μi) is adjusted for the observed crash frequency (yi). μ<sup>i</sup> may be interpreted as the expected crash frequency at a hypothetical new site with the same modeled characteristics as those of site i but without any observed crash count.
+
+Fitting the candidate models to three relatively large datasets (from Texas, Michigan, and Indiana), the following outcomes were observed:
+
+1) The differences between the λi's predicted by the three models grew larger as the over-dispersion in the data increased. While the differences between posterior λi's across models were significant at a few sites (especially in the highly over-dispersed Indiana dataset),
+
+<span id="page-12-1"></span>![](_page_12_Figure_2.jpeg)
+
+Fig. 7. Difference between Indiana dataset E(μ<sup>i</sup> | y)'s estimated using any two of the candidate models as a function of the E(μ<sup>i</sup> | y) from one of the models considered.
+
+<span id="page-12-0"></span>Table 6 Total (E(μi|y)) over all sites predicted by each model for each dataset.
+
+| Dataset  | n<br>∑ = E(μ<br>) y i<br>i 1 |           |                | Total Observed Crash<br>n<br>Frequency, ∑ = Y i |  |
+|----------|------------------------------|-----------|----------------|-------------------------------------------------|--|
+|          | PG Model                     | PLN Model | PIGam<br>Model | i 1                                             |  |
+| Texas    | 1412.0                       | 1428.9    | 1444.8         | 1415                                            |  |
+| Michigan | 387.8                        | 394.0     | 399.3          | 383                                             |  |
+| Indiana  | 5326.0                       | 6298.2    | 7430.9         | 5737                                            |  |
+
+the positive and negative differences balanced off and none of the models predicted an overall higher or lower crash frequency compared to another. Thus, the model choice would not make a practically important difference for predicitng the crash frequency expectation at sites in the dataset (with an available observed crash count).
+
+2) The differences between the μi's predicted by the three models were
+
+quite significant and grew larger as the over-dispersion in the data increased. The importance of the differences between the μi's was that they resulted in the PIGam model predicting higher posterior regression means (for all sites combined) compared to the PLN model and the PG model, in order. Thus, the model choice can make a critical difference when the calibrated models are used for predictions at new sites without observed crash counts. There is a clear link between the μ<sup>i</sup> predictions of alternative models and the shape of their respective mixing distribution; the lognormal distribution falls in between the gamma and inverse gamma distribution in terms of its important shape characteristics (e.g., tail thickness, skewness, etc.) and so do the PLN model's μ<sup>i</sup> predictions fall in between those of the PG and PIGam models.
+
+3) According to the most commonly used measure of fit for Bayesian hierarchical models, the DIC, the PIGam and PLN models provided a better fit to the highly over-dispersed Indiana dataset (compared to the PG model) but not to the other less over-dispersed datasets. The better performance of the PLN and PIGam models may have arisen
+
+<span id="page-13-39"></span>Table 7 Deviance Information Criterion (DIC) for each model-dataset.
+
+| Dataset  | DIC      |           |             |
+|----------|----------|-----------|-------------|
+|          | PG Model | PLN Model | PIGam Model |
+| Texas    | 1608.9   | 1614.2    | 1624.6      |
+| Michigan | 883.3    | 891.8     | 903.8       |
+| Indiana  | 1490.6   | 1465.3    | 1468.2      |
+
+from their thicker tails. However, the analysis illustrated a major deficiency of the DIC in comparing hierarchical models. For Indiana data, the PLN and PIGam models yielded very similar DIC values while predicting drastically different μi's, with the PIGam model predicting a higher μ<sup>i</sup> for most sites in the dataset. It was explained that this shortcoming of the DIC is due to the notion that its calculation is based solely on the conditional likelihood of the parameters in the lowest (i.e., observation) level of the hierarchy, ignoring the higher level(s). Consequently, the authors advise against using DIC (as the sole measure of GOF) for comparison of hierarchical models when the coefficients are highly dependent on the choice of model's distribution (often the case for highly over-dispersed datasets such as Indiana data).
+
+As described throughout this paper, the research did not seek to advise a technique for selecting the best performing model out of a pool of candidates, but rather to demonstrate the impact on the magnitude of predictions when one model is selected instead of another. The analysis rejected the suggestion that a simple measure of dispersion can be used to reliably predict the best-performing model. Research using simulated data and a multitude of field data with a wide range of characteristics is needed for such purpose. At the time of publication for this paper, new research was identified which has addressed this matter using heuristic methods [\(Shirazi et al., 2017](#page-14-16); [Shirazi and Lord, 2018\)](#page-14-19).
+
+In addition, further research is needed to assess the implications of this research's findings on different highway safety analysis applications, such as identification of hazardous sites and before-after analyses that employ FB models. Similar assessment can also be performed for recently introduced multi-parameter models such as the NB-L ([Geedipally et al., 2012](#page-13-0)), the NB-generalized exponential (NB-GE) ([Vangala et al., 2015\)](#page-14-20), or the NB- Dirichlet Process (NB-DP) [\(Shirazi](#page-14-14) [et al., 2016](#page-14-14)).
+
+### References
+
+- <span id="page-13-36"></span>[AASHTO, 2010. Highway Safety Manual, 1st ed. American Association of State and](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0005) [Highway Transportation O](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0005)fficials, Washington, D.C.
+- <span id="page-13-15"></span>[Aguero-Valverde, J., Jovanis, P.P., 2008. Analysis of road crash frequency with spatial](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0010) [models. Transp. Res. Rec. 2061, 55](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0010)–63.
+- <span id="page-13-16"></span>[Aguero-Valverde, J., 2013. Full Bayes Poisson gamma, Poisson lognormal, and zero in](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0015)flated random eff[ects models: comparing the precision of crash frequency estimates.](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0015) [Accid. Anal. Prev. 50, 289](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0015)–297.
+- <span id="page-13-10"></span>[Bonneson, J.A., McCoy, P.T., 1993. Estimation of safety at two-way stop-controlled in](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0020)[tersections on rural highways. Transp. Res. Rec. 1401, 83](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0020)–89.
+- <span id="page-13-42"></span>[Carlin, B.P., 2006. Comment on article by Celeux et al. Bayesian Anal. 1, 675](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0025)–676.
+- <span id="page-13-21"></span>Carriquiry, A., Pawlovich, M.D., 2004. From Empirical Bayes to Full Bayes: Methods for Analyzing Traffic Safety Data. (Accessed 7 March 2014). [http://www.iowadot.gov/](http://www.iowadot.gov/crashanalysis/pdfs/eb_fb_comparison_whitepaper_october2004.pdf) [crashanalysis/pdfs/eb\\_fb\\_comparison\\_whitepaper\\_october2004.pdf.](http://www.iowadot.gov/crashanalysis/pdfs/eb_fb_comparison_whitepaper_october2004.pdf)
+- <span id="page-13-41"></span>[Celeux, G., Forbes, F., Robert, C.P., Titterington, D.M., 2006. Deviance information cri](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0035)[teria for missing data models \(with discussion\). Bayesian Anal. 1, 651](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0035)–706.
+- <span id="page-13-47"></span>[Chib, S., 1995. Marginal likelihood from the Gibbs output. J. Am. Stat. Assoc. 90,](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0040)
+- <span id="page-13-20"></span>[Davis, G.A., Yang, S., 2001. Bayesian identi](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0045)fication of high-risk intersections for older [drivers via Gibbs sampling. Transp. Res. Rec. 1746, 84](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0045)–89.
+- <span id="page-13-22"></span>[Daziano, R.A., Miranda-Moreno, L., Heydari, S., 2013. Computational Bayesian statistics](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0050) [in transportation modeling: from road safety analysis to discrete transport reviews. J.](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0050) [Transp. Rev. 33 \(5\), 570](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0050)–592.
+- <span id="page-13-18"></span>[El-Basyouny, K., Sayed, T., 2009. Collision prediction models using multivariate poisson](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0055)[lognormal regression. Accid. Anal. Prev. 41, 820](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0055)–828.
+- <span id="page-13-28"></span>Fitzpatrick, K., Park, E.S., 2009. Safety eff[ectiveness of HAWK pedestrian treatment.](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0060) [Transp. Res. Rec. 2140, 214](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0060)–223.
+- <span id="page-13-0"></span>[Geedipally, S.R., Lord, D., Dhavala, S.S., 2012. The negative binomial-Lindley generalized](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0065)
+
+- [linear model: characteristics and application using crash data. Accid. Anal. Prev. 45](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0065) [\(2\), 258](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0065)–265.
+- <span id="page-13-34"></span>[Geedipally, S.R., Lord, D., Dhavala, S.S., 2013. A caution about using the deviance in](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0070)[formation criterion while modeling tra](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0070)ffic crashes. Saf. Sci. 62, 495–498.
+- <span id="page-13-19"></span>[Gelfand, A.E., Smith, A.F.M., 1990. Sampling-based approaches to calculating marginal](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0075) [densities. J. Am. Stat. Assoc. 85, 398](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0075)–409.
+- <span id="page-13-33"></span>[Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., Rubin, D.B., 2013.](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0080) [Bayesian Data Analysis, third edition. Chapman and Hall/CRC, New York](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0080).
+- <span id="page-13-45"></span>[Gelman, A., Meng, X.-L., 1998. Simulating normalizing constants: from importance](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0085) [sampling to bridge sampling to path sampling. Stat. Sci. 13, 163](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0085)–185.
+- <span id="page-13-32"></span>[Goldstein, H., 2010. Multilevel Statistical Models, 4th ed. John Wiley & Sons, West](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0090) [Sussex, England.](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0090)
+- <span id="page-13-46"></span>[Han, C., Carlin, B.P., 2001. Markov chain Monte Carlo methods for computing Bayes](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0095) [factors: a comparative review. J. Am. Stat. Assoc. 96, 1122](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0095)–1132.
+- <span id="page-13-27"></span>Harwood, [D.W., Bauer, K.M., Potts, I.B., Torbic, D.J., Richard, K.R., Kohlman Rabbani,](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0100) [E.R., Hauer, E., Elefteriadou, L., 2002. Safety E](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0100)ffectiveness of Intersection Left- and [Right-Turn Lanes, Report No. FHWA-RD-02-089. Federal Highway Administration](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0100) [\(FHWA\), Washington, D.C](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0100).
+- <span id="page-13-23"></span>[Hauer, E., Persaud, B.N., 1983. A common bias in before and after accident comparisons](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0105) [and its elimination. Transp. Res. Rec. 905, 164](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0105)–174.
+- <span id="page-13-24"></span>[Hauer, E., 1986. On the estimation of the expected number of accidents. Accid. Anal.](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0110) [Prev. 18, 1](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0110)–12.
+- <span id="page-13-9"></span>[Hauer, E., Ng, J.C.N., Lovell, J., 1989. Estimation of safety at signalized intersections.](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0115) [Transp. Res. Rec. 1185, 48](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0115)–61.
+- <span id="page-13-25"></span>[Hauer, E., 1992. Empirical Bayes approach to the estimation of unsafety: the multivariate](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0120) [regression approach. Accid. Anal. Prev. 24 \(5\), 456](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0120)–478.
+- <span id="page-13-26"></span>Hauer, E., 1997. Observational Before–[After Studies in Road Safety: Estimating the E](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0125)ffect of Highway and Traffi[c Engineering Measures on Road Safety. Pergamon Press,](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0125) [Elsevier Science, Ltd., Oxford, United Kingdom.](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0125)
+- <span id="page-13-1"></span>[Hauer, E., 2001. Overdispersion in modeling accidents on road sections and in empirical](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0130) [Bayes estimation. Accid. Anal. Prev. 33 \(6\), 799](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0130)–808.
+- <span id="page-13-29"></span>Heydecker, B.G., Wu, J., 2001. Identifi[cation of sites for accident remedial work by](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0135) [Bayesian statistical methods: an example of uncertain inference. Adv. Eng. Softw. 32,](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0135) 859–[869](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0135).
+- <span id="page-13-13"></span>[Hinde, J., 1982. Compound Poisson regression models. In: Gilchrist, R. \(Ed.\), GLIM 82:](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0140) [Proceedings of the International Conference on Generalized Linear Models. Springer-](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0140)[Verlag, New York](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0140).
+- <span id="page-13-38"></span>[Khazraee, S.H., 2016. Full Bayesian Poisson-Hierarchical Models for Crash Data Analysis:](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0145) [Investigating the Impact of Model Choice on Site-Speci](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0145)fic Predictions. PhD [Dissertation. Department of Civil Engineering, Texas A&M University, College](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0145) [Station, Texas](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0145).
+- <span id="page-13-37"></span>[Lee, P., 2012. Bayesian Statistics: An Introduction, 4th ed. John Wiley & Sons, West](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0150) [Sussex, England.](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0150)
+- <span id="page-13-35"></span>[Lord, D., Geedipally, S.R., Persaud, B.N., Washington, S.P., van Schalkwyk, I., Ivan, J.N.,](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0155) [Lyon, C., Jonsson, T., 2008. Methodology for Estimating the Safety Performance of](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0155) [Multilane Rural Highways. NCHRP Web-Only Document 126. National Cooperation](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0155) [Highway Research Program, Washington, D.C](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0155).
+- <span id="page-13-3"></span>[Lord, D., Mannering, F., 2010. The statistical analysis of crash-frequency data: a review](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0160) [and assessment of methodological alternatives. Transp. Res. Part A 44, 291](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0160)–305.
+- <span id="page-13-12"></span>Lord, D., Miranda-Moreno, L.F., 2008. Eff[ects of low sample mean values and small](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0165) sample size on the estimation of the fi[xed dispersion parameter of Poisson-gamma](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0165) [models for modeling motor vehicle crashes: a Bayesian perspective. Saf. Sci. 46 \(5\),](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0165) 751–[770](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0165).
+- <span id="page-13-31"></span>[Lord, D., Park, P.Y.-J., 2008. Investigating the e](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0170)ffects of the fixed and varying dispersion [parameters of Poisson-gamma models on empirical Bayes estimates. Accid. Anal.](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0170) [Prev. 40 \(4\), 1441](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0170)–1457.
+- <span id="page-13-6"></span>[Lord, D., Washington, S.P., Ivan, J.N., 2005. Poisson, Poisson-gamma and zero in](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0175)flated [regression models of motor vehicle crashes: balancing statistical](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0175) fit and theory. Accid. [Anal. Prev. 37 \(1\), 35](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0175)–46.
+- <span id="page-13-17"></span>[Ma, J., Kockelman, K., Damien, P., 2008. A multivariate Poisson-lognormal regression](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0180) [model for prediction of crash counts by severity, using Bayesian methods. Accid.](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0180) [Anal. Prev. 40, 964](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0180)–975.
+- <span id="page-13-4"></span>[Mannering, F.L., Bhat, C.R., 2014. Analytic methods in accident research: methodological](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0185) [frontier and future directions. Anal. Methods Accid. Res. 1, 1](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0185)–22.
+- <span id="page-13-5"></span>[Mannering, F.L., Shankar, V., Bhat, C.R., 2016. Unobserved heterogeneity and the sta](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0190)[tistical analysis of highway accident data. Anal. Methods Accid. Res. 11, 1](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0190)–16.
+- <span id="page-13-8"></span>[Maycock, G., Hall, R.D., 1984. Accidents at Four-Arm Roundabouts. Laboratory Report LR](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0195) [1120. Transport Research Laboratory, Crowthorne, Berkshire, U.K.](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0195)
+- <span id="page-13-43"></span>Meng, X.-L., Vaida, F., 2006. What'[s missing for DIC with missing data? \(Comment on](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0200) [article by Celeux et al.\). Bayesian Anal. 1, 687](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0200)–698.
+- <span id="page-13-7"></span>[Miaou, S.P., Lord, D., 2003. Modeling tra](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0205)ffic crash-flow relationships for intersections: [dispersion parameter, functional form, and Bayes versus empirical Bayes. Transp.](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0205) [Res. Rec. 1840, 31](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0205)–40.
+- <span id="page-13-14"></span>[Miaou, S.-P., Song, J.J., Mallick, B.K., 2003. Roadway tra](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0210)ffic crash mapping: a space time [modeling approach. J. Transp. Stat. 6 \(1\), 33](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0210)–57.
+- <span id="page-13-11"></span>[Miaou, S.-P., Song, J.J., 2005. Bayesian ranking of sites for engineering safety improve](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0215)[ments: decision parameter, treatability concept, statistical criterion and spatial de](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0215)[pendence. Accid. Anal. Prev. 37 \(4\), 699](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0215)–720.
+- <span id="page-13-44"></span>[Miller, 2009. Comparison of hierarchical Bayesian models for overdispersed count data](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0220) using DIC and Bayes' [factors. Biometrics 65, 962](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0220)–969.
+- <span id="page-13-30"></span>[Miranda-Moreno, L.F., Fu, L., Saccomanno, F.F., Labbe, A., 2005. Alternative risk models](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0225) [for ranking locations for safety improvement. Transp. Res. Rec. 1908, 1](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0225)–8.
+- <span id="page-13-2"></span>[Mitra, S., Washington, S., 2006. On the nature of over-dispersion in motor vehicle crash](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0230) [prediction models. Accid. Anal. Prev. 39, 459](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0230)–468.
+- <span id="page-13-40"></span>MRC Biostatistics Unit, Cambridge Biomedical Campus. DIC: Deviance Information Criterion. [http://www.mrc-bsu.cam.ac.uk/software/bugs/the-bugs-project-dic.](http://www.mrc-bsu.cam.ac.uk/software/bugs/the-bugs-project-dic)
+
+#### (Accessed 3 April 2016).
+
+- <span id="page-14-5"></span>[Park, E.S., Lord, D., 2007. Multivariate Poisson-lognormal models for jointly modeling](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0240) [crash frequency by severity. Transp. Res. Rec.: J. Transp. Res. Board. 2019, 1](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0240)–6.
+- <span id="page-14-12"></span>[Park, E.S., Park, J., Lomax, T.J., 2010. A fully Bayesian multivariate approach to before](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0245)[after safety evaluation. Accid. Anal. Prev. 42, 1118](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0245)–1127.
+- <span id="page-14-9"></span>Persaud, B.N., 1988. Do traffic signals aff[ect safety? Some methodological issues. Transp.](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0250) [Res. Rec. 1185, 37](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0250)–47.
+- <span id="page-14-11"></span>[Persaud, B., Lyon, C., Nguyen, T., 1999. Empirical Bayes procedure for ranking sites for](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0255) [safety investigation by potential for safety improvement. Transp. Res. Rec. 1665,](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0255) 7–[12](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0255).
+- <span id="page-14-10"></span>[Persaud, B., Lyon, C., 2007. Empirical Bayes before](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0260)–after safety studies: lessons learned [from two decades of experience and future directions. Accid. Anal. Prev. 39,](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0260) 546–[555](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0260).
+- <span id="page-14-17"></span><span id="page-14-1"></span>[Plummer, M., 2006. Comment on article by Celeux et al. Bayesian Anal. 1, 681](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0265)–686. [Poch, M., Mannering, F.L., 1996. Negative binomial analysis of intersection accident](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0270) [frequency. J. Transp. Eng. 122 \(2\), 105](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0270)–113.
+- <span id="page-14-4"></span>[Schluter, P.J., Deely, J.J., Nicholson, A.J., 1997. Ranking and selecting motor vehicle](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0275) [accident sites by using a hierarchical Bayesian model. Statistician 46, 293](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0275)–316.
+- <span id="page-14-16"></span>[Shirazi, M., Dhavala, S.S., Lord, D., Geedipally, S.R., 2017. A methodology to design](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0280) [heuristics for model selection based on characteristics of data: application to in](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0280)[vestigate when negative binomial Lindley \(NB-L\) is preferred over negative binomial](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0280) [\(NB\). Accid. Anal. Prev. 107, 186](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0280)–194.
+- <span id="page-14-19"></span>[Shirazi, M., Lord, D., 2018. Characteristics based heuristics to select a logical distribution](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0285) [between the Poisson gamma and the Poisson log-normal. In: Paper Presented at the](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0285) 97th [Annual Meeting of the Transportation Research Board. Washington, D.C](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0285).
+- <span id="page-14-14"></span>[Shirazi, M., Lord, D., Dhavala, S.S., Geedipally, S.R., 2016. A semiparametric negative](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0290) [binomial generalized linear model for modeling over dispersed count data with a](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0290) [heavy tail: characteristics and applications to crash data. Accid. Anal. Prev. 91,](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0290)
+
+- 10–[18](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0290).
+- <span id="page-14-18"></span>[Sinharay, S., Stern, H.S., 2003. Posterior predictive model checking in hierarchical](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0295) [models. J. Stat. Plan. Inference 111, 209](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0295)–221.
+- <span id="page-14-13"></span>[Spiegelhalter, D.J., Best, N.G., Carlin, B.P., van der Linde, A., 2002. Bayesian measures of](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0300) model complexity and fi[t. J. R. Stat. Soc. Ser. B 64, 583](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0300)–640.
+- <span id="page-14-7"></span>Spiegelhalter, D.J., Thomas, A., Best, N.G., Lun, D., 2003. WinBUGS Version 1.4.3 User Manual. MRC Biostatistics Unit, Cambridge. [http://www.mrc-bsu.cam.ac.uk/bugs/](http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/manual14.pdf) [winbugs/manual14.pdf](http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/manual14.pdf).
+- <span id="page-14-6"></span>[Tanner, M., Wong, W., 1987. The calculation of posterior distributions by data aug](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0310)[mentation \(with discussion\). J. Am. Stat. Assoc. 82, 528](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0310)–550.
+- <span id="page-14-20"></span>[Vangala, P., Lord, D., Geedipally, S.R., 2015. Exploring the application of the negative](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0315) [binomial-generalized exponential model for analyzing tra](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0315)ffic crash data with excess [zeros. Anal. Methods Accid. Res. 7, 29](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0315)–36.
+- <span id="page-14-3"></span>[Vogt, A., Bared, J., 1998. Accident models for two-lane rural segments and intersections.](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0320) [Transp. Res. Rec. 1635, 18](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0320)–29.
+- <span id="page-14-2"></span>[Washington, S.P., Karlaftis, M., Mannering, F.L., 2010. Statistical and Econometric](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0325) [Methods for Transportation Data Analysis, 2nd ed. Chapman and Hall, Boca Raton](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0325).
+- <span id="page-14-8"></span>Yang, M., Rasbash, J., Goldstein, H., Barbosa, M., 1999. MLwiN Macros for Advanced Multilevel Modelling. Version 2.0a. Multilevel Models Project, Institute of Education, University of London, UK. [http://www.bris.ac.uk/cmm/software/mlwin/download/](http://www.bris.ac.uk/cmm/software/mlwin/download/d-1-10/advmacma.pdf) [d-1-10/advmacma.pdf.](http://www.bris.ac.uk/cmm/software/mlwin/download/d-1-10/advmacma.pdf)
+- <span id="page-14-0"></span>[Zha, L., Lord, D., Zhou, Y., 2016. The Poisson inverse Gaussian \(PIG\) generalized linear](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0335) [regression model for analyzing motor vehicle crash data. J. Transp. Saf. Secur. 8 \(1\),](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0335) 18–[35](http://refhub.elsevier.com/S0001-4575(18)30154-4/sbref0335).
+- <span id="page-14-15"></span>Zou, Y., Lord, D., Zhang, Y., 2012. Analyzing Highly Dispersed Crash Data Using the Sichel Generalized Additive Models for Location, Scale and Shape. Working Paper. Zachry Department of Civil Engineering, Texas A&M University, College Station, TX. [https://ceprofs.civil.tamu.edu/dlord/Papers/Zou\\_et\\_al\\_Sichel\\_GAMLSS.pdf](https://ceprofs.civil.tamu.edu/dlord/Papers/Zou_et_al_Sichel_GAMLSS.pdf).

Zotero/001_artiklid/Road Safety Indicators/kmeturbanruraldifferencesmotor2006.md

@@ -0,0 +1,204 @@
+---
+category: literaturenote
+citekey: kmeturbanruraldifferencesmotor2006
+title: Urban–rural differences in motor vehicle crash fatality and hospitalization rates among children and youth
+authors: "Kmet, Leanne; Macarthur, Colin"
+year: 2006
+date: 2006-00-00 2006
+doi: 10.1016/j.aap.2005.07.007
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2005.07.007"
+zotero_key: UZF259QW
+zotero_storage: EQPWF8CF
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Kmet, Leanne"
+status: converted
+---
+![](_page_0_Picture_0.jpeg)
+
+Accident Analysis and Prevention 38 (2006) 122–127
+
+![](_page_0_Picture_2.jpeg)
+
+# Urban–rural differences in motor vehicle crash fatality and hospitalization rates among children and youth
+
+Leanne Kmet a, Colin Macarthur <sup>b</sup>,<sup>∗</sup>
+
+<sup>a</sup> *Department of Community Health Sciences, University of Calgary, 3330 Hospital Drive NW, Calgary, Alta., Canada T2N 4N1* <sup>b</sup> *University of Toronto, Department of Paediatrics, Bloorview MacMillan Children's Center, Bloorview Research Institute, 150 Kilgour Road, Toronto, Ont., Canada M4G 1R8*
+
+Received 29 April 2005; received in revised form 29 July 2005; accepted 29 July 2005
+
+#### **Abstract**
+
+This population-based study examined motor vehicle crash hospitalization rates and death rates among children and youth in rural and urban areas of the province of Alberta, Canada. Using police report data (1997–2002, inclusive), average annual motor vehicle crash hospitalization and death rates among those 0–19 years of age were calculated for rural and urban regions. Across all age and sex strata examined, both the hospitalization and the fatality rates were significantly higher in rural compared with urban regions. After adjusting for age, sex and calendar year, the relative risk of a motor vehicle crash hospitalization (rural versus urban) was 3.0 (95% CI: 2.8, 3.2). After adjusting for age, sex and calendar year, the relative risk of a motor vehicle crash fatality was 5.4 (95% CI: 4.2, 6.9). Motor vehicle crash injury hospitalization and fatality rates among children and youth in the province of Alberta are considerably higher in rural areas compared with urban areas. There is a need to identify social, demographic and environmental driving hazards associated with the rural environment. © 2005 Elsevier Ltd. All rights reserved.
+
+*Keywords:* Child; Motor vehicle; Mortality; Hospitalization; Urban–rural
+
+#### **1. Introduction**
+
+In Canada, motor vehicle traffic crashes are the leading cause of injury mortality and a leading cause of injury hospitalization among children and youth ([Child Injury](#page-4-0) [Division—Bureau of Reproductive and Child Health, 1999\).](#page-4-0) Such injuries accounted for 574 deaths and 6035 hospitalizations in the 0–19 year age group in Canada in 1997 ([Child](#page-4-0) [Injury Division—Bureau of Reproductive and Child Health,](#page-4-0) [1999\).](#page-4-0) Motor vehicle traffic crashes accounted for approximately 33%, 50% and 75% of all unintentional injury deaths among those 0–4, 5–14 and 15–19 years of age, respectively [\(Child Injury Division—Bureau of Reproductive and Child](#page-4-0) [Health, 1999\).](#page-4-0)
+
+Examination of geographic variation in motor vehicle crash injury rates may contribute to the effective targeting of injury countermeasures. The objective of this populationbased study was to examine motor vehicle crash hospitalization and death rates among children and youth in rural and urban areas of the province of Alberta, Canada.
+
+#### **2. Materials and methods**
+
+## *2.1. Motor vehicle crash records*
+
+Data on motor vehicle crash injuries resulting in hospitalization or death among children and youth (0–19 years) in the province of Alberta from January 1, 1997 to December 31, 2002 were obtained from police reports compiled in the Alberta Collision Information System. Legislation in Alberta requires that any traffic collision resulting in death, injury or property damage exceeding CDN\$ 1000 be reported to police. Fatal injuries were defined as those resulting in death within 30 days of the collision. (It has been estimated that only 2% of all motor vehicle crash-related fatalities occur
+
+<sup>∗</sup> Corresponding author. Tel.: +1 416 424 3862; fax: +1 416 425 9332. *E-mail address:* cmacarthur@bloorviewmacmillan.on.ca (C. Macarthur).
+
+after 30 days [\(Mao et al., 1997\).](#page-5-0)) The analysis was restricted to occupants of privately operated vehicles injured on public roadways.
+
+### *2.2. Urban/rural classification*
+
+During the study period, Alberta had 17 geographically defined Regional Health Authorities, each of which assumed local responsibility for injury prevention. For each motor vehicle crash injury, the crash location was determined by linking the police detachment responding to the crash to the respective Health Authority. The two regions with the largest population densities (about 160 persons/km2) were defined as 'urban'. The remaining 15 regions (all with population densities <11 persons/km2) were defined as 'rural'.
+
+### *2.3. Rate calculations and comparisons*
+
+Age- and sex-specific population estimates for each Regional Health Authority were obtained from the Alberta Health Care Population Registry. Assuming a Poisson distribution of both the hospitalizations and the deaths (appropriate given rare events in sufficiently large populations ([Selvin,](#page-5-0) [1995\)\)](#page-5-0), data from the 6-year study period were used to calculate average annual motor vehicle crash hospitalization and death rates for urban and rural regions, along with 95% Poisson confidence intervals. Rates were also calculated by age group (0–14, 15–19 years) and sex (male, female). Using these rates, the relative risk of a motor vehicle crash injury hospitalization (rural versus urban) or death (rural versus urban) along with 95% confidence intervals were calculated for each age- and sex-specific stratum. Poisson regression analysis was then used to calculate adjusted relative risks (rural versus urban), specifying either the number of deaths or the number of injury hospitalizations as the dependent variable and including population size (by age group and sex), location (rural versus urban) and calendar year as independent variables. All analyses were conducted using STATA 6.0 (College Station, TX).
+
+## **3. Results**
+
+A total of 4660 children and youth (0–19 years) in Alberta were involved in motor vehicle crashes resulting in hospitalization or death during the study period. Injuries to pedestrians (*n* = 440), bicyclists (*n* = 201), motorcyclists (*n* = 123), and injuries involving farm vehicles (*n* = 2), construction vehicles (*n* = 2), public transportation (*n* = 11), off-road snow, or all-terrain vehicles (*n* = 118) were excluded. Injuries for which the sex of the victim (*n* = 5), the type of vehicle involved (*n* = 5) or the crash location (*n* = 3) were not documented were also excluded, leaving 383 fatalities and 3367 hospitalizations for analysis.
+
+There was homogeneity of motor vehicle traffic crash fatality and hospitalization rates between the two urban regions and also across the 15 rural regions. Therefore, the data from the two urban regions were combined as were the data from the 15 rural regions. The pooled data were then used to estimate summary urban and rural rates for Alberta.
+
+As shown in [Table 1,](#page-2-0) the motor vehicle traffic crash fatality rates were significantly higher – across all age and sex strata – in rural Alberta, compared with urban Alberta. The relative risk of a motor vehicle traffic crash fatality (rural versus urban) – after adjusting for age, sex, and calendar year – was 5.4 (95% confidence interval (CI): 4.2, 6.9).
+
+Similar findings were noted for motor vehicle traffic crash hospitalization rates. As shown in [Table 2,](#page-2-0) hospitalization rates were significantly higher – across all age and sex strata – in rural Alberta, compared with urban Alberta. The relative risk of a motor vehicle traffic crash hospitalization (rural versus urban) – after adjusting for age, sex, and calendar year – was 3.0 (95% CI: 2.8, 3.2).
+
+To assess whether the higher relative risks in rural versus urban areas were influenced by multiple casualty crashes, we also determined rates based on the number of crashes. Data from 1997 (85 fatalities, 537 hospitalizations) were excluded from these calculations as case numbers (i.e., unique crash identifiers) were not available. The remaining 298 deaths resulted from 259 motor vehicle crashes. Forty-six of these occurred in urban areas, compared with 213 in rural areas, yielding average annual fatal motor vehicle crash rates (per 100,000 population) of 1.9 and 11.3, respectively. The relative risk (rural versus urban) of a fatal motor vehicle crash was 5.9 (95% CI: 4.3, 8.3). The 2830 injury hospitalizations occurred as a result of 693 urban crashes and 1503 rural crashes, yielding average annual crash rates (per 100,000 population) of 29.0 and 79.7, respectively. The relative risk (rural versus urban) of a motor vehicle crash resulting in hospitalization was 2.8 (95% CI: 2.5, 3.0).
+
+Finally, since a single crash could result in both fatalities and injury hospitalizations, the data were combined and the number of severe crashes (i.e., crashes resulting in death and/or hospitalization) was determined. There were 724 severe urban crashes and 1625 severe rural crashes, yielding average annual crash rates (per 100,000 population) of 30.3 and 86.2, respectively. The relative risk (rural versus urban) of a severe motor vehicle crash was 2.8 (95% CI: 2.6, 3.1).
+
+As would be expected, in both the rural and urban areas, motor vehicle crash injury rates (deaths and hospitalizations) were considerably higher among youth 15–19 years of age, compared with younger children (see [Tables 1 and 2\).](#page-2-0) In both areas, the rates were highest among 15–19-year-old males. For example, in rural areas, males in this age group were ten times more likely than younger children of both sexes to die in motor vehicle crashes.
+
+As shown in [Table 3,](#page-2-0) rural adolescent males were significantly more likely to be involved in a motor vehicle crash resulting in death or hospitalization, compared with rural adolescent females. Urban data showed an increased relative risk (male versus female) for motor vehicle crash fatalities among adolescents only.
+
+<span id="page-2-0"></span>Table 1 Fatal motor vehicle crash injury rates by rural and urban location among children and youth (0–19 years) in Alberta, Canada (1997–2002)
+
+| Age (years) | Sex    | Rural         |                              | Urban         |                              | Relative risk (rural vs. |
+|-------------|--------|---------------|------------------------------|---------------|------------------------------|--------------------------|
+|             |        | No. of deaths | Fatal injury ratea (95% CI)b | No. of deaths | Fatal injury ratea (95% CI)b | urban) (95% CI)b         |
+| 0–14        | Male   | 44            | 5.1 (3.7–6.9)                | 9             | 0.8 (0.4–1.6)                | 6.2 (3.0–12.7)           |
+|             | Female | 40            | 4.9 (3.5–6.7)                | 7             | 0.7 (0.3–1.4)                | 7.3 (3.3–16.3)           |
+| 15–19       | Male   | 142           | 47.7 (40.2–56.2)             | 41            | 11.1 (8.0–15.1)              | 4.3 (3.0–6.1)            |
+|             | Female | 85            | 29.9 (23.9–37.0)             | 15            | 4.3 (2.4–7.1)                | 7.0 (4.0–12.1)           |
+
+<sup>a</sup> Average annual rate, expressed as deaths/100,000.
+
+Table 2 Motor vehicle crash injury hospitalization rates by rural and urban location among children and youth (0–19 years) in Alberta, Canada (1997–2002)
+
+| Age (years) | Sex    | Rural                         |                     | Urban                         |                    | Relative risk (rural vs. |
+|-------------|--------|-------------------------------|---------------------|-------------------------------|--------------------|--------------------------|
+|             |        | No. of hospital<br>admissions | Ratea (95% CI)b     | No. of hospital<br>admissions | Ratea (95% CI)b    | urban) (95% CI)b         |
+| 0–14        | Male   | 309                           | 35.9 (32.0–40.2)    | 111                           | 10.1 (8.3–12.2)    | 3.5 (2.9–4.4)            |
+|             | Female | 273                           | 33.5 (29.7–37.7)    | 114                           | 10.9 (9.0–13.1)    | 3.1 (2.5–3.8)            |
+| 15–19       | Male   | 1004                          | 337.2 (316.6–358.7) | 395                           | 107.3 (97.0–118.5) | 3.1 (2.8–3.5)            |
+|             | Female | 786                           | 276.5 (257.5–296.5) | 375                           | 107.1 (96.5–118.5) | 2.6 (2.3–2.9)            |
+
+<sup>a</sup> Average annual rate, expressed as injury hospitalizations/100,000.
+
+Table 3 Relative risk (male vs. female) of motor vehicle crash injury (death and hospitalization) by rural and urban location among children and youth (0–19 years) in Alberta, Canada (1997–2002)
+
+| Age (years) | Location | Relative risk: male vs. female (95% CI)a |                 |  |
+|-------------|----------|------------------------------------------|-----------------|--|
+|             |          | Death                                    | Hospitalization |  |
+| 0–14        | Rural    | 1.0 (0.7–1.6)                            | 1.1 (0.9–1.3)   |  |
+| 0–14        | Urban    | 1.2 (0.5–3.3)                            | 0.9 (0.7–1.2)   |  |
+| 15–19       | Rural    | 1.6 (1.2–2.1)                            | 1.2 (1.1–1.3)   |  |
+| 15–19       | Urban    | 2.6 (1.4–4.7)                            | 1.0 (0.9–1.2)   |  |
+
+<sup>a</sup> 95% Poisson confidence interval.
+
+#### **4. Discussion**
+
+From 1997 to 2002, rural children and youth in Alberta, Canada were five times more likely than their urban counterparts to die in a motor vehicle crash and three times more likely to be admitted to hospital following a motor vehicle crash. These increases in risk (rural versus urban) were consistent across age groups and sex.
+
+An increased risk of fatal and severe motor vehicle crash injuries among children and youth in rural (versus urban) environments has been previously documented ([Hwang et al.,](#page-5-0) [1997; King et al., 1994; Lapidus et al., 1998; Niemcryk et al.,](#page-5-0) [1997; Svenson et al., 1996a,b\).](#page-5-0) For example, a Nevada study examining severe motor vehicle crash injuries [\(Niemcryk et](#page-5-0) [al., 1997\)](#page-5-0) showed that rates among children (0–14 years) were three times higher in rural areas, compared with urban areas. In contrast, however, to the two- to three-fold increased risk of fatal crash involvement in rural (versus urban) areas reported by others [\(Hwang et al., 1997; King et al., 1994](#page-5-0); [Svenson et al., 1996a,b\),](#page-5-0) we observed a five-fold increase in risk.
+
+The main strength of our study was the use of populationbased data collected over a 6-year period, resulting in a large sample size and stable estimates of motor vehicle crash injury rates. The same data source was used to identify motor vehicle crash injuries among both urban and rural children. When estimating relative risks (rural versus urban), Poisson regression analyses were used to control for confounding by age, sex and calendar year.
+
+The study, however, also had several limitations. For example, police data are known to underestimate the number of motor vehicle crash injuries. Comparisons of police, hospital and vital statistics data show that between 85% and 99% of crash fatalities are captured by police records ([Aptel et](#page-4-0) [al., 1999; Rosman and Knuiman, 1994\).](#page-4-0) Capture of non-fatal crash injuries requiring hospitalization is lower, ranging from 45% to 67% ([Alsop and Langley, 2001; Aptel et al., 1999;](#page-4-0) [Cryer et al., 2001; Rosman, 2001; Rosman and Knuiman,](#page-4-0) [1994\).](#page-4-0) Further, some ([Alsop and Langley, 2001; Rosman,](#page-4-0) [2001\),](#page-4-0) but not all ([Aptel et al., 1999; Rosman and Knuiman,](#page-4-0) [1994\)](#page-4-0) studies indicate that police underestimation of nonfatal injuries is greater among children and youth compared with adults. In addition, a single study reported greater underestimation in rural areas compared with urban areas ([Aptel](#page-4-0) [et al., 1999\).](#page-4-0) Taken together, these data suggest that both the absolute injury rates and the relative risks (rural versus urban) in our study may be conservative estimates, particularly for injury hospitalizations.
+
+<sup>b</sup> 95% Poisson confidence interval.
+
+<sup>b</sup> 95% Poisson confidence interval.
+
+Our study also had other limitations. Injury rates were calculated using the number of injuries occurring in a given region in the numerator and regional population estimates in the denominator. As such, each injury included in the numerator did not necessarily represent an injury to an 'at risk' individual in the denominator. While rates calculated in this manner succeed in identifying high risk locations, they are subject to error. For example, in areas with many visitors, motor vehicle crash rates may be exaggerated. Conversely, in areas where residents frequently travel outside the area, the rates may be underestimated.
+
+Further, as our data did not identify injured individuals by place of residence, our analysis was not specific to Alberta residents. While estimates of non-fatal motor vehicle crash injuries to out-of-province residents are not available in Alberta, data from the provincial Vital Statistics Death Registry indicate that out-of-province residents account for only 8% of all motor vehicle crash deaths in the province each year. Therefore, even if out-of-province deaths were systematically distributed to particular geographic areas, this bias would not have materially altered our relative risk estimates (rural versus urban) for crash fatalities.
+
+Finally, data on all variables of interest were not available to us. Regional and individual-level information on variables such as vehicle travel speed, type of collision, environmental conditions, alcohol use and safety restraint use would have permitted an exploration of the predictors contributing to the rural–urban differences in motor vehicle crash injury rates. In general, there is limited population-based information on the urban–rural distribution of motor vehicle crash injury risk factors.
+
+The higher motor vehicle crash injury rates observed in rural Alberta may reflect greater exposure to vehicle travel by individuals in outlying areas. With respect to travel to school, an observational study of 4000 children attending randomly selected elementary schools in South Carolina found that in both urban and suburban areas, 44% of children were driven to and from school in private vehicles ([Sirard et al., 2005\)](#page-5-0). Similarly, in a population-based survey in Georgia, 43% of parents of children 5–15 years reported driving their children to and from school most of the time ([Anon., 2002\).](#page-4-0) While the percentage of children who traveled in private vehicles or buses was not reported by place of residence, the percentage of children who regularly walked to school was similar in rural and urban areas (17% and 19%, respectively) ([Anon.,](#page-4-0) [2002\).](#page-4-0)
+
+While the frequency of travel to and from school by private vehicle may be similar in rural and urban areas, distances traveled (to and from school, and overall) are likely greater in rural areas. For example, a survey of drivers in Ontario, Canada, found that rural drivers, compared with urban drivers, spend more time on the road (55 min per day versus 47 min per day, respectively) and drive longer distances (44 km per day versus 31 km per day, respectively) ([Chipman et al., 1993\)](#page-4-0). In an ecologic analysis of United States Federal Highway Administration data, estimates of vehicle miles traveled per capita (in thousands) ranged from 10.0 to 52.5 in rural areas (median of 15.0) and from 5.3 to 12.2 in urban areas (median of 8.0) [\(Clark and Cushing,](#page-4-0) [2004\).](#page-4-0) Regardless, even a two-fold urban–rural differential in average exposure in Alberta would not explain the magnitude of the urban–rural differences in crash injury rates that were observed.
+
+In addition to absolute differences in distances traveled, it is important to note that a mile of rural travel is not necessarily equivalent to a mile of urban travel. For example, both crash and crash injury risks are influenced by traffic flow characteristics such as traffic volume, vehicle density and the traffic volume to road capacity ratio [\(Lord et al., 2005\).](#page-5-0) However, the combined contribution of these factors to road safety has not been well established and the need for separate analyses of urban and rural roadways has been noted ([Lord et al.,](#page-5-0) [2005\).](#page-5-0) Similarly, road safety features such as traffic control devices, graded curves, lighting and divided traffic streams are, for the most part, less common in rural compared with urban areas, which likely contributes to urban–rural differences in crash and crash injury risks. For example, in the United States, head-on collisions account for 12% of all rural fatal crashes but only 7% of all urban fatal crashes [\(National](#page-5-0) [Highway Traffic Safety Administration, 2002\). A](#page-5-0) before-andafter analysis found that the installation of rumble strips (i.e., raised or grooved patterns that produce audible or tactile warnings when crossed by vehicle tires) along the centerlines of undivided, two-lane rural roads was associated with a 14% reduction in all injury crashes and a 25% reduction in head-on and opposing direction sideswipe injury crashes ([Persaud et al., 2004\).](#page-5-0)
+
+In an ecological study, which considered police reported crashes and self-reported estimates of vehicle miles traveled in the U.S., fatal crash incidence density (i.e., fatal crashes/vehicle miles traveled) was determined for rural and urban areas [\(Zwerling et al., 2005\). F](#page-5-0)atal crash incidence density was calculated as the product of crash incidence density (all crashes/million vehicle miles traveled), the crash injury rate (injury crashes/all crashes) and the injury fatality rate (fatalities/injury crashes). While crash incidence density and crash injury rates were generally similar or slightly higher (rural versus urban) among drivers of both sexes and all ages, injury fatality rates were consistently 2–3.5 times higher in rural areas. Accordingly, the authors suggested that differences in crash and post-crash factors may contribute more to urban–rural differences in fatality risks than differences in driving exposure. For example, in the U.S., about 80% of vehicles involved in fatal rural crashes must be towed from the scene, compared with 66% of vehicles involved in fatal urban crashes ([National Highway Traffic Safety Adminis](#page-5-0)[tration, 2002\),](#page-5-0) suggesting higher crash speeds and/or more severe crashes in rural areas. Delayed access to emergency care and inadequate advanced trauma care, known to be predictive of worse outcomes among crash survivors [\(Grossman](#page-5-0) [et al., 1997; Muelleman and Mueller, 1996\)](#page-5-0), may also contribute to the higher crash fatality rates observed in rural <span id="page-4-0"></span>areas. In general, rural (versus urban) crash victims experience longer discovery times, less advanced on-scene care, and longer transport times to hospital (Esposito et al., 1995; Grossman et al., 1997; Svenson et al., 1996a,b). In Alberta, there are only two trauma centers (located in the urban areas of Calgary and Edmonton) to serve the entire province, which spans over 660,000 km2.
+
+Lastly, some data indicate less compliance with road safety behaviors in rural areas. A lower prevalence of seat belt use has been observed in rural areas of Alberta compared with the entire province. In 1999, the prevalence of seat belt use among drivers and front seat passengers of light-duty vehicles was about 69% in rural areas (Alberta Occupant Restraint Program, 2002), compared with 89% for the entire province [\(Transport Canada, 1999\).](#page-5-0) By 2001, the prevalence in rural areas had increased to 76% (Alberta Occupant Restraint Program, 2002) but remained below the provincial average of 85% [\(Transport Canada, 2001\)](#page-5-0). While Canadian data specific to children are not available, a U.S. population-based observational survey of restraint use at controlled intersections during daylight hours found that restraint use among child passengers 0–4 years was less common in rural areas compared with urban areas (72% and 96%, respectively) [\(National Highway Traffic Safety Administration, 2003\).](#page-5-0)
+
+Impaired driving citations are issued at a three-fold higher rate in rural (versus urban) areas in Alberta [\(Slavik, 1996\).](#page-5-0) A survey of Canadian police officers ([Jonah et al., 1999\)](#page-5-0); however, found that 70% of such citations were issued as a result of an impaired driver attracting attention, e.g., by driving erratically or following a collision attended by police. Only 11% were issued during focused enforcement campaigns. Therefore, given the low probability of encountering a police officer while driving, impaired driving citations likely underestimate the true impaired driving rate, and such underestimation is probably greater in remote rural areas compared with urban areas.
+
+A recent survey of Australian drivers showed that 33% admitted to exceeding posted speed limits by 10 km/h or more in urban zones with 60 km/h limits, and 46% admitted to this practice in rural zones with 100 km/h limits ([Mitchell-](#page-5-0)[Taverner et al., 2003\).](#page-5-0) Furthermore, drivers residing in some rural areas were more likely than other drivers (15% versus 10%) to report speeds of at least 120 km/h in areas with posted speed limits of 100 km/h ([Mitchell-Taverner et al.,](#page-5-0) [2003\).](#page-5-0) These higher speeds likely contribute to the higher injury crash rates in rural areas. A case–control study in rural Australia [\(Kloeden et al., 2001\) c](#page-5-0)ompared the pre-crash travel speeds of passenger vehicles involved in severe or fatal crashes with control vehicles (traveling in the same direction, at the same location, time of day, day of week and time of year) and found that travel speeds 10 km/h above the average speed of non-crash vehicles more than doubled the risk of injury crash involvement. Travel speeds 20 km/h above the average speed resulted in a near six-fold increase in risk.
+
+The logistical difficulties and financial constraints faced by police agencies responsible for enforcing traffic laws on low volume rural roads may contribute to these urban–rural differences in unsafe driving behaviors. For example, despite a lower self-reported prevalence of speeding in urban areas, urban drivers in Australia were more likely than rural drivers (20% versus 15%) to have received a citation for speeding in the previous two years [\(Mitchell-Taverner et al., 2003\).](#page-5-0)
+
+## **5. Conclusion**
+
+This study showed that motor vehicle crash fatality and injury hospitalization rates among children and youth in the province of Alberta were five times higher and three times higher, respectively, in rural compared with urban areas. Possible reasons for the elevated rates observed in rural areas include greater exposure to motor vehicle travel, less safe driving environments, reduced access to advanced trauma care and less compliance with road safety behaviors. There is a need to identify social, demographic and environmental driving hazards associated with the rural environment.
+
+## **Acknowledgements**
+
+The authors would like to thank Ms. L. Owens and Ms. M. Anderson (Transportation Safety Services Division, Alberta Transportation) and Mr. L. Svenson (Alberta Health and Wellness) for their assistance in data collection and interpretation.
+
+## **References**
+
+Alberta Occupant Restraint Program, 2002. Rural seat belt use observational study 1999 and 2001: Summary report. Edmonton, Alberta Centre for Injury Control and Research, Alberta's Regional Health Authorities, Transport Canada (available: [http://www.med.ualberta.ca/](http://www.med.ualberta.ca/acicr/download/2001seat.pdf) [acicr/download/2001seat.pdf\)](http://www.med.ualberta.ca/acicr/download/2001seat.pdf).
+
+Alsop, J., Langley, J., 2001. Under-reporting of motor vehicle traffic crash victims in New Zealand. Accid. Anal. Prev. 33, 353–359.
+
+Anon., 2000. School transportation modes—Georgia. MMWR Morb. Mortal Wkly. Rep. 51, 704–705.
+
+Aptel, I., Salmi, L.R., Masson, F., Bourde, A., Henrion, G., Erny, P., 1999. Road accident statistics: discrepancies between police and hospital data in a French island. Accid. Anal. Prev. 31, 101–108.
+
+Child Injury Division—Bureau of Reproductive and Child Health, 1999. Canadian Injury Data: Mortality—1997 and Hospitalizations— 1996–97. Ottawa, Laboratory Centre for Disease Control, Health Protection Branch, Health Canada (available: [http://www.phac-aspc.gc.ca/](http://www.phac-aspc.gc.ca/injury-bles/cid98-dbc98/injury_e.pdf) [injury-bles/cid98-dbc98/injury](http://www.phac-aspc.gc.ca/injury-bles/cid98-dbc98/injury_e.pdf) e.pdf).
+
+Chipman, M.L., MacGregor, C.G., Smiley, A.M., Lee-Gosselin, M., 1993. The role of exposure in comparisons of crash risk among different drivers and driving environments. Accid. Anal. Prev. 25, 207–211.
+
+Clark, D.E., Cushing, B.M., 2004. Rural and urban traffic fatalities, vehicle miles and population density. Accid. Anal. Prev. 36, 967–972.
+
+Cryer, P.C., Westrup, S., Cook, A.C., Ashwell, V., Bridger, P., Clarke, C., 2001. Investigation of bias after data linkage of hospital admissions data to police road traffic crash reports. Inj. Prev. 7, 234–241.
+
+Esposito, T.J., Maier, R.V., Rivara, F.P., Pilcher, S., Griffith, J., Lazear, S., Hogan, S., 1995. The impact of variation in trauma care times: urban versus rural. Prehospital Disaster Med. 10, 161–166.
+
+- <span id="page-5-0"></span>Grossman, D.C., Kim, A., Macdonald, S.C., Klein, P., Copass, M.K., Maier, R.V., 1997. Urban–rural differences in prehospital care of major trauma. J. Trauma 42, 723–729.
+- Hwang, H.C., Stallones, L., Keefe, T.J., 1997. Childhood injury deaths: rural and urban differences, Colorado 1980–8. Inj. Prev. 3, 35–37.
+- Jonah, B., Yuen, L., Au-Yeung, E., Paterson, D., Dawson, N., Thiessen, R., Arora, H., 1999. Front-line police officers' practices, perceptions and attitudes about the enforcement of impaired driving laws in Canada. Accid. Anal. Prev. 31, 421–443.
+- King, W.D., Nichols, M.H., Hardwick, W.E., Palmisano, P.A., 1994. Urban/rural differences in child passenger deaths. Pediatr. Emerg. Care 10, 34–36.
+- Kloeden, C.N., Ponte, G., McLean, A.J., 2001. Travelling Speed and the Risk of Crash Involvement on Rural Roads. Adelaide, Road Accident Research Unit, Adelaide University (available: [http://www.atsb.gov.au/](http://www.atsb.gov.au/road/rpts/cr204/index.cfm) [road/rpts/cr204/index.cfm](http://www.atsb.gov.au/road/rpts/cr204/index.cfm)).
+- Lapidus, G., Lerer, T., Zavoski, R., Banco, L., 1998. Childhood injury in Connecticut. Conn. Med. 62, 323–331.
+- Lord, D., Manar, A., Vizioli, A., 2005. Modeling crash-flow-density and crash-flow-V/C ratio relationships for rural and urban freeway segments. Accid. Anal. Prev. 37, 185–199.
+- Mao, Y., Zhang, J., Robbins, G., Clarke, K., Lam, M., Pickett, W., 1997. Factors affecting the severity of motor vehicle traffic crashes involving young drivers in Ontario. Inj. Prev. 3, 183–189.
+- Mitchell-Taverner, P., Zipparo, L., Goldsworthy, J., 2003. Survey on Speeding and Enforcement. Surry Hills, Taverner Research Company (available: [http://www.atsb.gov.au/road/pdf/cr214a](http://www.atsb.gov.au/road/pdf/cr214a_final_report.pdf) final report.pdf).
+- Muelleman, R.L., Mueller, K., 1996. Fatal motor vehicle crashes: variations of crash characteristics within rural regions of different population densities. J. Trauma 41, 315–320.
+- National Highway Traffic Safety Administration, 2002. Traffic safety facts, 2002: Rural/Urban Comparison. National Center for Statistics and Analysis, Washington, DC (available: [http://www-nrd.nhtsa.dot.](http://www-nrd.nhtsa.dot.gov/pdf/nrd-30/ncsa/tsf2002/2002ruralurban.pdf) [gov/pdf/nrd-30/NCSA/TSF2002/2002RuralUrban.pdf](http://www-nrd.nhtsa.dot.gov/pdf/nrd-30/ncsa/tsf2002/2002ruralurban.pdf)).
+- National Highway Traffic Safety Administration, 2003. Research Note: the Use of Child Restraints in 2002. National Center for Statistics and Analysis, Washington, DC (available: [http://www.nhtsa.dot.gov/](http://www.nhtsa.dot.gov/cps/childrestraints/childrestraints.html) [CPS/ChildRestraints/ChildRestraints.html](http://www.nhtsa.dot.gov/cps/childrestraints/childrestraints.html)).
+
+- Niemcryk, S.J., Kaufmann, C.R., Brawley, M., Yount, S.I., 1997. Motor vehicle crashes, restraint use, and severity of injury in children in Nevada. Am. J. Prev. Med. 13, 109–114.
+- Persaud, B.N., Retting, R.A., Lyon, C.A., 2004. Crash reduction following installation of centerline rumble strips on rural two-lane roads. Accid. Anal. Prev. 36, 1073–1079.
+- Rosman, D.L., 2001. The Western Australian Road Injury Database (1987–1996): ten years of linked police, hospital and death records of road crashes and injuries. Accid. Anal. Prev. 33, 81–88.
+- Rosman, D.L., Knuiman, M.W., 1994. A comparison of hospital and police road injury data. Accid. Anal. Prev. 26, 215–222.
+- Selvin, S., 1995. Poisson regression analysis. In: Practical Biostatistical Methods. Duxbury Press, Toronto, pp. 455–496.
+- Sirard, J.R., Ainsworth, B.E., McIver, K.L., Pate, R.R., 2005. Prevalence of active commuting at urban and suburban elementary schools in Columbia, SC. Am. J. Public Health 95, 236–237.
+- Slavik, W., 1996. Alberta Profile: Social and Health Indicators of Alcohol and Drug Abuse, 1996 Update Now Including Regional Health Authorities. Alberta Alcohol and Drug Abuse Commission, Edmonton, Alberta.
+- Svenson, J.E., Nypaver, M., Calhoun, R., 1996a. Pediatric prehospital care: epidemiology of use in a predominantly rural state. Pediatr. Emerg. Care 12, 173–179.
+- Svenson, J.E., Spurlock, C., Nypaver, M., 1996b. Factors associated with the higher traumatic death rate among rural children. Ann. Emerg. Med. 27, 625–632.
+- Transport Canada, 1999. Results of Transport Canada's June 1999 Survey of Seat Belt Use in Canada. Road Safety Programs Branch, Transport Canada, Ottawa (available: [http://www.tc.gc.ca/roadsafety/](http://www.tc.gc.ca/roadsafety/tp2436/cl9907/menu.htm) [tp2436/cl9907/menu.htm\)](http://www.tc.gc.ca/roadsafety/tp2436/cl9907/menu.htm).
+- Transport Canada, 2001. Results of Transport Canada's July 2001 Survey of Seat Belt Use in Canada. Road Safety Programs Branch, Transport Canada, Ottawa (available: [http://www.tc.gc.ca/roadsafety/](http://www.tc.gc.ca/roadsafety/tp2436/rs200107/en/menu.htm) [tp2436/rs200107/en/menu.htm\)](http://www.tc.gc.ca/roadsafety/tp2436/rs200107/en/menu.htm).
+- Zwerling, C., Peek-Asa, C., Whitten, P.S., Choi, S.-W., Sprince, N.L., Jones, M.P., 2005. Fatal motor vehicle crashes in rural and urban areas: decomposing rates into contributing factors. Inj. Prev. 11, 24–28.

Zotero/001_artiklid/Road Safety Indicators/laomodelinganimalvehiclecollisions2011.md

@@ -0,0 +1,434 @@
+---
+category: literaturenote
+citekey: laomodelinganimalvehiclecollisions2011
+title: Modeling animal–vehicle collisions considering animal–vehicle interactions
+authors: "Lao, Yunteng; Zhang, Guohui; Wu, Yao-Jan; Wang, Yinhai"
+year: 2011
+date: 2011-00-00 2011
+doi: 10.1016/j.aap.2011.05.017
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2011.05.017"
+zotero_key: C9WQTZZ7
+zotero_storage: 9MRZ9HDE
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Lao, Yunteng"
+status: converted
+---
+<span id="page-0-0"></span>
+
+Contents lists available at [ScienceDirect](http://www.sciencedirect.com/science/journal/00014575)
+
+# Accident Analysis and Prevention
+
+journal homepage: [www.elsevier.com/locate/aap](http://www.elsevier.com/locate/aap)
+
+![](_page_0_Picture_5.jpeg)
+
+# Modeling animal–vehicle collisions considering animal–vehicle interactions
+
+Yunteng Laoa, Guohui Zhang <sup>b</sup>,1, Yao-Jan Wu<sup>a</sup>,2, Yinhai Wang<sup>a</sup>,<sup>∗</sup>
+
+- <sup>a</sup> Department of Civil and Environmental Engineering, University of Washington, Seattle, WA 98195, USA
+- <sup>b</sup> Department of Civil, Architectural and Environmental Engineering, University of Texas at Austin, Austin, TX 78705, USA
+
+### a r t i c l e i n f o
+
+Article history: Received 18 October 2010 Received in revised form 2 May 2011 Accepted 17 May 2011
+
+Keywords: Accident modeling Animal–vehicle collision Carcass removal Roadway Safety Vehicle–animal interaction-based probability model Negative binomial (NB) regression
+
+### a b s t r a c t
+
+Animal–Vehicle Collisions (AVCs) have been a major safety problem in the United States over the past decades. Counter measures against AVCs are urgently needed for traffic safety and wildlife conservation. To better understand the AVCs, a variety of data analysis and statistical modeling techniques have been developed. However, these existing models seldom take human factors and animal attributes into account. This paper presents a new probability model which explicitly formulates the interactions between animals and drivers to better capture the relationship among drivers' and animals' attributes, roadway and environmental factors, and AVCs. Findings of this study show that speed limit, rural versus urban, and presence of white-tailed deer habitat have an increasing effect on AVC risk, whereas male animals, high truck percentage, and large number of lanes put a decreasing effect on AVC probability.
+
+Published by Elsevier Ltd.
+
+### **1. Introduction**
+
+Over the past decade, the number of Animal–vehicle collisions (AVCs) has been rising with the continued increase of motor vehicle traffic ([Curtis](#page-6-0) [and](#page-6-0) [Hedlund,](#page-6-0) [2005\).](#page-6-0) [Romin](#page-7-0) [and](#page-7-0) [Bissonette](#page-7-0) [\(1996\)](#page-7-0) reported that at least 1.5 million deer-vehicle collisions occurred annually nationwide. In Washington State, approximately 3000 collisions occur annually with deer and elk on state highways ([Wagner](#page-7-0) [and](#page-7-0) [Carey,](#page-7-0) [2006\).](#page-7-0) These increasing AVCs have caused significant damage to human safety, property, and wildlife in the past decades. These collisions caused about 200 human fatalities, and 20,000 human injuries annually in the United States ([Huijser](#page-7-0) et [al.,](#page-7-0) [2007\).](#page-7-0) Property damage related with AVCs exceeds one billion dollars each year. In most AVCs, the animal dies immediately or soon after [\(Allen](#page-6-0) [and](#page-6-0) [McCullough,](#page-6-0) [1976\).](#page-6-0) AVCs may also affect the population level of some precious species (e.g., [Van](#page-7-0) [der](#page-7-0) [Zee](#page-7-0) et [al.,](#page-7-0) [1992;](#page-7-0) [Huijser](#page-7-0) [and](#page-7-0) [Bergers,](#page-7-0) [2000\)](#page-7-0) or even lead to a serious decrease in the probability of population survival [\(Proctor,](#page-7-0) [2003\).](#page-7-0) Thus, a bet-
+
+To identify the contributing factors in general traffic accidents, a number of statistical modeling techniques have been developed based on the diverse characteristics of collisions in different circumstances. Poisson regression (e.g., [Jovanis](#page-7-0) [and](#page-7-0) [Chang,](#page-7-0) [1986;](#page-7-0) [Miaou](#page-7-0) [and](#page-7-0) [Lum,](#page-7-0) [1993;](#page-7-0) [Miaou,](#page-7-0) [1994\),](#page-7-0) negative binomial (NB) regression (or Poisson-gamma regression) ([Miaou,](#page-7-0) [1994;](#page-7-0) [Maher](#page-7-0) [and](#page-7-0) [Summersgill,](#page-7-0) [1996;](#page-7-0) [Milton](#page-7-0) [and](#page-7-0) [Mannering,](#page-7-0) [1998;](#page-7-0) [Chin](#page-7-0) [and](#page-7-0) [Quddus,](#page-7-0) [2003;](#page-7-0) [Wang](#page-7-0) et [al.,](#page-7-0) [2003;](#page-7-0) [Wang](#page-7-0) [and](#page-7-0) [Nihan,](#page-7-0) [2004;](#page-7-0) [El-](#page-7-0)Basyouny [and](#page-7-0) [Sayed,](#page-7-0) [2006;](#page-7-0) [Donnell](#page-7-0) [and](#page-7-0) [Mason,](#page-7-0) [2006;](#page-7-0) [Kim](#page-7-0) et [al.,](#page-7-0) [2007;](#page-7-0) [Malyshkina](#page-7-0) [and](#page-7-0) [Mannering,](#page-7-0) [2010;](#page-7-0) [Daniels](#page-7-0) et [al.,](#page-7-0) [2010\),](#page-7-0) and Poisson-lognormal models [\(Miaou](#page-7-0) et [al.,](#page-7-0) [2005;](#page-7-0) [Lord](#page-7-0) [and](#page-7-0) [Miranda-](#page-7-0)Moreno, [2008\)](#page-7-0) have been commonly used in accident modeling. Recently, some other innovative accident models, including finitemixture/Markov switching models, random parameter models, Bayesian neural networks, neural networks, and support vector machines, have been used in the collision analysis studies. A detail review ofthese recent accident models was elaborated in ([Lord](#page-7-0) [and](#page-7-0) [Mannering,](#page-7-0) [2010\).](#page-7-0)
+
+These regression models have been used for modeling vehicle–vehicle collisions and are able to provide insight into the contributing factors of accidents. For most AVC research, Poisson regression and negative binomial regression models are used for modeling deer-vehicle collisions to investigate the factors that influence the frequency and severity of deer-vehicle crashes
+
+ter understanding of the factors contributing to AVCs is critical for indentifying the high risk locations and prioritizing potential countermeasures, such as signs, fences, wildlife underpasses and overpasses, roadside reflectors, whistles, and diversionary feeding areas [\(Danielson](#page-6-0) [and](#page-6-0) [Hubbard,](#page-6-0) [1998\).](#page-6-0)
+
+<sup>∗</sup> Corresponding author. Tel.: +1 206 616 2696; fax: +1 206 543 5965. E-mail addresses: [laoy@u.washington.edu](mailto:laoy@u.washington.edu) (Y. Lao), [guohui@mail.utexas.edu,](mailto:guohui@mail.utexas.edu) [yaojan@slu.edu](mailto:yaojan@slu.edu) (G. Zhang), [yaojan@u.washington.edu,](mailto:yaojan@u.washington.edu) [guohui08@gmail.com](mailto:guohui08@gmail.com)
+
+<sup>(</sup>Y.-J. Wu), [yinhai@u.washington.edu](mailto:yinhai@u.washington.edu) (Y. Wang). <sup>1</sup> Address: Department of Civil Engineering, University of New Mexico, Albuquerque, NM 87106, USA.
+
+<sup>2</sup> Address: Department of Civil Engineering, Parks College of Engineering,Aviation and Technology, 3450 Lindell Boulevard, McDonnell Douglas Hall Room 2051, Saint Louis University, St. Louis, MO 63103, USA. Tel.: +1 314 977 8249; fax: +1 314 977 8 388.
+
+<span id="page-1-0"></span>([Gkritza](#page-7-0) et [al.,](#page-7-0) [2010\).](#page-7-0) However, most previous accident modeling studies did not reflect human factors, despite their critical roles in the crash mechanism ([Wang](#page-7-0) et [al.,](#page-7-0) [2003\).](#page-7-0) Even though [Wang](#page-7-0) [\(1998\)](#page-7-0) implemented a microscopic probability (MP) model to include drivers' responses as part of the collision model, the MP model is only designed to model vehicle collisions. In order to further investigate animal-related factors, such as animal population distribution and vehicle–animal interactions forAVCs, we are motivated to propose a vehicle–animal interaction-based probability (VAIP) model to characterize the responses of drivers and animals and the unique impacts of animal habitats on collisions to better understand AVCs and their associated contributing factors.
+
+The remainder of this paper is organized as follows. The MP and VAIP models are introduced in Section 2 before describing the details of the test data in Section [3.](#page-2-0) Section [4](#page-3-0) shows the model estimation results, and model interpretation and discussion are detailed in Section [5.](#page-3-0) The model spatial and temporal transferability tests are described in Section [6.](#page-5-0) In the end, this research effort will be concluded with findings and recommendations.
+
+## **2. Methodology**
+
+## 2.1. Extended application of the MP model
+
+### 2.1.1. MP model structure
+
+This study is based on the MP model proposed by [Wang](#page-7-0) [\(1998\).](#page-7-0) An overview of the MP model and its association with the AVC model are summarized in this section. The MP model describes the relationship between the presence of a leading vehicle and the ineffective response of a driver in the following vehicle. An important advantage of this approach is its capability of considering the mechanism of accident occurrence in risk modeling. This approach has been successfully applied in many subsequent studies of accident risks (see for example [Siddique,](#page-7-0) [2000,](#page-7-0) [Wang](#page-7-0) et [al.,](#page-7-0) [2003](#page-7-0) and [Kim](#page-7-0) et [al.,](#page-7-0) [2007\)](#page-7-0) and achieved favorable results. Although animals' behavior exhibit different patterns than drivers', we will apply Wang' MP model to formulate the VAC before the new VAIP model is developed and investigated. Their performance will be examined and analyzed as follows.
+
+In the MP model, the probability for a randomly selected vehicle tohave anaccident ona certainroadway sectionis theprobability of the driver's ineffective response Pvf to the obstacles conditioned on the probability of an obstacle presenting on the road Po. In other words, the probability for a driver to have an AVC(PAVC) can be expressed as the product of Po and Pvf [\(Wang,](#page-7-0) [1998\):](#page-7-0)
+
+$$P_{AVC} = P_o \cdot P_{vf} \tag{1}$$
+
+However, Po and Pvf are not directly observable, and require further estimation.
+
+### 2.1.2. Po formulation
+
+An animal becomes an obstacle for vehicles if its highwaycrossing movement interrupts the smooth movement of vehicles. When an animal highway-crossing movement occurs within a certain period, the animal may become an obstacle to the arriving vehicle. This period is called "effective time." As the arrival of an obstacle is discrete, nonnegative, and random, it is assumed to be a Poisson arrival process. In such a process, intervals between arrivals are independent and follow the same exponential distribution ([Pitman,](#page-7-0) [1993\).](#page-7-0) Assuming the arrival rate is <sup>j</sup> in the time interval j, the average number of Poisson disturbances during the entire time period is jjtj then probability of encountering an obstacle animal, Po, is equivalent to the probability that at least one disturbance occurs within the effective period. Therefore, Po can be formulated as:
+
+$$P_0 = 1 - e^{-\sum_j \eta_j t_j} \tag{2}$$
+
+In Eq. (2), jjtj should always be positive and dependent on a set of variables. Thus, an exponential link function can be employed to reflect the effects of the explanatory factors as shown as ([Wang,](#page-7-0) [1998;](#page-7-0) [Kim](#page-7-0) et [al.,](#page-7-0) [2007\):](#page-7-0)
+
+$$\sum_{i} \eta_{dj} t_{dj} = e^{\beta_0 \mathbf{x_0}} \tag{3}$$
+
+Po then becomes:
+
+$$P_o = 1 - e^{-e^{\beta_0 \mathbf{x_0}}} \tag{4}$$
+
+where **ˇ<sup>o</sup>** and **xo** are vectors of unknown parameters and explanatory variables of disturbance frequency, respectively. **ˇ<sup>o</sup>** does not change with location, while **xo** does. Animal habitat integrity, habitat size, and animal population are very likely contribution variables to **xo**.
+
+# 2.1.3. Pvf formulations
+
+It is assumed that a driver cannot avoid a collision if their Necessary Perception Reaction Time (NPRT) is longer than the Available Perception Reaction Time (APRT). The APRT refers to the time a driver has for completing their perception and response under a given condition. The NPRT is the ability-oriented minimum required perception reaction time and typically varies from person to person. Both the APRT and the NPRT are random variables and are assumed to follow normal distributions. Since a normal distribution does not have a closed form for cumulative probability calculation, the Weibull distribution is used instead. The NPRT is assumed to follow the Weibull (˛, ) distribution, and the APRT is assumed to follow the Weibull (˛, ) distribution. Here, and are the scale parameters. The Weibull distribution shape parameter ˛ is chosen to be 3.25 in this study because it has been empirically verified that when ˛= 3.25, the Weibull distribution is a very good approximation to the normal distribution ([Kao,](#page-7-0) [1960;](#page-7-0) [Plait,](#page-7-0) [1962\).](#page-7-0) Using the assumed distributions for the APRT and the NPRT, Pvf can be calculated as:
+
+$$P_{vf} = \int_0^\infty \int_{t_{av}}^\infty f(\lambda, t) f(\gamma, t_{av}) dt dt_{av}$$
+
+$$= \int_0^\infty e^{-\lambda t_{av}^\alpha} \alpha \gamma t_{av}^{\alpha - 1} e^{-\gamma t_{av}^\alpha} dt_{av} = \frac{1}{1 + \lambda/\gamma}$$
+(5)
+
+where tav is the variable used to represent the APRT. Eq. (5) shows that Pvf is only dependent on /, and has no relationship to ˛. Since the parameters and are positive variables, / can be related to various factors by using an exponential link function as shown in Eq. (5) [\(Wang,](#page-7-0) [1998;](#page-7-0) [Kim](#page-7-0) et [al.,](#page-7-0) [2007\).](#page-7-0) Correspondingly, Pvf can be written as.
+
+$$\frac{\lambda}{\gamma} = e^{-\beta_{vh}x_{vh}} \tag{6}$$
+
+$$P_{vf} = \frac{1}{1 + e^{-\beta_{vh}\kappa_{vh}}} \tag{7}$$
+
+where **ˇvh** and **xvh** are vectors of unknown parameters and explanatory variables, respectively, related to Pvf. Variables affecting drivers' task load and action complexity need to be included in **xvh**.
+
+### 2.1.4. Integrated MP model
+
+The application of [Wang's](#page-7-0) [\(1998\)](#page-7-0) MP model in AVC only has the terms of the probability of an animal being present on the road (Po) and the probability of an ineffective response by the driver (Pvf). Substituting Eqs. (4) and (7) into Eq. (1), the probability of an individual vehicle being involved in an AVC is formulated as:
+
+$$P_{AVC} = P_o P_{vf} = \frac{1 - e^{-e} \beta_o x_o}{1 + e^{-\beta_o y} x_o}$$
+
+$$\tag{8}$$
+
+# <span id="page-2-0"></span>2.2. Vehicle–animal interaction-based probability (VAIP) model
+
+As discussed in Section [1,](#page-0-0) the AVC process is difficult to accurately model and interpret because many subjective and objective factors, such as human and animal factors, cannot be properly reflected in the model. It is needed to have a modeling process that considers two significant AVC contributors: insufficient responses from drivers, such as a lack of deceleration, swerving and late responses from animals, such as freezing, running in the wrong direction. These two contributors interact witheachother so that an AVC may be caused by either one or both. Since the MP model was originally developed for vehicle-to-vehicle collisions,the responses of animals were not considered in the modeling structure. An AVC could be avoided if drivers can react early and quickly to the obstacle or if the animals can notice to oncoming vehicles in a timely manner. Therefore, a third item addressing animal's response is desired in the MP model to enhance model rationality and applicability on AVCs. Thus, we propose a vehicle–animal interaction-based probability (VAIP) model as an extension of the MP model.
+
+# 2.2.1. VAIP model structure
+
+In this study, we consider that the occurrence of an AVC is conditioned on the presence of an animal in the roadway, ineffective response of the arriving vehicle driver, and the animal's failure to escape. Therefore, the vehicle–animal interaction probability can be formulated as
+
+$$P_{AVC} = P_o \cdot P_{vf} \cdot P_{af} \tag{9}$$
+
+where Po is the probability of a hazardous crossing presence of an animal when vehicles travel along roadways, Pvf is the probability of ineffective response of the driver, and Paf is the probability of the animal failing to escape being hit. Thus the probability for a randomly selected vehicle to have an AVC on a certain roadway section is the product of Po, Pvf, and Paf: In this VAIP model, Po and Pvf are defined according to the MP model and Paf describes animals' responses in a collision. In this study we simplify animals' responses by following a similar model structure of Pvf by comparing the animals' necessary perception reaction time with the available perception reaction time. Here, the available preception reaction time refers to the time an animal has for noticing and escaping from the approaching vehicle. The necessary perception reaction time is the minimum required perception reaction time depending on factors such as animal species and characteristics. Both are random variables and are assumed to follow normal distributions. By following the same modeling process with Pvf in Section [2.1.3](#page-1-0) "Pvf Formulation", Paf can be written as:
+
+$$P_{af} = \frac{1}{1 + e^{-\beta_{ah} \kappa_{ah}}} \tag{10}$$
+
+where **ˇah** and **xah** are vectors of unknown parameters and explanatory variables, respectively, related to Paf. Variables affecting animal' action need to be included in **xah**.
+
+### 2.2.2. Integrated VAIP model
+
+By substituting Eqs. [\(4\),](#page-1-0) [\(7\),](#page-1-0) and (10) into Eq. (9), the integrated VAIP risk model for each roadway section can be rewritten as:
+
+$$P_{AVC} = P_o P_{af} P_{vf} = \frac{1 - e^{-e^{\beta_o \mathbf{x}_o}}}{(1 + e^{-\beta_o \mathbf{f} \cdot \mathbf{x}_o} \mathbf{f})(1 + e^{-\beta_o \mathbf{f} \cdot \mathbf{x}_o} \mathbf{f})}$$
+(11)
+
+where Po is the probability of an animal being present on the road, Paf is the failure probability by the animal to escape from being hit, and Pvf is the probability of an ineffective response by the driver. One can see that the model contains not only road environment related factors, but also factors related to human and animal behaviors. The inclusion of human and animal factors is one of the major distinctions between the proposed model and most existing AVC models. Note that if the animals' reactions are dispensable as stationary objects, the probability, Paf = 1, and the VAIP model reduces to the MP model.
+
+# 2.3. PAVC formulation
+
+It is assumed that vehicles within a traffic flow have an average AVC risk, PAVCi. Since AVCs are very rare, PAVCi should be very small while traffic volume fi is very large for the given span of time. Thus, the Poisson distribution is good approximations to the binomial distribution ([Pitman,](#page-7-0) [1993\):](#page-7-0)
+
+$$P(n_i) = \frac{m_i n_i \cdot e^{-m_i}}{n_i !} \tag{12}$$
+
+with Poisson distribution parameter:
+
+$$m_i = E(n_i) = f_i \cdot P_{AVCi} \tag{13}$$
+
+where fi is the annual traffic volume that can be calculated from the annual average daily traffic (AADT) for roadway section i, and ni is the number of AVC occurred within fi.
+
+The mean and variance in a Poisson distribution need to be the same. However, in most cases, accident data are over-dispersed. An easy way to overcome this difficulty is to add an independently distributed error term, εi, to the log transformation of Eq. (13). That is:
+
+$$\ln m_i = \ln(f_i P_{AVCi}) + \varepsilon_i$$
+(14)
+
+We assume exp(εi) is a Gamma distributed variable with mean 1 and variance ı. Substituting Eq. (14) into Eq. (12) yields:
+
+$$P(n_i|\varepsilon_i) = \frac{e^{(-f_i P_{AVCI} \exp(\varepsilon_i)) \cdot (f_i P_{AVCI} \exp(\varepsilon_i))^{n_i}}}{n_i!}$$
+(15)
+
+Integrating ε<sup>i</sup> out of Eq. (15), we can directly derive a negative binomial distribution model as the following:
+
+$$P(n_i) = \frac{\Gamma(n_i + \theta)}{\Gamma(n_i + 1)\Gamma(\theta)} \left(\frac{\theta}{f_i \cdot P_{AVCI} + \theta}\right)^{\theta} \left(\frac{f_i P_{AVCI}}{f_i \cdot P_{AVCI} + \theta}\right)^{n_i}$$
+(16)
+
+where = 1/ı. The expectation of this negative binomial distribution equals to the expectation of the Poisson distribution shown in Eq. (14). The variance is now:
+
+$$V(n_{ik}) = E(n_{ik})[1 + \delta E(n_{ik})]$$
+(17)
+
+Note that the Poisson regression model is regarded as a limiting NB regression model when ı approaches zero [\(Washington](#page-7-0) et [al.,](#page-7-0) [2003\).](#page-7-0)
+
+### **3. Data description**
+
+Three major data sources are used in this study:
+
+• Carcass removal data by Washington State Department of Transportation (WSDOT) stores the information of animal carcass being collected. The information includes location (by milepost), date, weather, animal type, sex, age, etc. Carcass removal data have been commonly used in AVC research [\(Reilley](#page-7-0) [and](#page-7-0) [Green,](#page-7-0) [1974;](#page-7-0) [Allen](#page-7-0) [and](#page-7-0) [McCullough,](#page-7-0) [1976;](#page-7-0) [Knapp](#page-7-0) [and](#page-7-0) [Yi,](#page-7-0) [2004;](#page-7-0) [Lao](#page-7-0) et [al.,](#page-7-0) [submitted](#page-7-0) [for](#page-7-0) [publication\).](#page-7-0) This study used two years (2005–2006) of carcass removal data from ten highway routes (US 2, SR 8, US 12, SR 20, I-90, US 97, US 101, US 395, SR 525 and
+
+<span id="page-3-0"></span>**Table 1** Description of explanatory variables in the models.
+
+|     | Variable                                                      | Min  | Max   | Mean  | S.D.  |
+|-----|---------------------------------------------------------------|------|-------|-------|-------|
+| Ya  | Number of carcasses per<br>segmentb                           | 0    | 16    | 0.095 | 0.564 |
+| z1  | Annual average daily traffic<br>(in thousands)                | 0.31 | 148.8 | 15.11 | 21.07 |
+| z2  | Restrictive access control<br>(Yes: 1; No: 0)                 |      |       | 0.24  |       |
+| z3  | Speed level (>50 mph: 1;<br>otherwise: 0)                     |      |       | 0.68  |       |
+| z4  | Truck percentage level<br>(>5%: 1; otherwise: 0)              |      |       | 0.78  |       |
+| z5  | Median width (>6 feet: 1;<br>others: 0)                       |      |       | 0.33  |       |
+| z6  | Total number of lanes (in<br>both directions)                 | 2    | 9     | 2.96  | 1.32  |
+| z7  | Roadway length (mile)                                         | 0.01 | 6.99  | 0.22  | 0.4   |
+| z8  | Terrain type (Rolling: 1;<br>Otherwise: 0)                    |      |       | 0.72  |       |
+| z9  | Terrain type<br>(Mountainous: 1;<br>Otherwise: 0)             |      |       | 0.095 |       |
+| z10 | Lane width (feet)                                             | 10   | 20    | 12.5  | 1.88  |
+| z11 | Left shoulder width (feet)                                    | 0    | 18    | 2.44  | 2.04  |
+| z12 | Right shoulder width (feet)                                   | 0    | 20    | 4.03  | 3.52  |
+| z13 | Rural area (Rural: 1;<br>Urban: 0)                            |      |       | 0.76  |       |
+| z14 | White-tailed deer habitat<br>(Yes: 1; No: 0)                  |      |       | 0.31  |       |
+| z15 | Mule deer habitat (Yes: 1;<br>No: 0)                          |      |       | 0.51  |       |
+| z16 | Elk habitat (Yes: 1; No: 0)                                   |      |       | 0.31  |       |
+| z17 | Sex of animal (Male: 1;<br>Female: 0)                         |      |       | 0.328 |       |
+| z18 | Horizontal curve (Curve<br>degree > 3: 1; otherwise: 0)       |      |       | 0.16  |       |
+| z19 | Vertical curve (Grade<br>percentage > 3%: 1;<br>otherwise: 0) |      |       | 0.22  |       |
+
+aSpecific to carcass removal data only; bdependent variable, number of carcasses within two years (2005–2006); Min: Minimum; Max: Maximum; S.D.: standard deviation.
+
+SR 970) as the study routes following the recommendation from WSDOT experts.
+
+- Deer distribution data by Washington Department of Fish Wildlife (WDFW) is in the form of GIS-based maps for mule deer, white-tailed deer, and elks.
+- Roadlog data by Highway Safety Information System (HSIS) provides geometric information for the roadway, such as median width, number of lanes and shoulder width.
+
+Table 1 lists all explanatory variables used in the modeling process. Most of the quantitative and dummy variables were directly selected from the combined dataset. Several variables were created based on the observed data. For example, the variable "Speed Level" was created based on posted speed limits. This variable is a dummy variable. The variable is set to 1 when the posted speed limit is greater than 50 mph and 0 otherwise. This is because a dramatic increase in AVCs was found when the speed limit > 50 mph. Other examples, such as variables z14, z15, and z16, were created for representing habitats of different types of animal.
+
+The minimum, maximum, mean, and standard deviation (S.D.) of each variable are shown in Table 1. One can find thatthe reported collision data is over-dispersed as indicated by the variance being higher than the mean.
+
+# **4. Model estimation**
+
+For the purpose of comparison, both a Poisson regression model (Eq. [\(12\),](#page-2-0) when ı approaches zero in Eq. [\(16\)\)](#page-2-0) and a negative binomial regression model (Eq. [\(16\)\)](#page-2-0) were produced for the MP and VAIP model estimation using the carcass removal data. An open source statistical analysis package, R ([http://www.r-project.org/,](#page-7-0) [2010\),](#page-7-0) was used for model estimation in this research.
+
+In order to evaluate the explanatory and predictive power of the model, two measures of goodness-of-fit (GOF) are adopted here for model comparisons: Adjusted <sup>2</sup> ([Ben-Akiva](#page-6-0) [and](#page-6-0) [Lerman,](#page-6-0) [1985\),](#page-6-0) and Akaike's Information Criterion (AIC) ([Akaike,](#page-6-0) [1974\).](#page-6-0) Adjusted <sup>2</sup> (rho-squared) is the log-likelihood ratio index, and is used to evaluate model's GOF for random, discrete, and sporadic count data [\(Ben-Akiva](#page-6-0) [and](#page-6-0) [Lerman,](#page-6-0) [1985;](#page-6-0) [Chin](#page-6-0) [and](#page-6-0) [Quddus,](#page-6-0) [2003;Washington](#page-6-0) et [al.,](#page-6-0) [2003\).](#page-6-0) The index is formulated as:
+
+$$\rho^2 = 1 - \frac{\ln L(\hat{\beta}) - K}{\ln L(0)}$$
+ (18)
+
+where L(-ˆ ) is the maximum likelihood estimation of the compared model, L(0) is the initial maximum likelihood estimation of the same model with only the constant term, and K is the number of parameters estimated in the model.
+
+AIC is another measure of GOF for a statistical model [\(Akaike,](#page-6-0) [1974\).](#page-6-0) AIC is often used for model selection. The model with the lowest AIC is considered the best model. In general, AIC is formulated as:
+
+$$AIC = 2K - 2\ln(L) \tag{19}$$
+
+where L is the maximum likelihood estimation of the model.
+
+The variables were firstly assigned based on preliminary analysis, then the assignment of variables was adjusted based on their significance. The final model is selected based on the AIC value. [Table](#page-4-0) 2 shows the coefficients of explanatory variables and statisticaltest results ofthe convergence MP model, estimated by negative binomial regression. Variables significantly associated with the probability of a hazardous crossing of an animal, Po and the probability of the driver's ineffective response, Pvf, are shown as the explanatory variables in the models.
+
+Similarly, the coefficients of the explanatory variables and their significance are shown for the VAIP model in [Table](#page-4-0) 3. The traditional NB regression model with the standard structure was also estimated in [Table](#page-4-0) 3. Compared with the traditional NB regression model, the VAIP model has made (19,484 − 17,177)/19,484 = 12% improvement on the AIC value. In addition to the probabilities, Po and Pvf, the probability of the animal's failure to escape from being hit, Paf, is explicitly formulated. One variable, the sex of animal, is identified significant by Paf. Additionally, to fully understand the marginal effects of each independent variable, their elasticity values are calculated as [\(Shankar](#page-7-0) et [al.,](#page-7-0) [1995;](#page-7-0) [Abdel-Aty](#page-7-0) [and](#page-7-0) [Radwan,](#page-7-0) [2000;](#page-7-0) [Washington](#page-7-0) et [al.,](#page-7-0) [2003\):](#page-7-0)
+
+$$E_{x_{ik}}^{\lambda_i} = \frac{\partial \lambda_i}{\partial x_{ik}} \frac{x_{ik}}{\lambda_i} = \beta_k x_{ik} \tag{20}$$
+
+where <sup>i</sup> is the expected number of accidents for roadway segment i, xik is the k-th variable in the vector of explanatory variables for roadway segmenti, and ˇ<sup>k</sup> is the corresponding coefficient ofthe kth variable. The elasticity in Eq. (20) applies when the explanatory variable xik is continuous. In case of an indicator variable, pseudoelasticity is estimated as an approximate elasticity of this variable [\(Washington](#page-7-0) et [al.,](#page-7-0) [2003\):](#page-7-0)
+
+$$E_{X_{ik}}^{\lambda_i} = \frac{\exp(\beta_k) - 1}{\exp(\beta_k)} \tag{21}$$
+
+### **5. Model interpretation**
+
+The estimated coefficients, their t-values, and GOF for the MP model and the VAIP model are shown in [Tables](#page-4-0) 2 and 3 respectively. Comparing the estimation results from [Tables](#page-4-0) 2 and 3, one can find
+
+<span id="page-4-0"></span>**Table 2** Description of explanatory variables in the MP model.
+
+| Explanatory variables                                                           | Coeffa  | st. errb | t-value |
+|---------------------------------------------------------------------------------|---------|----------|---------|
+| Variables affecting the probability of a hazardous crossing of an animal (Po)   |         |          |         |
+| Constant                                                                        | −16.359 | 0.268    | −60.945 |
+| Median width (>6 feet: 1; others: 0)                                            | −1.016  | 0.137    | −7.444  |
+| Total number of lanes                                                           | −0.290  | 0.057    | −5.119  |
+| Terrain type (Rolling: 1; Otherwise: 0)                                         | 0.248   | 0.070    | 3.525   |
+| Rural area (Rural: 1; Urban: 0)                                                 | 1.890   | 0.133    | 14.197  |
+| White-tailed deer habitat (Yes: 1; No: 0)                                       | 1.516   | 0.056    | 26.963  |
+| Animal sex (Male: 1; Female: 0)                                                 | −0.720  | 0.056    | −12.876 |
+| Variables affecting the probability of ineffective response of the driver (Pvf) |         |          |         |
+| Speed level (>50 mph: 1; otherwise: 0)                                          | 1.954   | 0.277    | 7.042   |
+| Truck percentage Level (>5%: 1; otherwise: 0)                                   | −1.219  | 0.183    | −6.646  |
+| Model evaluation                                                                |         |          |         |
+| AIC at base model#                                                              |         |          | 26,861  |
+| AIC at convergence with Poisson regression                                      |         |          | 19,653  |
+| AIC at convergence with NB regression (ı = 1.66)                                |         |          | 17,177  |
+| 2                                                                               |         |          | 0.36    |
+
+aCoefficients in the model; bstandard error; <sup>2</sup> was calculated by comparing the log-likelihood with the base model; base model#; ı approaches zero and ˇ = 0.
+
+that the GOF of these two models are almost the same: both the adjusted <sup>2</sup> values are 0.36, and the AIC values are undistinguished. Based on the AIC values within Table 2 or 3, the negative binomial regression outperformed the corresponding Poisson regression. The estimate results show that the ı value is 1.66 in both MP and VAIP models and their p value is 0.00, which verifies that ı is significantly greater than 0, and the carcass removal data are overdispersed. In this case,the model estimated with Poisson regression should not be used because it requires the mean and variance ofthe carcass removal data to be the same. Model estimated with the NB regression is a better choice for this study.
+
+For both the MP and VAIP models estimated by the NB regression, a total of eight variables are identified as significant, including the number oflanes,terrain type, rural area, white-tailed deer habitat, median width, sex of animals, "truck percentage level", and "speed level". Among them, two variables, "truck percentage level" and "speed level", have significant impacts on Pvf, the probability of drivers' ineffective response, and the other six variables play significant roles in determining Po, the probability of encountering a disturbance animal in the MP model. However, in the VAIP model, one variable, sex of animal, is explicitly identified as significant by Paf, the probability of the animal's failure to escape from being hit, instead of Po in the MP model. Although both models show the similar GOF, further analyses show that the VAIP model demonstrates more capability of interpreting the AVC process and the impacts of explanatory variables. Therefore, the detailed explanations and discussions regarding the VAIP model follows.
+
+# 5.1. Interpretation of estimation results for Po
+
+The five significant variables affecting the probability of an animal's presence reflects both roadway geometric characteristics and animal distribution features as shown in Table 3. Compared with the level terrain type, rolling terrain tends to have an increasing effect on the possibility of the presence of an animal on the road Po (Coef. = 0.248, t = 3.525, E = 0.220). This may be because rolling terrain has a higher animal population than that of level terrain. The elasticity value here shows that an incremental change of 0.22% to the AVC accident risk is caused by the changes from level terrain to rolling terrain. Similarly, compared to the highways in urban areas, those in rural areas also tend to have a higher Po (Coef. = 1.890, t = 14.197, E = 0.849). This may also be due to the higher animal population and activity levels in rural areas. The elasticity value here shows that an incremental change of 0.849% to the AVC accident risk is caused by the changes from urban area to rural area.
+
+**Table 3** Description of explanatory variables in the VAIP model.
+
+| Explanatory variables                                                                    | Coeffa  | st. errb | t-value | Ec     |
+|------------------------------------------------------------------------------------------|---------|----------|---------|--------|
+| Variables affecting the probability of a hazardous crossing of an animal (Po)            |         |          |         |        |
+| Constant                                                                                 | −15.666 | 0.268    | −58.363 |        |
+| Median width (>6 feet: 1; others: 0)                                                     | −1.016  | 0.137    | −7.444  | −1.762 |
+| Number of lanes                                                                          | −0.290  | 0.057    | −5.119  | −0.336 |
+| Terrain type (Rolling: 1; otherwise: 0)                                                  | 0.248   | 0.070    | 3.525   | 0.220  |
+| Rural area (Rural: 1; Urban: 0)                                                          | 1.890   | 0.133    | 14.197  | 0.849  |
+| White-tailed deer habitat (Yes: 1; No: 0)                                                | 1.516   | 0.056    | 26.963  | 0.780  |
+| Variables affecting the probability of the animal failure to escape from being hit (Paf) |         |          |         |        |
+| Animal sex (Male: 1; Female: 0)                                                          | −1.134  | 0.074    | −15.347 | −2.108 |
+| Variables affecting the probability of ineffective response of the driver (Pvf)          |         |          |         |        |
+| Speed level (>50 mph: 1; otherwise: 0)                                                   | 1.954   | 0.277    | 7.042   | 0.858  |
+| Truck percentage level (>5%: 1; otherwise: 0)                                            | −1.219  | 0.183    | −6.646  | −2.384 |
+| Model evaluation                                                                         |         |          |         |        |
+| AIC at base modele                                                                       |         |          | 26,861  |        |
+| AIC at convergence with Poisson regression                                               |         |          | 19,653  |        |
+| AIC at convergence with standard NB regressiond                                          |         |          | 19,484  |        |
+| AIC at convergence with NB regression (ı = 1.66)                                         |         |          | 17,177  |        |
+| 2                                                                                        |         |          | 0.36    |        |
+
+aCoefficients in the model; bstandard error; caverage elasticity value; ı is referred to as the overdispersion parameter; <sup>2</sup> was calculated by comparing the log-likelihood with the base model; base modele; ı approaches zero and ˇ = 0; standard NB regressiond; the traditional NB regression model with the standard structure.
+
+<span id="page-5-0"></span>Among all the variables, white-tailed deer habitat was found to be the most significant explanatory variable affecting AVCs (Coef. = 1.516, t = 26.963, E = 0.780). This may be due to the higher animal population in the white-tailed deer habitat, contributing to the increased probability of animal crossing Po. If a highway section segments a white-tailed deer habitat area, a driver using this section will have a higher probability of encountering an animal. Compared with white-tailed deer habitats, the variable of elk habitats is not significant at 95% significance level. This can be explained by the fact that the total number of collisions with elk only contributes a small part of the whole AVC records for the study period. Mule deer habitat also was not significant in the model. The reason for this may be because the mule deer habitat distribution is relatively uniformly in Washington State and covers a large portion of the study routes. The elasticity value for the white-tailed deer habitat indicates an incremental change of 0.780% on the AVC accident risk caused by the changes from other areas to white-tailed deer habitat areas. The finding is consistent with another AVC study ([Lao](#page-7-0) et [al.,](#page-7-0) [2011\).](#page-7-0)
+
+The number of lanes is the significant factor having a negative effect on the presence of animals, Po (Coef. = −0.290, t = −5.119, E = −0.336). With an increase in the total number of lanes,the probability of animals present on the road tends to be lower. This is understandable because roadway sections with more travel lanes are typically wider, which might increase the crossing difficulty for animals. Therefore, animals would be reluctant to cross a wider segment and thus the Po is lower. The elasticity value here shows that a 1% increase in the number of lane decreases the AVC accident risk by 0.336%.
+
+The variable of median width is related to roadway geometric design elements.A median width of greater than 6 feet was found to have a significant decreasing effect on Po (Coef. = −1.016,t = −7.444, E = −1.762). This variable is similar to the number of lanes in that a wider median will increase the crossing hesitation for animals, and hence reduce the likelihood ofAVCs. The elasticity valuehere shows a decrement change of 1.762% on the AVC accident risk caused by the changes from median width less than 6 feet to median width more than 6 feet.
+
+## 5.2. Interpretation of Pvf
+
+Among the factors affecting the probability of the driver's ineffective response, Pvf, two explanatory variables, "Speedlevel" and "Truck percentage level", were found to be significant. The speed limit level has a positive estimated coefficient (Coef. = 1.954, t = 7.042, E = 0.858). This implies that when a highway segment had a speed limit greater than 50 mph, the probability of a driver's ineffective response wouldincrease.Avehicle running at ahigher speed requires a longer stopping distance. Hence, when an animal is perceived,the reaction time for a faster vehicle is shorter. This explains why speed limit has an increasing effect on Pvf. This finding is consistent with many previous AVC related studies, e.g. [Rolley](#page-7-0) [and](#page-7-0) [Lehman](#page-7-0) [\(1992\)](#page-7-0) and [Allen](#page-6-0) [and](#page-6-0) [McCullough](#page-6-0) [\(1976\).](#page-6-0) The elasticity value here indicates an incremental change of 0.858% to the AVC accident risk is caused by the changes from the highways of speed limit lower than 50mph to the highways of speed limit higher than 50mph.
+
+The truck percentage level was found to have an increasing impact on the probability of driver's effective response (decreased failure to avoid collision, Coef. = −1.219, t = −6.646, E = −2.384). This is presumably because truck drivers drive at relatively lower speeds. High-profile trucks have taller profiles, providing the drivers with longer sight distances and most truck drivers are professionally trained and well experienced. This resultis supported by the motor vehicle accident research [\(Milton](#page-7-0) [and](#page-7-0) [Mannering,](#page-7-0) [1998\)](#page-7-0) in which the increase in the percentage of trucks may decrease the accident probabilities. The elasticity value here indicates a decrement change of 2.384% to the AVC accident risk is caused by the changes from the areas with lower truck percentage to the areas with higher truck percentage. Additionally, it could be because truck-animal collisions were underreported due to the possibly smaller risks of property damage and people injury. Further studies are desirable to consolidate this finding.
+
+# 5.3. Interpretation of Paf
+
+Turning to the factors affecting the probability of animal's response, Paf, one variable, sex of animal, were found to affect Paf significantly. Compared with female animals, male animals tend to have lower collision risk (Coef. = −1.025, t = −12.877, E = −1.787). This may be because male animals require less response time than female animals. However, further study is still needed for this argument. The elasticity value here indicates a decrement change of 1.787% on the AVC accident risk caused by the change from female animals to male animals. The modeling capability of the MP model is extended by the item, Paf to explicitly explain unique animal response behavior with different attributes. For instance, animal species and gender may play significant roles in determining their reactions when a vehicle is approaching. Some animal species may detect the approaching vehicles much earlier than the others. Male animals may respond and run faster than females. The proposed VAIP model is capable of capturing specific animal responses in an AVC and enhances the MP model's ability in data interpretation. Due to data constraints, animal species data are not available for model calibration and estimation. [Table](#page-4-0) 3 shows that sex of animal is considered significant in describing animal response behavior.
+
+### 5.4. Findings and practical implications
+
+The model estimation results indicate the proposed VAIP model extends the MP modeling capability and enables better formulation of an AVC process and identification of its significant contributing factors.
+
+Among all the significant variables, rural area type, speed limit, white-tailed deer habitat, and rolling terrain type have positive effects on AVC risk. The remaining four variables, median width, sex of animal, truck percentage level, and number of lanes may reduce the probability of AVC risk when the values of these variables increase. Results from this model are useful to compile countermeasures against AVCs. For example, in the areas where the highway crosses the habitat of non-domestic animals, such as white-tailed deer, transportation agencies should further examine some key associated variables, such as speed limit, and develop suitable countermeasures.
+
+### **6. Spatial and temporal transferability test**
+
+The relationship between AVCs and their associated factors may change temporally and spatially. Thus, a concern with the model is whether its estimated coefficients are transferable spatially or temporally. When testing spatial and temporal transferability, the following likelihood ratio test can be conducted [\(Washington](#page-7-0) et [al.,](#page-7-0) [2003\):](#page-7-0)
+
+$$X^{2} = -2[LL(\beta_{T}) - LL(\beta_{a}) - LL(\beta_{b})]$$
+(22)
+
+where LL(ˇT)is the log likelihood at convergence ofthe model using the data from both regions (or time periods), LL(ˇa) is the log likelihood at convergence of the model using the data from a region (or time period a), LL(ˇb) is the log likelihood at convergence of the model using the data from b region (or time period b). This X<sup>2</sup> statistic is a <sup>2</sup> distribution with degrees of freedom equal to the
+
+<span id="page-6-0"></span>**Table 4** Spatial and temporal transferability test results for AI model 3.
+
+|                               | # of segments                                                 | # of accidents | log-likelihood |  |
+|-------------------------------|---------------------------------------------------------------|----------------|----------------|--|
+| Spatial transferability test  |                                                               |                |                |  |
+| *First five routes            | 10,415                                                        | 1,290          | −3,369         |  |
+| #Second five routes           | 9,993                                                         | 2,607          | −5,132         |  |
+| Overall data                  | 20,408                                                        | 3,897          | −8,585         |  |
+| X2                            | = −2[LL(ˇT) − LL(ˇa) − LL(ˇb)] =−2[−8585 + 5132 + 3369] = 168 |                |                |  |
+| Temporal transferability test |                                                               |                |                |  |
+| 2005                          | 9,942                                                         | 2,110          | −4,572         |  |
+| 2006                          | 10,466                                                        | 1,787          | −3,992         |  |
+| Overall data                  | 20,408                                                        | 3,897          | −8,585         |  |
+| X2                            | = −2[LL(ˇT) − LL(ˇa) − LL(ˇb)] =−2[−8585 + 4572 + 3992] = 42  |                |                |  |
+
+\*First five routes: SR 8, US 12, I-90, US 101, and SR970; #Second five routes: US 2, SR 20, US 97, US 395, and SR 525.
+
+summation of the number of coefficients in region a and region b minus the number of coefficients in the overall model.
+
+We statistically tested spatial and temporal transferability for the model. Table 4 shows the transferability test results. The number of segments, number of accidents, and the log-likelihoods for different data sets are also show in Table 4.
+
+For both transferability tests, the null hypothesis was that the coefficients are transferable. For the spatial test, the first data set was routes SR 8, US 12, I-90, US 101, and SR 970, with the second data set having the remaining five routes. For the temporal test, the first data set was the year 2005, and the second the year 2006. Following Eq. [\(22\),](#page-5-0) the data sets were estimated separately and then together. For the spatial test, X<sup>2</sup> was 168 with 9 degrees of freedom, which is greater than 16.92 at a 95% confidence level. For the temporal test, X<sup>2</sup> was 42 with 9 degrees of freedom, which is greater than 16.92 at a 95% confidence level. Thus, the coefficients were found to not be transferable between routes or years.
+
+Although the estimated coefficients could nottransfer from year to year or from location or location,the significant explanatory variables and their sign (positive or negative) converged from these different data sets are basically exact the same. The poor transferability may be attributed to the unsatisfied data quality and availability or may be a reflection of the performance and characteristic differences among drivers and animals in different time period or location. Thus, if we want to estimate a more accurate elasticity for different explanatory variables, we need to recalibrate the model using the data set in a particular time and location. However, the impacts from those variables, either being with a decreasing or an increasing factor on the AVCs, remain the same in different time periods and different locations. This implies that the model can still be applied to develop AVC countermeasures in practice.
+
+### **7. Conclusions**
+
+A series of count data models have been used in AVC analysis in many previous studies. However, most of these models used in vehicle collisions seldom include human factors or animal characteristics in their analysis process, although these attributes are critical to the occurrence of AVCs. Thus, the previous models could not be directly used in the AVC analysis process.
+
+This paper presents the MP and VAIP models and their estimation results. Both models consider the probability of drivers' ineffective response and animals' presence. As an improvement, the VAIP model includes a third term, the probability of an animals' response failure to escape, to capture animals' reaction characteristics in AVCs. The test results show that this model can provide a reasonably good explanation ofthe relationship among human factors, animal distributions, roadway design factors, and AVCs. Key research findings are summarized as follow:
+
+- Compared with urban areas, the probability for a vehicle to encounter an animal is high in rural areas. This is likely due to the animal population difference between the two areas.
+- The probability for a vehicle to hit a deer is much higher when driving on a highway through a white-tailed deer habitat.
+- The probability of a driver's ineffective response will increase with speed limit. It goes up significantly when speed limit is greater than 50 mph.
+- Compared with female animals, male animals are more alert and have a better chance to escape from potential AVCs.
+
+Results from this model are helpful to transportation agencies for determining countermeasures against AVCs. The authors recommend that transportation agencies should further examine some key associated variables, such as speed limit, and develop suitable countermeasures in the areas where the highway crosses the habitat of non-domestic animals, such as white-tailed deer. For better application of this method, the spatial and temporal transferability tests are still needed in future study.
+
+# **Acknowledgements**
+
+The authors are grateful for the financial support to this project fromtheWashingtonState Department of Transportation(WSDOT) and Transportation Northwest (TRANSNow). The authors wish to express sincere appreciation to WSDOT's Environmental Services Office and Research Office personnel, specifically Kelly McAllister and Rhonda Brooks, for their help on the data collection. The authors also wantto acknowledge Highway Safety Information System (HSIS) staff member Yusuf Mohamedshah for his help with data collection. Special thanks also go to Barrett Welford Taylor at the University of Washington for his editing work, and Heather Turner from University of Warwick for her suggestions on model calculation.
+
+### **References**
+
+Abdel-Aty, M.A., Radwan, A.E., 2000. Modeling traffic accident occurrence and involvement. Accid. Anal. Prev. 32 (5), 633–642.
+
+Akaike,H., 1974.Anew look atthe statisticalmodelidentification.IEEE Trans.Autom. Control 19 (6), 716–723.
+
+Allen, R.E., McCullough, D.R., 1976. Deer-car accidents in Southern Michigan. J.Wildl. Manage. 40, 317–325.
+
+Ben-Akiva, M.E., Lerman, S.R., 1985. Discrete Choice Analysis: Theory and Application to Travel Demand. MIT Press Series in Transportation Studies, 9. MIT Press, Cambridge, MA, pp. 167–168.
+
+Chin, H.C., Quddus, M.A., 2003. Applying the random effect negative binomial model to examine traffic accident occurrence at signalized intersections. Accid. Anal. Prev. 35, 253–259.
+
+Curtis, P.D., Hedlund, J.H., 2005. Reducing Deer-Vehicle Crashes. Wildlife Damage Management Fact Sheet Series. Cornell Cooperative Extension, Ithaca, NY.
+
+Danielson, B.J., Hubbard, M.W., 1998. A Literature Review for Assessing the Status of Current Methods of Reducing Deer-Vehicle Collisions. A report prepared for The Task Force on Animal Collisions, Iowa Department of Transportation and Iowa Department of Natural Resources, 25.
+
+- <span id="page-7-0"></span>Daniels, S., Brijs, T., Nuyts, E., Wets, G., 2010. Explaining variation in safety performance of roundabouts. Accid. Anal. Prev. 42 (2), 292–402.
+- Donnell, E.T., Mason, J.M., 2006. Predicting the frequency of median barrier crashes on Pennsylvania interstate highways. Accid. Anal. Prev. 38 (3), 590–599.
+- El-Basyouny, K., Sayed, T., 2006. Comparison of two negative binomial regression techniques in developing accident prediction models. Transp. Res. Rec. 1950, 9–16.
+- Gkritza, K., Baird, M., Hans, Z.N., 2010. Deer-vehicle collisions, deer density, and land use in Iowa's urban deer herd management zones. Accid. Anal. Prev. 42 (6), 1916–1925.
+- [http://www.r-project.org/.](http://www.r-project.org/) Accessed July 5, 2010.
+- Huijser, M.P., Bergers, P.J.M., 2000. The effect of roads and traffic on hedgehog (Erinaceus europaeus) populations. Biol. Conserv. 95, 111–116.
+- Huijser, M.P., Fuller, J., Wagner, M.E., Hardy, A., Clevenger, A.P., 2007. Animal-vehicle Collision Data Collection: A Synthesis of Highway Practice. National Cooperative Highway Research Board Program: Synthesis 370. Transportation Research Board, Washington, DC.
+- Jovanis, P.P., Chang, H., 1986. Modeling the relationship of accidents to miles traveled. Transp. Res. Rec. 1068, 42–51.
+- Kao, J.H.K., 1960. A summary of some new techniques in failure analysis. In: Proceedings of Sixth National Symposium on Reliability and Quality Control , Washington, DC, pp. 190–201.
+- Kim, J., Wang, Y., Ulfarsson, G., 2007. Modeling the probability of freeway rear-end crash occurrence. ASCE J. Transp. Eng. 133 (1), 11–19.
+- Knapp, K.K., Yi, X., 2004. Deer-Vehicle Crash Patterns and Proposed Warning Sign Installation Guidelines. Transportation Research Board, Washington, DC.
+- Lao, Y., Wu, Y., Corey, J., Wang, Y., 2011. Modeling animal-vehicle collisions using diagonal inflated bivariate Poisson regression. Accid. Anal. Prev. 43 (1), 220–227.
+- Lao, Y., Wu, Y., Wang, Y., McAllister, K. submitted for publication. Fuzzy Logic-based Mapping Algorithm for Improving Animal-Vehicle Collision Data. Working paper, Department of Civil and Environmental Engineering, University of Washington, submitted for publication.
+- Lord, D., Miranda-Moreno, L.F., 2008. Effects of low sample mean values and small sample size on the estimation of the fixed dispersion parameter of Poissongamma models for modeling motor vehicle crashes: a Bayesian perspective. Saf.
+- Sci. 46 (5), 751–770. Lord, D., Mannering, F., 2010. The statistical analysis of crash-frequency data: a review and assessment of methodological alternatives. Accid. Anal. Prev. 44 (5),
+- 291–305. Maher, M.J., Summersgill, I.A., 1996. Comprehensive methodology for the fitting of
+- predictive accident models. Accid. Anal. Prev. 28 (3), 281–296. Malyshkina, N., Mannering, F., 2010. Empirical assessment of the impact of highway designexceptions onthe frequency and severity of vehicle accidents.Accid.Anal. Prev. 42.
+
+- Miaou, S.P., Lum,H., 1993.Modeling vehicle accidents andhighway geometricdesign relationships. Accid. Anal. Prev. 25 (6), 689–709.
+- Miaou, S.P., 1994. The relationship between truck accidents and geometric design of road sections: Poisson versus negative binomial regressions. Accid. Anal. Prev. 26 (4), 471–482.
+- Miaou, S.-P., Bligh, R.P., Lord, D., 2005. Developing median barrier installation guidelines: a benefit/cost analysis using Texas data. Transp. Res. Rec. 1904, 3–19.
+- Milton, J., Mannering, F., 1998. The relationship among highway geometrics, trafficrelated elements and motor-vehicle accidentfrequencies. Transportation 25 (4), 395–413.
+- Pitman, J., 1993. Probability. Springer-Verlag, New York, Inc.
+  - Plait, A., 1962. The Weibull distribution with the tables. Ind. Qual. Control (19), 19–26.
+- Proctor, M.F., 2003. Genetic Analysis of Movement, Dispersal, and Population Fragmentation of Grizzly Bears in Southwestern Canada, Ph.D. dissertation, The University of Calgary, Calgary, AB, Canada.
+- Reilley, R.E., Green, H.E., 1974. Deer mortality on a Michigan Interstate Highway. J. Wildl. Manage. 38, 16–19.
+- Rolley, R.E., Lehman, L.E., 1992. Relationships among raccoon road kill surveys, harvests, and traffic. Wildl. Soc. Bull. 20, 313–318.
+- Romin, L.A., Bissonette, J.A., 1996. Deer–vehicle collisions: status of state monitoring activities and mitigation efforts. Wildl. Soc. Bull. 24 (2), 276–283.
+- Shankar, V., Mannering, F., Barfield, W., 1995. Effect of roadway geometric and environment factors on rural freeway accident frequencies. Accid. Anal. Prev. 27 (3), 371–389.
+- Siddique, Z.Q., 2000. Accident Risk Modeling of Vehicle-to-Bicycle and Vehicle-to-Pedestrian Accidents at Four-Legged Signalized Intersections, Master Thesis, Asian Institute of Technology.
+- Van der Zee, F.F., Wiertz, J., ter Braak, C.J.F., van Apeldoorn, R.C., Vink, J., 1992. Landscape change as a possible cause of the Badger Meles meles L. decline in The Netherlands. Biol. Conserv. 61, 17–22.
+- Wagner, P. and M. Carey. 2006. Connecting Habitats and Improving Safety. WSDOT 2007-09 Transportation Research Project Problem Statements.
+- Wang, Y., 1998. Modeling Vehicle-to-Vehicle Accident Risks Considering the Occurrence Mechanism at Four-Legged Signalized Intersections. Ph.D. Dissertation, University of Tokyo.
+- Wang, Y., Ieda, H., Mannering, F.L., 2003. Estimating rear-end accident probabilities at signalized intersections: an occurrence-mechanism approach. J. Transp. Eng. 129 (4), 1–8.
+- Wang, Y., Nihan, N.L., 2004. Estimating the risk of collisions between bicycles and automobiles at signalized intersections. Accid. Anal. Prev. 36 (3), 313–321.
+- Washington, S.P., Karlaftis, M.G., Mannering, F.L., 2003. Statistical and Econometric Methods for Transportation Data Analysis. Chapman & Hall/carcass removal C, Boca Raton, Florida, pp. 242–243.

Zotero/001_artiklid/Road Safety Indicators/latorredevelopmentaccidentpredictionmodel2019.md

@@ -0,0 +1,635 @@
+---
+category: literaturenote
+citekey: latorredevelopmentaccidentpredictionmodel2019
+title: Development of an accident prediction model for Italian freeways
+authors: "La Torre, Francesca; Meocci, Monica; Domenichini, Lorenzo; Branzi, Valentina; Tanzi, Niccolo’; Paliotto, Andrea"
+year: 2019
+date: 2019-00-00 2019
+doi: 10.1016/j.aap.2018.12.023
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2018.12.023"
+zotero_key: FJ62TCKP
+zotero_storage: D4I5N7K9
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "La Torre, Francesca"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+Contents lists available at [ScienceDirect](http://www.sciencedirect.com/science/journal/00014575)
+
+# Accident Analysis and Prevention
+
+journal homepage: [www.elsevier.com/locate/aap](https://www.elsevier.com/locate/aap)
+
+![](_page_0_Picture_5.jpeg)
+
+# Development of an accident prediction model for Italian freeways
+
+![](_page_0_Picture_7.jpeg)
+
+Francesca La Torre, Monica Meocci[⁎](#page-0-0) , Lorenzo Domenichini, Valentina Branzi, Niccolo' Tanzi, Andrea Paliotto
+
+University of Florence, Civil and Environmental Engineering Department, Via S. Marta, 3, 50139, Firenze, Italy
+
+#### ARTICLE INFO
+
+Keywords: Freeways Accident prediction models Safety performance function Highway safety manual Crash modification factors
+
+#### ABSTRACT
+
+The roadway safety management process plays an important role in the national efforts for improving road safety along the Italian freeway network. In 2016, 8.3% of the overall Italian road deaths and 6.3% of the overall road injuries occurred along the 6700 km-long freeway network. Accident Prediction Models (APMs) represent one of the best tools to perform a road safety quantitative assessment. With the aim of providing the Italian freeway agencies with a tool that allows to deal with potential safety issues, this paper defines two APMs for single- and multiple-vehicle fatal-and-injury crashes to be applied on Italian rural freeway segments, based on jurisdictional specific Safety Performance Functions (SPFs) developed in the PRACT project.
+
+The proposed procedure is based on the Highway Safety Manual (HSM) approach, and it introduces a new methodology to transfer the HSM to European motorways.
+
+In order to improve the prediction accuracy, the proposed APMs consist in a jurisdictional specific base SPF, developed for the base data set as a function of Annual Average Daily Traffic (AADT) and segment length, combined with Crash Modification Factors (CMFs), in order to account for differences between each site and the base conditions. The full models are then calibrated based on the total number of accidents observed in the wide data set. For both full models (one for single-vehicle and one for multiple-vehicle crashes), the goodness of fit is evaluated in terms of chi square test, root mean square error, observed Vs predicted diagram and predicted Vs residual diagram.
+
+The results show a good aptitude of both models to describe the analysis data set. The proposed models represent a solid and reliable tool for practitioners to perform accident predictions along the Italian freeway network.
+
+# 1. Introduction and objectives
+
+In 2016, 175,791 road accidents occurred in Italy causing 3283 deaths and 249,175 injuries ([ACI-ISTAT, 2017](#page-10-0)). Along the Italian freeways, 9360 accidents occurred (5.3% of the overall accidents) with 274 deaths (8.3%) and 15,790 injuries (6.3%).
+
+Despite a general increase in the number of road accidents (+ 2% on freeways compared to 2015), the number of fatalities is decreasing (-10.2% on freeways compared to 2015). Even though the crash rate in freeways is considerably lower than in other types of infrastructure, freeway accidents are characterized by a mortality index equal to 2.9, among the highest recorded in the different types of Italian roads [\(ACI-](#page-10-0)[ISTAT, 2017](#page-10-0)).
+
+With the aim of reaching the 2020 target (halve the number of road deaths, compared to 2010) [\(United Nations Road Safety Collaboration,](#page-10-1) [2011\)](#page-10-1), in the period 2017–2020 the number of road deaths in Italy needs a further reduction of about 11%.
+
+The Italian freeway network is about 6700 km long, therefore the reduction in freeways crashes is a priority for National Road Authorities (NRAs) and freeways agencies.
+
+The roadway safety management process plays an important role in the national efforts for improving traffic safety; within this topic, Accident Prediction Models (APMs) represent one of the best available tools for performing a quantitative safety assessment. APMs are mathematical equations that allow road engineers and National Road Authorities to relate the number of crashes expected on a site to its specific geometric and environmental characteristics ([Yannis et al.,](#page-10-2) [2016a,](#page-10-2)[b](#page-10-3)). The use of traditional approaches, such as crash frequency and crash rate estimation, allows the identification of sites with high frequency of crashes by means of a minimum amount of data. However, these methods alone do not allow a quantitative safety assessment of specific features. On the other hand, despite the need of more detailed
+
+E-mail addresses: [francesca.latorre@uni](mailto:francesca.latorre@unifi.it)fi.it (F. La Torre), [monica.meocci@uni](mailto:monica.meocci@unifi.it)fi.it (M. Meocci), [lorenzo.domenichini@uni](mailto:lorenzo.domenichini@unifi.it)fi.it (L. Domenichini), [valentina.branzi@uni](mailto:valentina.branzi@unifi.it)fi.it (V. Branzi), [niccolo.tanzi@uni](mailto:niccolo.tanzi@unifi.it)fi.it (N. Tanzi), [andrea.paliotto@uni](mailto:andrea.paliotto@unifi.it)fi.it (A. Paliotto).
+
+<span id="page-0-0"></span><sup>⁎</sup> Corresponding author.
+
+data, the use of APMs as part of the network screening process, allows a road agency to have a good insight in the safety level of its roads ([Eenink et al., 2005](#page-10-4)). Indeed, APMs allow to identify locations that may benefit the most from a safety treatment, and therefore to better define strategies in safety actions and to set priorities ([Srinivasan and Bauer,](#page-10-5) [2013\)](#page-10-5). Another application of APMs is to determine the safety impact of design changes at project level, in order to evaluate existing alternatives or proposed new solutions in the decision making phase [\(Srinivasan and](#page-10-5) [Bauer, 2013](#page-10-5)). APMs can also be used to evaluate the safety effects of engineering treatments, as part of a before-after study. In order to properly determine the effect of the implementation of specific safety countermeasures on crashes. APMs are often used in combination with the empirical Bayes procedure ([Hauer, 1997\)](#page-10-6), which allows to account for the potential bias due to the regression to the mean phenomenon.
+
+In order to provide the Italian freeway agencies with a tool that allows to deal with potential safety issues identifying the sites with the largest potential safety improvements and helping in choosing the best treatments, this paper provides two APMs for single- and multiple-vehicle fatal-and-injury crashes to be applied on Italian rural freeways based on jurisdictional specific Safety Performance Functions (SPFs) as developed in the PRACT project (La Torre F., Domenichini L., Meocci M. et al. 2016). The proposed procedure is developed based on the approach presented in the Highway Safety Manual ([AASHTO, 2010](#page-10-7)), and it introduces a new methodology to transfer the HSM Part C project-level models to European motorways.
+
+The general level of crash frequencies may vary substantially from one jurisdiction to another for a variety of reasons, including crash reporting thresholds and crash reporting system procedures; even when a proper calibration is performed, transferring a model developed for one jurisdiction to another one may lead to misleading results. On the other hand, the SPFs developed with data for a specific jurisdiction may provide improvements in the prediction accuracy; the HSM indicates that jurisdiction-specific SPFs "are likely to enhance the reliability of the HSM predictive method" ([AASHTO, 2010;](#page-10-7) [Srinivasan and Bauer,](#page-10-5) [2013;](#page-10-5) [Srinivasan et al., 2013](#page-10-8)). The accident prediction models here proposed for single-vehicle (SV) and multiple-vehicle (MV) fatal-andinjury crashes occurred on rural freeway segments consist in a jurisdictional specific base SPF, developed for the base data set as a function of Annual Average Daily Traffic and segment length, combined with Crash Modification Factors (CMFs), to account for differences between the base conditions and the specific conditions in each analysed section. The full models have been then calibrated based on the total number of crashes observed in the full data set.
+
+### 2. Background
+
+Nowadays the Highway Safety Manual (HSM) prediction model ([AASHTO, 2010\)](#page-10-7) represents the most widely used approach for road safety assessment. The HSM provides a predictive method for estimating the expected average crash frequency of a network, facility or individual site.
+
+According to the procedure proposed by the HSM, the "reference" crash frequency of a segment can be estimated by means of the application of a regression model developed from observed crash data on a large number of sites. The regression models, called Safety Performance Functions (SPFs), are presented for different facility types with "base conditions" that are the specific geometric design and traffic control features of a "base" site. The base functions depend only on the annual average daily traffic and on the segment length, with linear or nonlinear traffic effects (depending on the type of road). In order to account for the geometric design and traffic control features of the site under investigation that differ from the base conditions, Crash Modification Factors (CMFs) are used as multiplicative factors to adjust the predicted "base" crash frequency. CMFs are generally developed estimating the safety impact following an intervention or a design change, using a before-after approach. As the general level of crash frequency may vary substantially from one jurisdiction to another for a variety of reasons, including climate, driver population, crash reporting threshold and crash reporting system procedures, a calibration factor (C) is usually adopted to improve the prediction accuracy of the HSM model when transferred to different road networks ([AASHTO, 2010;](#page-10-7) [Bahar and](#page-10-9) [Hauer, 2014\)](#page-10-9).
+
+Safety prediction methodologies for freeways, not included in the 2010 version of the HSM, have been published in the HSM supplement ([AASHTO, 2014\)](#page-10-10). Likewise to the first approach, the new predictive models for freeways, provided for rural and urban freeways with four/ six/eight divided lanes, speed-change lanes, and ramps, consist of a "base" safety performance function combined with a set of crash modification factors.
+
+Other procedures have been developed and applied to estimate road safety levels in road networks. A recent review of APM models applied worldwide has recently been published by [Yannis et al. \(2017\)](#page-10-11).
+
+The formulation of an APM by means of a Complete Safety Performance Function consists in the development of a statistical model that provides estimates of the average accident frequency of a unit (road segments, intersections, grade crossing etc.) as a function of its traits (traffic, geometry, operation). Recent researches, developed by Behnood and Mannering in 2015 and Cheng et al. in 2018 by means of developing complete SPFs, evaluate the effect of spatiotemporal stability of factors affecting crashes prediction. This approach is based on the assumption of independence of the explanatory variables included in the analysis. Historically, various types of statistical techniques have been used for modelling roadway crashes: Multivariate Analysis, Empirical Bayes Method [\(Hauer, 2001;](#page-10-12) [Persaud et al., 1999](#page-10-13)), Fuzzy Logic and Neural Network ([Abdelwahab and Abdel-Aty, 2001](#page-10-14); [Adeli](#page-10-15) and [Karim, 2000](#page-10-15)). Among these methods, the multivariate analysis has a long and consolidated use in accident analysis. The early models developed with this technique were based on Multiple Linear Regression while, the currently used approach is based on the Generalized Linear Model technique (GLM), which allows to extend the linear modelling to stochastic variables that are not normally distributed with a constant variance. Within the stochastic processes, accident frequency in a specific location is usually treated as a random variable distributed with a Poisson distribution and a negative binomial error structure ([Roque and Cardoso, 2014\)](#page-10-16). The definition of a Complete Safety Performance function provides very robust and reliable predictions for the analysed sites [\(Behnood and Mannering, 2015](#page-10-17); [Cheng et al., 2018](#page-10-18)); however, these functions are hardly transferable to different conditions from the ones analysed in their formulation.
+
+In Italy, one of the first approaches regarding the development of APMs referred to the national road network is dated 2007, when Caliendo et al. developed a prediction model for Italian four-lane median-divided motorways based on accident and road geometry data monitored over a period of 5 years, between 1999 and 2003. The model proposed allowed to obtain crash frequency as a function of traffic flow, infrastructure characteristics, pavement surface characteristics as well as wet or dry conditions, and sight distance. It was developed by means of a stepwise forward procedure based on the Generalized Likelihood Ratio Test [\(Caliendo et al., 2007\)](#page-10-19).
+
+In 2008, Montella et al. developed two separate crash prediction models for total crashes and severe crashes (fatal plus all injury) occurred in the Italian motorway network. The models were developed by means of a Generalized Linear Modelling approach, assuming an error structure described by a negative binomial distribution. A sample of 2245 total crashes occurred from 2001 to 2005 in the A16 motorway was used in order to develop a model considering the following variables: curvature, operating speed reduction, straight length, traffic, friction characteristics, downhill and uphill slope ([Montella et al.,](#page-10-20) [2008\)](#page-10-20).
+
+The first implementation of the HSM method in Italy was conducted in 2009 ([Martinelli et al., 2009\)](#page-10-21) but it was not referred to freeways. The research focused on the application of the calibration procedure to twolane two-way rural roads in the Italian province of Arezzo in order to evaluate the practical transferability of the HSM methodology to a region characterized by a different environment and different road characteristics, driver behaviour, and crash reporting systems as compared to those on which the HSM models had been developed.
+
+A few years later, Cafiso and D'[Agostino \(2012\)](#page-10-22) focused their research on "ranking of high accident concentration sections". A Safety Performance Function based on the Italian road infrastructure characteristics was developed using the General Estimating Equation, including in the analysis a 76.85 km long stretch of road from the A18 Messina-Catania, with 314 crashes occurred in a 4-years period. A comparison with the HSM methodology was also performed. Statistical evaluations demonstrated that the estimations obtained with the GEE were more reliable than the HSM calibration process but the HSM model applied was the "divided highways" and not the "freeways" model (not available at the time). (Cafiso and D'[Agostino, 2012](#page-10-22)).
+
+In order to assess the potential issues that occur applying the HSM freeway model to a different road context characterized by diverse environmental conditions and a distinct crash reporting system, [La](#page-10-23) [Torre et al. \(2014\)](#page-10-23) evaluated the transferability of the HSM ay model to the Italian freeway network. The goodness of fit assessment was carried out by means of statistical tests such as mean absolute deviation, calibrated overdispersion parameter, root mean square errors and residual plots. The results obtained allowed to observe a good transferability of the HSM predictive models to the Italian network, especially for freeways fatal and injury models ([La Torre et al., 2014](#page-10-23)).
+
+With the aim of making a step forward in accident prediction methodologies and providing the European roadway agencies a tool that allows to obtain a reliable crash frequency estimation on freeway segments as a function of its geometric design, traffic control features, and traffic volumes, a new methodology was developed in the PRACT project to transfer to the European road network the HSM approach (La Torre F., Domenichini L., Meocci M. et al. 2016). This approach was considered more solid and transferable for accident prediction modelling than the use of a complete safety performance function.
+
+In this paper, the implementation on the Italian freeways network is described in detail.
+
+# 3. Database structure and data availability
+
+The Italian freeway network is about 6700 km long with a divided carriageway cross-section of 2, 3 or 4 lanes per direction. Any accident occurring on the freeway network, including property damage only (PDO) is recorded by the Police and/or a specific traffic assistant group.
+
+The data collected on site are recorded in an accident database managed by the management company of the freeway trunk.
+
+The accident database available for this study refers to the data collected by "Autostrade per l'Italia S.p.A." (ASPI), company that manages a 3020 km long freeway network across Italy. [Fig. 1](#page-2-0) shows the Italian freeway network, with highlighted in green the sections managed by ASPI.
+
+The accident data analysed within this research refers to 884 km of freeway segments in one direction of travel, covering a 5-year period, from 2009 to 2013.
+
+All the models developed within the PRACT project refer only to fatal and injury crashes as the data availability for property damage only in the other countries for which the PRACT models are developed (UK, Germany, Greece, Netherlands) is limited and often unreliable.
+
+Geometric data and traffic data have also been collected to perform the research.
+
+The geometric database provided by ASPI, containing curvature and gradient information discretised into 50-meter intervals, has been integrated with additional cross-sectional data collected by the University of Florence. The traffic database contains the average daily vehicle passages in each section between two interchanges, available on a monthly basis from 2009 to 2013, for each direction of travel.
+
+<span id="page-2-0"></span>![](_page_2_Picture_15.jpeg)
+
+Fig. 1. Italian freeways managed by Autostrade per l'Italia S.p.A. (source: Autostrade.it).
+
+The information contained in each database is summarised in [Table 1](#page-3-0).
+
+# 4. Methodology
+
+### 4.1. APM modelling methodology
+
+The accident prediction models (APMs) have been developed in the following scenario:
+
+- road type: dual carriageway;
+- site type: freeway segment (not including intersections, interchanges, driveways, etc.);
+- area type: rural;
+- cross section: two or more lanes for each direction of travel;
+- crash type: single-vehicle (SV) and multiple-vehicle (MV) crashes (two separate models);
+- crash severity: fatal and injury crashes.
+
+The models to estimate the predicted average crash frequency (Npredicted) were developed following the HSM approach ([AASHTO,](#page-10-10) [2014\)](#page-10-10), which combines a good flexibility and adaptability to local conditions with a reliable accident prediction. The predictive model consists of a base SPF, crash modification factors, and a calibration factor "C", according to the Eq. [\(1\)](#page-2-1):
+
+<span id="page-2-1"></span>
+$$N_{predicted,x} = N_{spf,x} \times (CMF_{1x} \times CMF_{2x} \times ... \times CMF_{ix}) \times C_x$$
+ (1)
+
+where:
+
+- x represents a given infrastructure (rural/urban) and lane configuration (2/3 or more for each direction) and crash type (SV or MV);
+- *Npredicted*,*<sup>x</sup>* is the predicted average crash frequency for a specific year for infrastructure/crash type x;
+- *Nspf* ,*<sup>x</sup>* is the predicted average crash frequency determined for base conditions for infrastructure/crash type x;
+- *CMFix* is the crash modification factor specific to SPF for site infrastructure/crash x;
+- *Cx* is the calibration factor to adjust the SPF to local conditions for infrastructure/crash site type x.
+
+Within this study, the "base" crash frequencies (*Nspf x*, ) have been determined by means of regression models developed from data from a number of "base" sites. These base safety performance functions were fitted using only sites with "base conditions", which are the specific geometric design and traffic control features of a "base" site in the analysed scenario. The "base" conditions have been defined for each dataset as the "standard" conditions that can differ from the HSM "base" conditions. The SPFs were fitted with negative binomial (NB) regression models estimated using a generalised linear model procedure
+
+<span id="page-3-0"></span>Table 1 Database information.
+
+| Geometric data                                                                    | Traffic data                                  | Accident data                                        |
+|-----------------------------------------------------------------------------------|-----------------------------------------------|------------------------------------------------------|
+| ASPI/UNIFI database                                                               | ASPI database                                 | ASPI database                                        |
+| •<br>Freeway ID                                                                   | •<br>Freeway ID                               | •<br>Accident ID                                     |
+| •<br>Direction                                                                    | •<br>Direction                                | •<br>Date                                            |
+| •<br>Beginning of the trait                                                       | •<br>Beginning of the section                 | •<br>Time                                            |
+| •<br>End of the trait                                                             | •<br>End of the section                       | •<br>Freeway ID                                      |
+| •<br>Geometric element: presence of straight or bend (for the bends, radius       | •<br>Date                                     | •<br>Direction                                       |
+| and direction: left/right)                                                        | •<br>Passages divided by vehicle class        | •<br>Milestone (km)                                  |
+| •<br>Longitudinal slope                                                           | •<br>Presence of an average speed enforcement | •<br>Localisation on the carriageway                 |
+| •<br>Outside shoulder width                                                       | (section control)                             | •<br>Type of accident                                |
+| •<br>Median Width                                                                 |                                               | •<br>Pavement conditions                             |
+| •<br>Median and outside barrier                                                   |                                               | •<br>Speed limit                                     |
+| •<br>Presence of a bridge, tunnel, junction, service area, parking                |                                               | •<br>Weather conditions                              |
+| •<br>Number of lanes                                                              |                                               | •<br>Presence of work zone                           |
+| •<br>Presence of a Speed limit<br>•<br>Presence of a high friction wearing course |                                               | •<br>Cause of the accident (not always<br>available) |
+|                                                                                   |                                               | •<br>Number of vehicles involved                     |
+|                                                                                   |                                               | •<br>Number of injuries                              |
+|                                                                                   |                                               | •<br>Number of fatalities                            |
+|                                                                                   |                                               | •<br>Type of damaged elements                        |
+|                                                                                   |                                               | •<br>For each vehicle involved:                      |
+|                                                                                   |                                               | •<br>Vehicle id                                      |
+|                                                                                   |                                               | •<br>Type of vehicle                                 |
+|                                                                                   |                                               | •<br>Condition of the driver after the accident      |
+|                                                                                   |                                               | •<br>Vehicle damage (Yes/No)                         |
+|                                                                                   |                                               | •<br>Number of people involved                       |
+|                                                                                   |                                               | •<br>Number of injuries                              |
+|                                                                                   |                                               | •<br>Number of fatalities                            |
+
+as a function of annual average daily traffic (AADT) volume and roadway segment length.
+
+Crash modification factors were then used as multiplicative factors to account for the specific site conditions, which are different from the base conditions; finally, the full Accident Prediction Models were calibrated to the entire data set in order to obtain the calibration factor to account for differences in the general level of crash frequency and enhance the reliability of the predictive method. The methodology adopted for modelling the APMs is presented in [Fig. 2](#page-3-1).
+
+### <span id="page-3-2"></span>4.2. Definition of the relevant variables included in the analysis
+
+As proposed by the HSM procedure, the APMs developed consider separately two types of accidents occurred on rural freeway segments:
+
+• single-vehicle fatal-and-injury crashes (SV-fi);
+
+<span id="page-3-1"></span>![](_page_3_Figure_10.jpeg)
+
+Fig. 2. Flowchart for the APM modelling methodology.
+
+• multiple-vehicle fatal-and-injury crashes (MV-fi).
+
+Two different accident prediction models provide, as dependent variable, the number of single-vehicle (or multiple-vehicle) fatal-and-injury crashes per km per year, obtained following the investigation in a 5-year period (from 2009 to 2013). Differently from the HSM, it was chosen to collect data and develop models specifically for one-direction of travel. This allowed more flexibility in applying the model, such as the possibility to account for differences in AADT [\(Roque and Cardoso, 2014](#page-10-16)), number of observed accidents and geometry in the two direction. This also provides more reliability in identifying the segments with the largest potential for safety improvements.
+
+In order to account for data over-dispersion, the response variable is assumed to have a negative binomial distribution, better suited than the Poisson distribution for modelling crash data ([AASHTO, 2014](#page-10-10)).
+
+The variables used in the APMs, have been extracted from road alignment, cross-sectional and traffic data bases in order to evaluate the crash modification factors (CMFs) that are part of the HSM predictive process for freeways ([AASHTO, 2014\)](#page-10-10) and the CMFs developed within the PRACT project [\(Karathodorou et al., 2016](#page-10-24); [Yannis et al., 2016a](#page-10-2)), such as the presence of a high friction wearing course and the presence of an average speed enforcement (section control) or of a stationary work zone ([La Torre et al., 2016, 2017, 2018\)](#page-10-25). The characteristics accounted for this analysis are here summarised:
+
+- segment length;
+- number of lanes;
+- horizontal curve radius and length;
+- lane width;
+- inside shoulder width;
+- portion of segment with a barrier in the median;
+- median width;
+- high volume;
+- portion of segment with a barrier in the outside edge;
+- outside clearance;
+- presence of a high friction wearing course;
+- presence of average speed enforcement (section control).
+
+Section with important work zones were removed from the analysis to develop the APM.
+
+### 4.3. Description of the road network used for modelling
+
+The sections used for the APM development are the result of the segmentation of the motorway network in 1-km-long homogeneous segments. Each segment has been assigned a weighted average of the geometric and functional variables of interest. [Table 2](#page-5-0) presents a summary of the data collected.
+
+The experience with crash data indicates that very shorts segments or sections associated with a very low traffic volume, tend to have no reported crashes during a period of several years ([Bonneson et al.,](#page-10-26) [2007\)](#page-10-26). When one of these segments is associated with one or more crashes, it has a tendency to exhibit undue leverage on the regression model coefficients and increases the Pearson *χ* <sup>2</sup> goodness of fit statistic in a disproportionate manner [\(Bonneson et al., 2007\)](#page-10-26). To avoid these issues, data has been screened and only the segments with the following characteristics were included in the analysis:
+
+- rural environment;
+- mean AADT recorded in the analysis period in the specific section above the AADT 10th percentile of the sample (9279 veh/day, rounded to 10,000 veh/day);
+- minimum level of exposure E <sup>≥</sup> Emin, where Emin is the segment exposure associated with prediction ratio PR = 3.0, million-vehiclemiles (mvm) and E is the exposure for all segments (mvm) according to the procedure defined by [Bonneson et al. \(2007\).](#page-10-26)
+
+In [Tables 3 and 4](#page-5-1) the descriptive statistics of the samples used for calibrating the accident prediction single- and multiple-vehicle models are shown.
+
+# 4.4. Road network analysis: identification of the "base" conditions
+
+In order to perform an effective integration of the base models with the HSM predictive method, the final base SPFs should have the same base conditions as the ones adopted in the HSM. This will allow to apply the HSM CMFs, without the need for any further transformations. However, because the use of the same base conditions adopted by the HSM did not allow to identify a sufficient number of base segments within the data set, the base models were developed using data for a different set of conditions as compared to the HSM than the base conditions, as suggested also by the HSM procedure [\(AASHTO, 2010,](#page-10-7) [2014\)](#page-10-7).
+
+Analysing the descriptive statistics of the full data set, "standard" conditions were identified in the dataset by means of the use of frequency distribution plots for each analysis variable. Each CMF was evaluated for the mode value of each analysed variable; then, sections where the CMF is within ± 5% of the value calculated for the mode value are considered as "standard" and are used for developing the base models. A sample of the frequency distribution plots used for the identification of the "standard" conditions is shown in [Fig. 3](#page-6-0).
+
+Once the standard conditions have been defined for each analysed variable and the corresponding CMF values have been estimated, a corrective coefficient was evaluated to adjust the SPFs for conditions different from the HSM base ones, according to Eq. [\(2\):](#page-4-0)
+
+<span id="page-4-0"></span>
+$$\prod_{i=1}^{n} CMF_{i}$$
+(2)
+
+where,CMFi are the crash modification factors computed for each variable i that has a standard condition different from the HSM base condition.
+
+In [Table 5](#page-6-1) the variable ranges used to identify the "standard" segments usable for fitting the SPF base models are summarized.
+
+## 4.5. Fitting of the base models
+
+The base SPFs were fitted with negative binomial (NB) regression models estimated using a generalised linear model (GLM) procedure. A statistical software, i.e. SPSS ver. 23 ([IBM, 2015\)](#page-10-27) was used for this purpose.
+
+The base models were developed according to the guidelines presented in the HSM Appendix B ([AASHTO, 2014](#page-10-10)). Assuming as the response variable fatal-and-injury single- (or multiple-) vehicle crashes per year in one direction of travel, the base SPFs include the effect of traffic volume in terms of annual average daily traffic. The predicted average crash frequency is directly proportional to the segment length, as shown in the Eq. [\(3\)](#page-4-1).
+
+<span id="page-4-1"></span>
+$$N_{spf} = L \times e^{[a+b\times ln(c\times AADT)]}$$
+(3)
+
+where:
+
+- Nspf is the predicted average crash frequency (fatal-and-injury accidents) in one direction of travel determined for base conditions;
+- L is the segment length (in km);
+- AADT is the annual average daily traffic in one direction of travel [vehicles/day];
+- <sup>a</sup>, <sup>b</sup>, <sup>c</sup> are the safety performance function coefficients.
+
+According to the HSM ([AASHTO, 2010](#page-10-7)), the response variable was assumed to follow a negative binomial distribution, which allowed to account for data over-dispersion. While the logarithm was assumed as a link function, the parameters of the models were estimated using the maximum likelihood method, selecting the set of parameters that
+
+<span id="page-5-0"></span>Table 2 Summary of data collected.
+
+| Crash frequency distribution | Multiple-vehicle [segments]        | 23 MV-fi-crashes | 1    |
+|------------------------------|------------------------------------|------------------|------|
+|                              |                                    | 17 MV-fi-crashes | 2    |
+|                              |                                    | 16 MV-fi-crashes | 1    |
+|                              |                                    | 15 MV-fi-crashes | 2    |
+|                              |                                    | 14 MV-fi-crashes | 1    |
+|                              |                                    | 11 MV-fi-crashes | 3    |
+|                              |                                    | 10 MV-fi-crashes | 5    |
+|                              |                                    | 9 MV-fi-crashes  | 6    |
+|                              |                                    | 8 MV-fi-crashes  | 16   |
+|                              |                                    | 7 MV-fi-crashes  | 14   |
+|                              |                                    | 6 MV-fi-crashes  | 39   |
+|                              |                                    | 5 MV-fi-crashes  | 45   |
+|                              |                                    | 4 MV-fi-crashes  | 58   |
+|                              |                                    | 3 MV-fi-crashes  | 84   |
+|                              |                                    | 2 MV-fi-crashes  | 141  |
+|                              |                                    | 1 MV-fi-crashes  | 201  |
+|                              |                                    | 0 MV-fi-crashes  | 265  |
+|                              | Single-vehicle [segments]          | 16 SV-fi-crashes | 1    |
+|                              |                                    | 10 SV-fi-crashes | 1    |
+|                              |                                    | 9 SV-fi-crashes  | 2    |
+|                              |                                    | 8 SV-fi-crashes  | 6    |
+|                              |                                    | 7 SV-fi-crashes  | 9    |
+|                              |                                    | 6 SV-fi-crashes  | 5    |
+|                              |                                    | 5 SV-fi-crashes  | 18   |
+|                              |                                    | 4 SV-fi-crashes  | 28   |
+|                              |                                    | 3 SV-fi-crashes  | 71   |
+|                              |                                    | 2 SV-fi-crashes  | 124  |
+|                              |                                    | 1 SV-fi-crashes  | 267  |
+|                              |                                    | 0 SV-fi-crashes  | 352  |
+| Fatal and Injury accidents   | Total [# accidents]                |                  | 3021 |
+|                              | Multiple-vehicle [#accidents]      |                  | 1906 |
+|                              | Single-vehicle [#accidents]        |                  | 1115 |
+| Segmentation process         | Total length [km]                  |                  | 884  |
+|                              | Sample size [segments]             |                  | 884  |
+| Accident severity            | Fatal-injury crashes               |                  |      |
+| Target accident group        | Multiple vehicle<br>Single vehicle |                  |      |
+| Observed period              | 2009-2013 (5 years)                |                  |      |
+
+<span id="page-5-1"></span>Table 3 Description of the calibration dataset for Italian freeways.
+
+| Number of segments in the samples                   |       |  |
+|-----------------------------------------------------|-------|--|
+| 2-lane segments                                     | 370   |  |
+| 3 or more lane segments                             | 348   |  |
+| Total length covered [km]                           | 718   |  |
+| Number of single-vehicle fatal-and-injury crashes   | 1042  |  |
+| Number of multiple-vehicle fatal-and-injury crashes | 1826  |  |
+| Total number of accidents                           | 2,868 |  |
+
+maximise the log-likelihood function.
+
+In order to build a model consistent with the bi-directional one developed within the HSM, the response variable for the model was set as twice the single- (or multiple-) vehicle fatal-and-injury crashes occurred in the segment in the analysis period; the predictors were set for number of lanes (developing models for 2 and 3 or more lanes) as 'factor', and ln (c\*AADT) as 'covariate', where c is a coefficient set equal to 0.002. The use of an offset variable allowed to take into account the standard conditions different from the HSM base ones and to obtain a predicted average crash frequency directly proportional to segment length and number of years of observation. The offset variable was therefore set as described in the Eq. [\(4\).](#page-5-2)
+
+<span id="page-5-2"></span>offsetvariable: 
+$$\ln(L \times CMF_{tot} \times n)$$
+ (4)
+
+where:
+
+- *<sup>L</sup>* is the segment length (km);
+- <sup>=</sup> <sup>∏</sup> <sup>=</sup> *CMF CMF tot <sup>i</sup> <sup>i</sup>* <sup>1</sup> is the corrective coefficient to account for base conditions different from the HSM base conditions;
+- n is the number of years of the observation data.
+
+Table 4 Descriptive statistics of the calibration dataset for Italian freeways.
+
+|                                                                          | Minimum | Maximum | Mean   | Std. Dev. |
+|--------------------------------------------------------------------------|---------|---------|--------|-----------|
+| AADT [veh/day]                                                           | 10,371  | 49,572  | 25,962 | 9,904     |
+| L [km]                                                                   | 1       | 1       | 1      | 0         |
+| Horizontal curvature [-]                                                 | 0.00    | 32.83   | 1.83   | 5.05      |
+| Lane width [m] *                                                         | 3.75    | 3.75    | 3.75   | 0         |
+| Inside shoulder width [m] **                                             | 0.60    | 0.60    | 0.60   | 0         |
+| Portion of the segment with a<br>barrier in the median *                 | 100%    | 100%    | 100%   | 0%        |
+| Proportion of AADT during hours<br>where volume exceeds 1000<br>veh/h/ln | 0%      | 41%     | 11%    | 10%       |
+| Outside shoulder [m]                                                     | 1.22    | 4.27    | 2.35   | 0.79      |
+| Portion of segment with a barrier in<br>the roadside                     | 0%      | 100%    | 84%    | 25%       |
+| Heavy Good Vehicle (HGVs)                                                | 15%     | 32%     | 25%    | 4%        |
+
+<span id="page-5-4"></span><sup>\*</sup> due to a lack of data in this variable, it was assumed a default value according to the Ministry Decree No. 6792 of 05.11.2001 (mandatory only for new roads).
+
+<span id="page-5-3"></span>The models' coefficients have been "corrected" to obtain a predicted average crash frequency in only one direction of travel, according to the Eq. [\(5\).](#page-5-3)
+
+$$a = \bar{a} + \ln(0.5) \tag{5}$$
+
+where, *a*¯ is the coefficient estimated from the fitting process.
+
+[Table 6](#page-6-2) presents the descriptive statistics of the samples used for
+
+<span id="page-5-5"></span><sup>\*\*</sup> due to the lack of data in this variable, the value 0.6 m represents the minimum value provided by the HSM for the CMF3 (inside shoulder width): 2 ft = 0.61 m.
+
+<span id="page-6-0"></span>![](_page_6_Figure_2.jpeg)
+
+Fig. 3. Sample of the frequency distribution plots for the identification of standard conditions.
+
+<span id="page-6-1"></span>Table 5 Summary of standard conditions adopted.
+
+|                                                                         | MIN   | MAX  |
+|-------------------------------------------------------------------------|-------|------|
+| Horizontal curvature [-]                                                | 0     | 0.19 |
+| Average Radius [m]                                                      | 4000  | –    |
+| Lane Width [m] *                                                        | 3.75  | 3.75 |
+| Inside Shoulder Width [m] *                                             | 0.60  | 0.60 |
+| Portion of segment with a barrier in the median [%] *                   | 100   | 100  |
+| Distance from edge to barrier face [m]                                  | 0     | 0.20 |
+| Portion of AADT during hours where volume exceeds 1000/veh/<br>h/ln [%] | 0     | 30   |
+| Outside shoulder width [m] **                                           | 2.90  | 3.10 |
+| Portion of segment with a barrier in the roadside [%] **                | 90    | 100  |
+| Distance to upstream entrance [km] ***                                  | 0.800 | –    |
+| Distance to downstream exit [km] ***                                    | 0.800 | –    |
+
+- <span id="page-6-4"></span>\* variable value assumed as constant.
+- <span id="page-6-5"></span>\*\* only for single-vehicle models.
+- <span id="page-6-6"></span>\*\*\* only for multiple-vehicle models.
+
+fitting the base SPFs.
+
+### 4.6. Selection of crash modification factors
+
+In order to account for the specific site conditions that differ from the base conditions, Crash Modification Factors, or Functions, are included in the prediction models considering the relevant variables listed in Section [4.2](#page-3-2) of the manuscript.
+
+Consequently to the base SPF formulation, the CMFs included in the full model have the same base conditions as the base models. According to this, when a variable assumes the base condition value, the relative CMF coefficient assumes a value equal to 1, without the need of any transformations due to different base condition systems.
+
+If the value of a relevant variable was out of range of applicability for a given Crash Modification Factor, the threshold value was selected to compute the CMF to be used.
+
+Additional CMFs, not used in this study can be applied to the freeways base model as proposed within the PRACT project [\(La Torre et al.,](#page-10-25) [2016, 2017, 2018](#page-10-25); [Karathodorou et al., 2016](#page-10-24)).
+
+# 4.7. Estimation of the calibration factor
+
+A calibration factor "C" is usually evaluated to improve the accuracy of the prediction of the HSM model transferred to road networks different from the one for which the model was developed. This allows to account for differences in the general level of crash frequency, which may change substantially from one country to another because of differences in climate, driver populations and trip purposes, complexity of the geometric layout, driver attitude (rate of compliance with road code rules), vehicle fleet characteristics, crash reporting threshold, and crash reporting system procedures ([AASHTO, 2010;](#page-10-7) [Bahar and Hauer, 2014](#page-10-9)).
+
+<span id="page-6-3"></span>The calibration factor C is defined as the ratio between the total number of crashes observed (*Nobs*) and the total number of crashes predicted ( ) *Npred* by the uncalibrated model as shown in Eq. [\(6\).](#page-6-3)
+
+$$C = \frac{\sum_{i=1}^{allsites} \sum_{j=1}^{allyears} N_{obs,i,j}}{\sum_{i=1}^{allsites} \sum_{j=1}^{allyears} N_{pred,i,j}}$$
+(6)
+
+The effect of several causal factors, such as driver habits, enforcement levels, vehicle technologies or safety systems, may improve over time and cause changes in crash counts unrelated to the safety
+
+<span id="page-6-2"></span>Table 6 Descriptive statistics of the sample used for modelling the base SPF.
+
+| Single-vehicle fatal-and-injury crashes |                 |                       | Multiple-vehicle fatal-and-injury crashes |                              |  |
+|-----------------------------------------|-----------------|-----------------------|-------------------------------------------|------------------------------|--|
+| Analysis period                         |                 | 2009 - 2013 (5 years) |                                           |                              |  |
+| Number of segments                      | 116             |                       |                                           | 2009 - 2013 (5 years)<br>310 |  |
+| 2 lanes                                 | 58              |                       |                                           | 160                          |  |
+| 3 or more lanes                         | 58              |                       |                                           |                              |  |
+| Total length covered [km]               | 116             |                       |                                           |                              |  |
+| Number of crashes                       | 93              |                       |                                           |                              |  |
+|                                         | Min - Max       | Mean (Std. Dev.)      | Min - Max                                 | Mean (Std. Dev.)             |  |
+| AADT [veh/day]                          | 10,371 – 49,572 | 29,742 (11,792)       | 10,371 – 49,572                           | 26,460 (10,252)              |  |
+
+treatment. The models used within the PRACT project for developing new CMFs were calibrated separately for each year of analysis to reflect these time trends ([La Torre et al., 2018](#page-10-28)).
+
+# 4.8. Evaluation of the goodness of fit and residual analysis
+
+Several statistics have been developed in accident analysis to assess how well a model fits the data. In this research, the goodness-of-fit (GoF) of the base and full models has been assessed by means of the following indicators:
+
+- Pearson'<sup>s</sup> <sup>χ</sup><sup>2</sup> statistic: the <sup>χ</sup><sup>2</sup> distribution with <sup>n</sup>–<sup>p</sup> degrees of freedom (df), where n is the number of observations (i.e., segments), and p is the number of variables; it is asymptotic to the χ<sup>2</sup> distribution for larger sample sizes and exact for normally distributed error structures ([Bonneson et al., 2007](#page-10-26); [McCullagh and Nelder,](#page-10-29) [1993](#page-10-29)). The Pearson χ<sup>2</sup> statistic evaluated for a model is usually compared to the *χ* <sup>2</sup> - *value* computed for n–p degrees of freedom and a p-value equal to 0.05 ( <sup>−</sup> *χ*0.05,( ) *n p* <sup>2</sup> ). If the χ<sup>2</sup> statistic of the model is less than the *χ* <sup>2</sup> - *value*, the hypothesis that the model fits the data cannot be rejected.
+- root mean square error (Se): this is an useful statistic for assessing the precision of a model estimate and is often used to compare the GoF of different models. It is computed using the equation suggested by [Bonneson et al. \(2012\)](#page-10-30).
+- R-squared coefficient of determination (R<sup>2</sup> ): the coefficient usually adopted to estimate the percentage of variation explained by a regression model. It may yield to controversial results for non-linear models, as typically accident prediction models are; however, it has some useful interpretation when computed using the equation suggested by [Kvalseth \(1985\).](#page-10-31) According to this criterion, the closer to 1 the *R*<sup>2</sup> coefficient is, the better the model explains the data set.
+- cumulative residuals plot (CURE): the graph consists in plotting the cumulative residuals[1](#page-7-0) versus a variable of interest, showing at a glance how good a fit is, and what the remaining concerns are.
+
+While the first three indicators are the most used single-number measures for generalised linear models able to describe a model overall goodness of fit, the last method (CURE plot) is commonly used for residual analysis in order to identify anomalies in a model, such as the presence of outliers or bias. A good model should have a CURE plot that oscillates around 0 and encroaches the ± 2*σ* limits only rarely ([Hauer,](#page-10-32) [2015\)](#page-10-32). Also, according to Hauer, "long up or down runs indicate regions of bias which demand model improvement", while "large vertical drops invite the examination of outliers".
+
+# 5. Results and discussion
+
+# 5.1. Base models
+
+The base models developed for Italian rural freeway segments allow to estimate single- (or multiple-) vehicle fatal-and-injury crashes per year in one direction of travel for sections with base conditions as defined in the HSM model for freeway segments [\(AASHTO, 2014\)](#page-10-10) by means of the equations [\(8\)](#page-7-1) and [\(9\).](#page-7-2)
+
+<span id="page-7-1"></span>SV fatal-injury crashes:
+
+$$N_{spf,sv,fi} = L \times e^{[a+1.393 \times ln(0.002 \times AADT)]}$$
+ (8)
+
+$$\varepsilon_{i,N} = N_{obs,i,N} - N_{pred,i,N}$$
+ (7)
+
+- *<sup>a</sup>* = −7.924 for 2-lane segments
+- *<sup>a</sup>* = −8.008 for 3(or more)-lane segments
+
+<span id="page-7-2"></span>MV fatal-injury crashes:
+
+$$N_{spf,mv,fi} = L \times e^{[a+1.797 \times ln(0.002 \times AADT)]}$$
+ (9)
+
+- *<sup>a</sup>* = −8.304 for 2-lane segments
+- *<sup>a</sup>* = −8.528 for 3(or more)-lane segments
+
+### where:
+
+- Nspf,sv,fi is the predicted average crash frequency (fatal-and-injury accidents) for single-vehicle crashes in rural freeway segments with base conditions, one direction of travel;
+- Nspf,mv,fi is the predicted average crash frequency (fatal-and-injury accidents) for multiple-vehicle crashes in rural freeway segments with base conditions, one direction of travel.
+
+The over-dispersion parameter is k = 0.789 for the SV base model and k = 0.678 for the MV base model.
+
+In order to check the soundness of the estimation, the regression parameters computed for base SPFs single-vehicle and multiple-vehicle fatal-and-injury crashes in one direction of travel are compared to the correspondent ones found in the published literature. In [Table 7](#page-8-0), the estimated regression coefficients are shown together with the ones developed in the HSM. In order to compare the coefficients developed for one-directional models with the HSM bidirectional ones, the Italian a coefficients are corrected inverting Eq. [\(5\).](#page-5-3)
+
+As shown in [Table 7](#page-8-0), the coefficients are consistent with the correspondent ones found in the published literature. The differences in absolute values reflect the different contexts in which the models have been developed.
+
+# 5.2. Evaluation of the goodness of fit of the base models
+
+The overall goodness of fit indicators for the base models are presented in [Table 8](#page-8-1).
+
+For both SV and MV models, the Pearson's χ<sup>2</sup> statistic is less than the corresponding value of χ<sup>2</sup> computed for the respective n–p degrees of freedom, assuming a p-value equal to 0.05. According to these results, the hypothesis that both models fit the data cannot be rejected.
+
+The root mean square error (Se) shows a good capability to describe the respective dataset, for both models.
+
+The evaluation of the coefficient of determinations R2 yields to controversial results. Being both coefficient of determinations R<sup>2</sup> not close to 1, the ability of the models to explain the respective data set is in doubt. However, several authors [\(Fridstrøm et al., 1995;](#page-10-33) [Miaou et al.,](#page-10-34) [1996\)](#page-10-34) expressed reservations about the use of the coefficient of determination as instrument of goodness of fit measure for negative binomial models.
+
+# 5.3. Cumulative residual analysis
+
+In order to perform an additional analysis on the models goodness of fit, identifying the areas which demand for further investigation, the cumulative residuals (CURE) for the base SV and MV models were plotted. The results are presented in [Figs. 4 and 5](#page-8-2).
+
+The graphs show that the functional form assumed for the base models is correct, and that there is no need to include other factors in the fitting process. This was expected as the sections used to develop the base models were selected with similar characteristic and standard conditions; indeed, only the exposure variable was expected to influence the crash occurrence. Both plots correctly oscillate around 0; the SV model encroaches the ± 2*σ* limit only in one point, and for a very small difference, indicating the substantial absence of any bias in both
+
+<span id="page-7-0"></span><sup>1</sup> The residuals are defined as the difference between the number of crashes observed in a segment and the number of crashes predicted by the model. The residual (*εi*,*<sup>N</sup>* ) for the segment i in the analysis period N is calculated with the Eq. (7):
+
+<span id="page-8-0"></span>Table 7 Interpretation of SV and MV regression parameters.
+
+|                                         | Italian SPF freeway model (bidirectional)                                                  |                             | HSM freeway model (bidirectional) |  |  |
+|-----------------------------------------|--------------------------------------------------------------------------------------------|-----------------------------|-----------------------------------|--|--|
+|                                         | SPF Coefficients for SV fatal-injury crashes on bidirectional Rural Freeway                |                             |                                   |  |  |
+| Segments<br>Number of through lanes = 4 |                                                                                            | Number of through lanes = 6 |                                   |  |  |
+| Italy                                   | HSM                                                                                        | Italy                       | HSM                               |  |  |
+| a (*) = -7.231                          | a = -2.126                                                                                 | a (*) = -7.315              | a = -2.055                        |  |  |
+| b = 1.393                               | b = 0.646                                                                                  | b = 1.393                   | b = 0.646                         |  |  |
+|                                         | SPF Coefficients for MV fatal-injury crashes on bidirectional Rural Freeway                |                             |                                   |  |  |
+| Segments                                |                                                                                            |                             |                                   |  |  |
+| Number of through lanes = 4             |                                                                                            | Number of through lanes = 6 |                                   |  |  |
+| Italy                                   | HSM                                                                                        | Italy                       | HSM                               |  |  |
+| a (*) = -7.611                          | a = -5.975                                                                                 | a (*) = -7.835              | a = -6.092                        |  |  |
+| b = 1.797                               | b = 1.492                                                                                  | b = 1.797                   | b = 1.492                         |  |  |
+|                                         | (*) The coefficient a is corrected for the evaluation of a bidirectional segment inverting |                             |                                   |  |  |
+
+caused by the presence of outliers.
+
+## 5.4. Calibration of the full models
+
+The combination of the base models with the CMFs presented in the HSM [\(AASTHO, 2014](#page-10-10)) and the ones developed within the project PRACT allowed to estimate the number of predicted crashes (uncalibrated) for each segment during the evaluation period. In order to account for differences in the general level of crash frequency between the base data set and the calibration data set, two calibration factors were then computed for the SV and MV models. The results of the calibrations are presented in [Table 9.](#page-8-3)
+
+The calibration factors with values greater than "1″ show a tendency of the uncalibrated full models to underestimate the total number of crashes in the full data set, more in the SV model than in the MV model.
+
+These factors are applied to the "uncalibrated" full models to obtain
+
+<span id="page-8-1"></span>Table 8 Goodness of fit of the base models.
+
+Eq. [\(5\)](#page-5-3).
+
+<span id="page-8-2"></span>
+
+|    | Pearson's χ2 statistic<br>χ2 | degrees of freedom<br>df | Reference Pearson's χ2 statistic<br>χ2<br>(0.05,df) | root mean square error<br>Se | coefficient of determination<br>R2 |
+|----|------------------------------|--------------------------|-----------------------------------------------------|------------------------------|------------------------------------|
+| SV | 64.18                        | 112                      | 137.68                                              | 0.177                        | 0.138                              |
+| MV | 291.10                       | 306                      | 347.85                                              | 0.447                        | 0.244                              |
+
+![](_page_8_Figure_11.jpeg)
+
+Fig. 4. CURE plot for SV base model.
+
+![](_page_8_Figure_13.jpeg)
+
+Fig. 5. CURE plot for MV base model.
+
+<span id="page-8-3"></span>Table 9 Calibration factors for the SV and MV full models.
+
+| Npredicted uncalibrated to t | Nobserved tot _ | _<br>Nobserved tot<br>C<br>=<br>_<br>_<br>Npredicted uncalibrated tot |
+|------------------------------|-----------------|-----------------------------------------------------------------------|
+| 614                          | 1042            | 1.696                                                                 |
+| 1,545                        | 1826            | 1.190                                                                 |
+|                              |                 |                                                                       |
+
+models. However, while the CURE plot of the MV base model does not indicate any presence of outliers, the vertical drop recorded in the SV base model in correspondence to an AADT of 40,000 veh/day could be the "calibrated" predictions.
+
+# 5.5. Evaluation of the goodness of fit of the full models
+
+The overall goodness of fit indicators for the calibrated full models are presented in [Table 10.](#page-9-0)
+
+For both SV and MV full models, the Pearson's χ<sup>2</sup> statistic is less than the corresponding value of χ<sup>2</sup> computed for the respective n–p degrees of freedom, assuming a p-value equal to 0.05. According to these results, the hypothesis that both models fit the data cannot be rejected.
+
+<span id="page-9-0"></span>Table 10 Goodness of fit of the full models.
+
+|    | Pearson's χ2 statistic<br>χ2 | degrees of freedom<br>df | Reference Pearson's χ2 statistic<br>χ2<br>(0.05,df) | root mean square error<br>Se | coefficient of determination<br>R2 |
+|----|------------------------------|--------------------------|-----------------------------------------------------|------------------------------|------------------------------------|
+| SV | 563.55                       | 717                      | 780.47                                              | 0.306                        | 0.218                              |
+| MV | 560.48                       | 717                      | 780.47                                              | 0.460                        | 0.271                              |
+
+<span id="page-9-1"></span>![](_page_9_Figure_4.jpeg)
+
+Fig. 6. Observed vs Predicted diagram for SV model.
+
+![](_page_9_Figure_6.jpeg)
+
+Fig. 7. Observed vs Predicted diagram for MV full model.
+
+The root mean square error (Se) shows also a good capability to describe the respective dataset, for both models. The final results of the full model calibrations are shown in [Figs. 6 and 7](#page-9-1) by means of observed vs predicted diagrams.
+
+The graphs show a tendency of both models to under-predict the number of crashes for those sections, where record a number of observed crashes higher than the mean values is recorded, this tendency is more clear in the MV model. These groups of segments, which represent outliers of the calibration data set, should be investigated in more details (e.g. by means of a disaggregate analysis) to understand if variables not considered in this study are involved in the high frequency records.
+
+Another graphical representation that allows to evaluate the presence of some sources of anomalies in the model is the plot of the residuals versus the number of accidents predicted. The residual plots in [Figs. 8 and 9](#page-9-2) show a particular point distribution that is due to the presence of heteroscedasticity, which means that a non-random
+
+<span id="page-9-2"></span>![](_page_9_Figure_11.jpeg)
+
+Fig. 8. Predicted vs Residuals diagram for SV full model.
+
+![](_page_9_Figure_13.jpeg)
+
+Fig. 9. Predicted vs Residuals diagram for MV full model.
+
+distribution of residuals is observed ([La Torre et al., 2014](#page-10-23)), more evident in the SV model. As a matter of fact, increasing in the number of expected crashes tends to increase the residuals. According to [Shankar](#page-10-35) [et al. \(1995\),](#page-10-35) heteroscedasticity is an important issue often associated with the negative binomial models.
+
+# 6. Conclusions
+
+In order to provide the Italian freeway agencies with a tool that allows to deal with potential safety assessments, identifying the sites with the largest potential safety improvements and helping in choosing the best treatment to apply, this paper describes two accident prediction models (APMs) for single- and multiple-vehicle fatal-and-injury crashes. Following the HSM approach, these models are based on jurisdictional specific Safety Performance Functions (SPFs) and a set of CMFs, applying the procedure defined in the PRACT Project.
+
+The accident prediction models were developed using a dataset including information about fatal and injury crashes occurred on 884 km of freeway segments in a 5-year period (2009–2013). The models allowed to estimate fatal-and-injury single- (or multiple-) vehicle crashes per year in one direction of travel. The base SPFs developed have the same base conditions than the ones used in the HSM; this allows to account for differences between a given section and the HSM base conditions by means of CMFs without the need of any further transformations due to different base conditions. In order to improve the reliability of the prediction accounting for differences in sites where the CMFs were developed, a calibration factors (C) was then computed for each APM.
+
+The goodness of fit of the base models and of the full models was investigated by means of Pearson's χ<sup>2</sup> statistic, root mean square error, and residual analysis.
+
+For both SV and MV full models, the Pearson's χ<sup>2</sup> statistic is less than the corresponding value of χ<sup>2</sup> computed for the respective n–p degrees of freedom, assuming a p-value equal to 0.05. According to these results, the hypothesis that both models fit the data cannot be rejected. The root mean square error (Se) shows a good capability to describe the respective dataset, for both models. The observed vs predicted diagrams show a tendency of both full models to under-predict the number of crashes in those sections where the number of observed crashes are higher than the mean values; this tendency is more evident in the MV model. These groups of segments, which represent outliers of the calibration data set, will be investigated in further research by means of a disaggregate analysis to understand if variables not considered in this study are involved in the high frequency records.
+
+The residuals versus the number of predicted accidents diagrams show a particular point distribution that is due to the presence of heteroscedasticity, which means that a non-random distribution of residuals is observed. This tendency, typical of Negative Binomial models, is more evident in the SV model.
+
+The results show a good aptitude of both models to describe the respective data set. The models proposed represent a solid and reliable tool for practitioners to perform accident prediction along the Italian freeway network.
+
+### Acknowledgments
+
+The authors wish to thank the PRACT Project (Predicting Road ACcidents - a Transferable methodology across Europe) funded under the CEDR Cross-border Transnational Road Research Programme 2013: Safety, by the national directors of roads in Belgium/Flanders, Germany, Ireland, Norway, Sweden and the United Kingdom.
+
+### References
+
+- <span id="page-10-7"></span>AASHTO, 2010. Highway Safety Manual, fi[rst edition. American Association of State and](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0005) [Highway Transportation O](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0005)fficials.
+- <span id="page-10-10"></span>[AASHTO, 2014. Highway Safety Manual,](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0010) first edition. 2014 Supplement. American [Association of State and Highway Transportation O](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0010)fficials.
+- <span id="page-10-14"></span>Abdelwahab, H.T., Abdel-Aty, M.A., 2001. Development of artificial neural network models to predict driver injury severity in traffic accident at signalized intersection. Transp. Res. Rec. 1746, 6–13. [https://doi.org/10.3141/1746-02.](https://doi.org/10.3141/1746-02) Transportation [Research Board of National Academies, Washington DC, 2001](https://doi.org/10.3141/1746-02).
+- <span id="page-10-15"></span><span id="page-10-0"></span>[ACI-ISTAT, 2017. Incidenti stradali - anno 2016. Luglio 2017.](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0020)
+- Adeli, H., Karim, A., 2000. Fuzzy-wavelet RBFNN model for freeway incident detection. J. Transp. Eng. 126 (6), 464–471. [https://doi.org/10.1061/\(ASCE\)0733-947X\(2000\)](https://doi.org/10.1061/(ASCE)0733-947X(2000)126:6(464)) [126:6\(464\)](https://doi.org/10.1061/(ASCE)0733-947X(2000)126:6(464)).
+- <span id="page-10-9"></span>Bahar, G., Hauer, E., 2014. User'[s Guide to Develop Highway Safety Manual Safety](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0030) [Performance Function Calibration Factors. NCHRP Project 20-07.](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0030)
+- <span id="page-10-17"></span>Behnood, A., Mannering, F., 2015. The temporal stability of factors affecting driver-injury severities in single-vehicle crashes: some empirical evidence. Anal. Methods Accid. Res. 8, 7–32. <https://doi.org/10.1016/j.amar.2015.08.001>.
+- <span id="page-10-26"></span>[Bonneson, J., Lord, J.D., Zimmerman, K., Fitzpatrick, K., Pratt, M., 2007. Development of](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0040) [Tools or Evaluating the Safety Implications of Highway Design Decisions. Project](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0040) [Report 0-4703-4. Texas Transportation Institute, Texas.](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0040)
+- <span id="page-10-30"></span>[Bonneson, J., Geedipally, S., Pratt, M.P., Lord, D., 2012. Safety Prediction Methodology](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0045) [and Analysis Tool for Freeways and Interchanges. NCHRP Project, pp. 17](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0045)–45 Final
+- <span id="page-10-22"></span>Cafiso, S., D'Agostino, C., 2012. Safety performance function for motorways using generalized estimation equations. Procedia – Social Behave. Sci. 53, 900–909. [https://](https://doi.org/10.1016/j.sbspro.2012.09.939) [doi.org/10.1016/j.sbspro.2012.09.939.](https://doi.org/10.1016/j.sbspro.2012.09.939)
+- <span id="page-10-19"></span>Caliendo, C., Guida, M., Paris, A., 2007. A crash-prediction model for multilane roads. Accid. Anal. Prev. 39, 107–115. [https://doi.org/10.1016/j.aap.2006.10.012.](https://doi.org/10.1016/j.aap.2006.10.012)
+- <span id="page-10-18"></span>Cheng, W., Gill, G.S., Ensch, J.L., Kwong, J., Jia, X., 2018. Multimodal crash frequency
+
+- modelling: multivariate space-time models with alternate spatiotemporal interactions. Accid. Anal. Prev. 113, 159–170. [https://doi.org/10.1016/j.aap.2018.01.034.](https://doi.org/10.1016/j.aap.2018.01.034)
+- <span id="page-10-4"></span>[Eenink, R., Reurings, M., Elvik, R., Cardoso, J., Wichert, S., Christian, S., 2005. Accident](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0065) [prediction models and road safety impact assessment: recommendations for using](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0065) [these tools. Ripcord 506184 \(January\), 1](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0065)–20.
+- <span id="page-10-33"></span>Fridstrøm, L., Ifver, J., Ingebrigtsen, S., Kulmala, R., Thomsen, L.K., 1995. Measuring the contribution of randomness, exposure, weather, and daylight to the variation in road accident counts. Accid. Anal. Prev. 27 (1), 1–20. [https://doi.org/10.1016/0001-](https://doi.org/10.1016/0001-4575(94)E0023-E) [4575\(94\)E0023-E.](https://doi.org/10.1016/0001-4575(94)E0023-E)
+- <span id="page-10-6"></span>[Hauer, E., 1997. Observational Before-After Studies in Road Safety: Estimating the E](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0075)ffect of Highway and Traffic [Engineering Measures on Road Safety. Emerald ISBN:](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0075) [9780080430539](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0075).
+- <span id="page-10-12"></span>Hauer, E., 2001. Overdispersion in modelling accidents on road sections and in Empirical Bayes estimation. Accid. Anal. Prev. 33, 799–808. [https://doi.org/10.1016/S0001-](https://doi.org/10.1016/S0001-4575(00)00094-4) [4575\(00\)00094-4](https://doi.org/10.1016/S0001-4575(00)00094-4).
+- <span id="page-10-32"></span>[Hauer, E., 2015. The Art of Regression Modeling in Road Safety. Ed. Springer ISBN: 978-](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0085)
+- <span id="page-10-27"></span>IBM, 2015. SPSS Statistic 23. Available at:. , last access 30/10/2017. [http://www-01.](http://www-01.ibm.com/support/docview.wss?uid=swg24038592) [ibm.com/support/docview.wss?uid=swg24038592](http://www-01.ibm.com/support/docview.wss?uid=swg24038592).
+- <span id="page-10-24"></span>Karathodorou, N., Graham, D., Hu, J., Richter, T., Ruhl, S., Yannis, G., Dragomanovits, A., Laiou, A., La Torre, F., Domenichini, L., 2016. Development of a crash modification factors model within the PRACT project. Proceedings of the Road Safety in 5 Continents Conference, Brasil. Available at. [https://www.vti.se/en/Publications/](https://www.vti.se/en/Publications/Publication/development-of-a-crash-modification-factors-model-_926147) [Publication/development-of-a-crash-modi](https://www.vti.se/en/Publications/Publication/development-of-a-crash-modification-factors-model-_926147)fication-factors-model-\_926147.
+- <span id="page-10-31"></span>Kvalseth, T.O., 1985. Cautionary note about R2. Am. Stat. 39, 279–285. [https://doi.org/](https://doi.org/10.2307/2683704) [10.2307/2683704.](https://doi.org/10.2307/2683704) ISSN: 0003-1305.
+- <span id="page-10-23"></span>La Torre, F., Domenichini, L., Corsi, F., Fanfani, F., 2014. Transferability of the highway safety manual freeway model to the italian motorway network. Transp. Res. Rec. (2435), 61–243571. [https://doi.org/10.3141/2435-08.](https://doi.org/10.3141/2435-08) Transportation Research [Board of National Academies, Washington DC, 2014.](https://doi.org/10.3141/2435-08)
+- <span id="page-10-25"></span>La Torre, F., Domenichini, L., Meocci, M., Graham, D., Karathodorou, N., Richter, T., Ruhl, S., Yannis, G., Dragomanovits, A., Laiou, A., 2016. Development of a transnational accident prediction model. Transp. Res. Procedia 14, 1772–1781. [https://doi.](https://doi.org/10.1016/j.trpro.2016.05.143) [org/10.1016/j.trpro.2016.05.143.](https://doi.org/10.1016/j.trpro.2016.05.143)
+- La Torre, F., Domenichini, L., Nocentini, A., 2017. Effect of stationary work zones on motorway crashes. Saf. Sci. 92, 148–159. [https://doi.org/10.1016/j.ssci.2016.10.](https://doi.org/10.1016/j.ssci.2016.10.008) [008](https://doi.org/10.1016/j.ssci.2016.10.008).
+- <span id="page-10-28"></span>[La Torre, F., Meocci, M., Domenichini, L., Nocentini, A., 2018. Safety e](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0120)ffects of automatic [section speed control on italian motorway network. Submitted to Journal of Safety](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0120) [Research August, 2018.](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0120)
+- <span id="page-10-21"></span>Martinelli, F., La Torre, F., Vadi, P., 2009. Calibration of the highway safety manual's accident prediction model for italian secondary road network. Transp. Res. Rec. 2103, 1–9. [https://doi.org/10.3141/2103-01.](https://doi.org/10.3141/2103-01) Transportation Research Board of [National Academies, Washington DC, 2009](https://doi.org/10.3141/2103-01).
+- <span id="page-10-29"></span>[McCullagh, P., Nelder, J.A., 1993. Generalized Linear Models, second editions. London](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0130) [New York Chapman & Hall ISBN: 13:9780412317606.](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0130)
+- <span id="page-10-34"></span>Miaou, S.-P., Lu, A., Lum, H., 1996. Pitfalls of using r 2 to evaluate goodness of fit of accident prediction models. Transp. Res. Rec. 1542, 6–13. [https://doi.org/10.3141/](https://doi.org/10.3141/1542-02) 1542-02. [Transportation Research Board of National Academies, Washington DC,](https://doi.org/10.3141/1542-02) [1996.](https://doi.org/10.3141/1542-02)
+- <span id="page-10-20"></span>Montella, A., Colantuoni, L., Lamberti, R., 2008. Crash prediction models for rural motorways. Transp. Res. Rec. 2083, 180–189. [https://doi.org/10.3141/2083-21.](https://doi.org/10.3141/2083-21) [Transportation Research Board of National Academies, Washington DC, 2008](https://doi.org/10.3141/2083-21).
+- <span id="page-10-13"></span>Persaud, B., Lyon, C., Nguyen, T., 1999. Empirical Bayes procedure for ranking sites for safety investigation by potential for safety improvement. Transp. Res. Rec. 1665, 7–12. [https://doi.org/10.3141/1665-02.](https://doi.org/10.3141/1665-02) Transportation Research Board of National [Academies, Washington DC, 1999](https://doi.org/10.3141/1665-02).
+- <span id="page-10-16"></span>Roque, C., Cardoso, J.L., 2014. Investigating the relationship between run-off-the-road crash frequency and traffic flow through different functional form. Accid. Anal. Prev. 63, 121–132. <https://doi.org/10.1016/j.aap.2013.10.034>.
+- <span id="page-10-35"></span>Shankar, V., Mannering, F., Barfield, W., 1995. Effect of roadway geometrics and environmental factors on rural freeway accident frequencies. Accid. Anal. Prev. 27, 371–389. [https://doi.org/10.1016/0001-4575\(94\)00078-Z](https://doi.org/10.1016/0001-4575(94)00078-Z).
+- <span id="page-10-5"></span>[Srinivasan, R., Bauer, K., 2013. Safety Performance Function Development Guide:](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0160) Developing Jurisdiction-Specifi[c SPFs, Final Report. Report No. FSWA-SA-14-005.](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0160)
+- <span id="page-10-8"></span>[Federal Highway Administration September 2013.](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0160) [Srinivasan, R., Carter, D., Bauer, K., 2013. Safety Performance Function Decision Guide:](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0165) [SPF Calibration Vs SPF Development. U.S. Department of Transportation. FHWA-SA-](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0165)[14-004. Federal Highway Administration September 2013.](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0165)
+- <span id="page-10-1"></span>United Nations Road Safety Collaboration, 2011. Global Plan for the Decade of Action for Road Safety 2011–2020. Available at:. Geneva: WHO, pp. 25. [http://www.who.int/](http://www.who.int/roadsafety/decade_of_action/plan/en/) [roadsafety/decade\\_of\\_action/plan/en/](http://www.who.int/roadsafety/decade_of_action/plan/en/).
+- <span id="page-10-2"></span>Yannis, G., Dragomanovits, A., Laiou, A., Richter, T., Ruhl, S., La Torre, F., Domenichini, L., Graham, D., Karathodorou, N., Li, H., 2016a. Use of Prediction model in road safety management – an international inquiry. Transportation Research Procbahahaedia 14, 4257–4266. [https://doi.org/10.1016/j.trpro.2016.05.397.](https://doi.org/10.1016/j.trpro.2016.05.397)
+- <span id="page-10-3"></span>[Yannis, G., Dragomanovits, A., Laiou, A., La Torre, F., Domenichini, L., Richter, T., Ruhl,](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0180) [S., Graham, D., Karathodorou, N., 2016b. Development of an online repository of](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0180) [accident prediction models and crash modi](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0180)fication factors. 1st European Road [Infrastructure Congress.](http://refhub.elsevier.com/S0001-4575(18)31220-X/sbref0180)
+- <span id="page-10-11"></span>Yannis, G., Dragomanovits, A., Laiou, A., La Torre, F., Domenichini, L., Richter, T., Ruhl, S., Graham, D., Karathodorou, N., 2017. Road traffic accident prediction modelling: a literature review. Proceedings of the Institution of Civil Engineers – Transport, Vol. 170, Issue 5 245–254. <https://doi.org/10.1680/jtran.16.00067>.

Zotero/001_artiklid/Road Safety Indicators/liexploringimpactbuiltenvironment2025.md

@@ -0,0 +1,556 @@
+---
+category: literaturenote
+citekey: liexploringimpactbuiltenvironment2025
+title: Exploring the impact of built environment on crash risks at transportation hubs
+authors: "Li, Chuanyao; Chen, Li"
+year: 2025
+date: 2025-00-00 2025
+doi: 10.1016/j.aap.2025.108079
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2025.108079"
+zotero_key: C5P7NFTK
+zotero_storage: E2W6TAR9
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Li, Chuanyao"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+Contents lists available at [ScienceDirect](www.sciencedirect.com/science/journal/00014575)
+
+# Accident Analysis and Prevention
+
+journal homepage: [www.elsevier.com/locate/aap](https://www.elsevier.com/locate/aap)
+
+![](_page_0_Picture_5.jpeg)
+
+![](_page_0_Picture_6.jpeg)
+
+# Exploring the impact of built environment on crash risks at transportation hubs
+
+Chuanyao Li \* , Li Chen
+
+*School of Traffic and Transportation Engineering, Central South University, Changsha 410075, China*
+
+ARTICLE INFO
+
+*Keywords:* Transportation hub Built environment Crash risk Geographically weighted regression
+
+#### ABSTRACT
+
+This study investigates the impact mechanism of the built environment surrounding transportation hubs on crash risks (CR). Three buffer zones (300 m, 500 m, and 800 m) are defined as the spatial analysis units, and Ordinary Least Squares (OLS), Geographically Weighted Regression (GWR), and Multiscale Geographically Weighted Regression (MGWR) are utilized in this study. The results reveals that the 800 m buffer zone provides deeper insights into the factors affecting CR related to the built environment surrounding transportation hubs. Additionally, MGWR demonstrates superior performance in explaining the built environment's impact on CR compared to the other two methods, with an explanation rate of 83.7 %. To reduce CR near transportation hubs, rationally planning the surrounding land use layout and reducing population density per unit area are recommended. Moreover, the density of road networks surrounding airports and railway stations should be kept at a lower level to reduce CR. The findings of this study contribute to a deeper understanding of the relationship between the built environment surrounding transportation hubs and crashes, providing planning guidance and creating a friendly environment surrounding transportation hubs.
+
+#### **1. Introduction**
+
+The transportation hubs, including airports, railway stations, and bus stations, serve as vital nodes within the urban transportation system. They facilitate the connection between various modes of transport and the movement of passengers and goods (Zhu et al. [2021,](#page-13-0) Yin et al. 2023). Nevertheless, integrating diverse transport modes and complex land uses surrounding transportation hubs leads to heightened traffic density and conflicts among heterogeneous traffic flows. Consequently, this increases the likelihood of vehicle crashes ([Yasmin](#page-13-0) et al. 2018, Ali et al. [2023\)](#page-13-0). In 2017–––2018, 6042 crashes in Maryland occurred 300 m near transportation hubs (MVA [2023\)](#page-13-0). These findings prompt consideration of the impact of the built environment surrounding transportation hubs on crash occurrences, in addition to traditional factors that scholars often focus on, such as road users, road design, and vehicle safety ([Jin](#page-13-0) et al. [2024,](#page-13-0) Zou et al. 2024).
+
+Scholars have extensively studied the influence of the built environment, denoting the overall surroundings of a developed city or area, on road safety (Xin et al. [2017,](#page-13-0) Lyu et al. 2023). Particularly, focused attention has been given to how the built environment affects crash risk, including the impact of factors like intersection density, building density, and mixed land use on bicycle accidents and overall road network safety (Asadi et al. [2022,](#page-13-0) Xiao et al. 2024). Interestingly, scholarly investigations on transportation hub safety have mainly concentrated on broader indoor safety aspects, encompassing fire safety and pedestrian evacuation procedures, among other considerations (Salarian et al. 2020, [Shiwakoti](#page-13-0) et al. 2020). Nevertheless, the impact of the external built environment at transportation hubs on the frequency and severity of traffic accidents still needs to be explored, revealing significant research gaps. Understanding how the external built environment surrounding transportation hubs influences crash risk is crucial, as this knowledge can empower authorities to devise evidencebased interventions and policies to reduce crash probability and enhance external traffic safety at these hubs.
+
+Given the existing research limitations, this study aims to investigate how the built environment influences crash risk (CR) at transportation hubs. The Maryland open crash dataset and additional open datasets delineated various buffer zones surrounding transportation hubs. Accident frequencies within these zones and built environment variables were analyzed. Three models—Ordinary Least Squares (OLS), Geographically Weighted Regression (GWR), and Multiscale Geographically Weighted Regression (MGWR)—were employed and
+
+*E-mail addresses:* [chuanyaoli@csu.edu.cn](mailto:chuanyaoli@csu.edu.cn) (C. Li), [chenlijt@csu.edu.cn](mailto:chenlijt@csu.edu.cn) (L. Chen).
+
+<sup>\*</sup> Corresponding author.
+
+compared across the buffer zones. The discussion focuses on the MGWR regression coefficients, offering insights from diverse spatial perspectives and comparing different hub types. The primary objective of this research is to pinpoint specific built environment factors contributing to crash risk at transportation hubs. By comprehensively examining how the built environment of transportation hubs influences peripheral crash risk, this study provides valuable insights for urban planners, policymakers, and traffic management authorities.
+
+This study is outlined as follows. Section 2 presents the literature review, acknowledging the relationship between the built environment and traffic safety, along with the common methodologies employed to establish this connection. Section 3 encompasses the methodology, detailing the study region, data sources, data processing methods, and an in-depth explanation of both global and local regression models utilized. Section 4 consists of the results, which incorporates spatial analysis of crash risk and the outcomes from the modeling. [Section](#page-8-0) 5 is discussion, including the contributions, policy implications, and limitations of this study. [Section](#page-12-0) 6 is conclusion,summarizes the main findings of the study and suggests directions for future research.
+
+# **2. Literature review**
+
+#### *2.1. The impact of built environments on traffic safety*
+
+Numerous studies have established strong correlations between various built environments and traffic safety ([Merlin](#page-13-0) et al. 2020, Asadi et al. [2022](#page-13-0)). These correlations involve bicycle crashes and the built environment of streets, student commuting safety and its built surroundings, and roadway network safety and the built environment. Specifically, Costa et al. [\(2024\)](#page-13-0) demonstrated the connection between bicycle accidents and the type of built environment using a latent class discrete outcome model. Their findings highlighted that busy intersections and a small percentage of road surface area increase the likelihood of crashes and influence crash severity; Wu et al. [\(2023\)](#page-13-0) delved into the interrelation between the built environment and crash risks for school-commuting children. Their research emphasized the importance of factors like lower road network densities, rational educational facility layouts, reduced bus routes, and increased on-street parking spaces in mitigating crash risks; Xiao et al. [\(2024\)](#page-13-0) explored how the built environment and traffic flow impact crash frequency on urban road networks. Their study revealed that variables such as employment density, road density, land use diversity, and public transport accessibility significantly impact crash occurrences. Moreover, attention has been directed towards examining the correlation between transportation safety and the built environment for older people ([Lee](#page-13-0) et al. [2020\)](#page-13-0) and investigating the impact of the built environment on children's traffic security ([Macedo](#page-13-0) et al. 2023).
+
+While existing studies have extensively explored the correlation between various types of built environments and crash risk, there needs to be more research investigating the link between the built environment surrounding transportation hubs and crashes. Most prominently, research on transportation hub safety has predominantly focused on the indoor aspects of the hub, encompassing passenger safety [\(Xie](#page-13-0) et al. [2022\)](#page-13-0), safety facility design (Xie et al. [2023](#page-13-0)), and emergency evacuation and escape of crowds ([Alam](#page-13-0) et al. 2023). Therefore, there remains a substantial knowledge gap concerning the external safety of transport hubs, specifically regarding how the surrounding built environment influences crash risk.
+
+#### *2.2. Modeling approaches to link the built environments and traffic safety*
+
+There are two primary modelling categories in exploring the link between the built environment and traffic safety: machine learning methods and traditional regression modelling. Machine learning methods leverage extensive datasets for training models to comprehend the influence of built environment factors on traffic safety. This method offers the advantage of handling the intricate, non-linear connections and uncovering nuanced patterns between the built environment and traffic safety (Bi et al. 2023, [Galvez-P](#page-13-0) ´ ´erez et al. 2023). These models can forecast the impact of changes in the built environment on traffic safety to inform decision-making processes ([Chang](#page-13-0) et al. 2022, Costa et al. [2024\)](#page-13-0). However, machine learning methods necessitate substantial quantities of high-quality data for effective model training, and the absence of inadequacy of data may compromise model performance (Zhu et al. [2023\)](#page-13-0). Furthermore, specific deep learning models exhibit a profound black-box nature that hinders a comprehensive understanding of the correlation between the built environment and traffic safety, including the spatial relationships between the built environment and crash risk.
+
+Another category is traditional regression modeling. The first is a global model that does not take scale effects into account, including OLS, Generalized Linear Models (GLM), and binary logit models [\(Umair](#page-13-0) et al. 2022, AlKheder 2023, Soltani and Roohani [Qadikolaei](#page-13-0) 2024). Although these models can identify potential control factors impacting traffic safety due to the built environment, they fail to address spatial dependencies. To address this limitation, GWR models are recognized for their spatial analysis capability, effectively illustrating the spatial relationships between dependent and independent variables [\(Gao](#page-13-0) et al. [2024,](#page-13-0) Liu et al. 2024). However, standard GWR models assume uniform spatial interactions at the same scale, applying a single bandwidth across all variables. This uniformity restricts the model's ability to adequately capture variations across multiple scales. Therefore, [Fotheringham](#page-13-0) et al. [\(2017\)](#page-13-0) introduced MGWR based on a generalized additive model. MGWR allows for individual spatial smoothing levels for each covariate, utilizing optimal bandwidths for each independent variable. This customization of spatial scales optimizes the modeling of each spatial process, enhancing the model's explanatory power and realism [\(Duan](#page-13-0) et al. [2024](#page-13-0)). In MGWR, the model employs the golden search algorithm to determine the most suitable bandwidths for diverse variables in regression modeling. This approach minimizes discrepancies between bandwidths and variable action scales, thereby reducing noise in parameter estimation (Zhao et al. [2024\)](#page-13-0). MGWR has found applications in interdisciplinary studies encompassing environmental protection, urban landscapes, and public transport (Tang et al. [2023,](#page-13-0) Wang et al. [2023,](#page-13-0) Ma et al. 2024).
+
+#### *2.3. Research summary*
+
+In conclusion, while existing studies have extensively examined the relationship between built environments and traffic safety, there has been limited focus on the external safety of transportation hubs, particularly regarding how the surrounding built environment influences crash risk. Furthermore, most prior research has relied on either local or global spatial models, with insufficient attention to multi-scale spatial interactions. This study's contribution is to addresses these gaps by employing Multi-Scale Geographically Weighted Regression (MGWR) to analyze the Maryland crash dataset and other open-source datasets. By investigating the built environment's multi-scale spatial effects on crashes around transportation hubs, this study not only advances the theoretical understanding of the spatial dynamics between built environments and traffic safety but also provides practical recommendations for optimizing the external built environment of transportation hubs. These findings contribute to the existing body of knowledge and offer decision-makers actionable insights to enhance transportation safety in critical urban areas.
+
+# **3. Methodology**
+
+### *3.1. Study area and data source*
+
+The State of Maryland is in the Mid-Atlantic region of the United States and is also known as the "Free State". The state includes 23 <span id="page-2-0"></span>counties and one independent city, with the largest city being Baltimore and the capital being Annapolis. According to the U.S. Census Bureau's 2022 data, the state has 6, 165, 000 residents, making it one of the smallest states in the U.S. in terms of land area and one of the most densely populated.<sup>1</sup> Maryland's economy is predominantly tertiary, with transportation, centered on the Port of Baltimore being the most critical component of Maryland's economy.2 As a result, Maryland is not only home to world-class ports such as the Port of Baltimore and a welldeveloped network of roadways but also includes the Baltimore-Washington International Thurgood Marshall Airport (BWI), as well as significant air and rail carriers such as Amtrak, the Maryland Area Regional Commuter Railroad (MARC), and Washington, D.C.'s Metropolitan Area Rapid Transit (MART) system.<sup>3</sup> Overall, Maryland is very well connected.
+
+This study uses transportation hub data, administrative district data, and built environment data from Open Street Map (OSM).4 OSM is an open and accessible geographic information dataset from volunteers worldwide and is widely used in many studies. The data at the national level of the United States was acquired from OSM and subsequently subjected to data processing using ArcGIS 10.5 software. Finally, the dataset about the state of Maryland was exported. This includes multiple layers such as buildings, land use, POIs, transport, etc. We extracted geographic data from the transport layer for 208 transportation hubs, specifically 110 railway stations, 69 airports, 24 ferry terminals, and 5 bus stations. Their geographic distribution in Maryland is shown in [Fig.](#page-3-0) 1.
+
+In addition, a substantial dataset was obtained from the Open Data Portal of the Maryland State Government,5 which provides official and publicly accessible data for developing and studying traffic crashes. This dataset comprises 187, 405 crash records from January 1, 2017, to December 31, 2018. The original dataset consisted of 38 fields, including the location (down to the lane level), type of crash, and time of day each accident occurred. For this study, we selected the necessary fields, which include five key variables: date of the crash, type of crash, and geographic coordinates (including latitude and longitude), and assigned a unique identifier to each data record. The locations of the crash points are presented in [Fig.](#page-3-0) 2.
+
+### *3.2. Spatial scale*
+
+This study investigated how the built environment surrounding a transportation hub influences its CR. Therefore, we used the buffer zone as the basic spatial unit of analysis. Specifically, a buffer zone is established by designating a specific radius centered on the transportation hub. However, there is an important issue that needs to be considered: the modifiable areal unit problem (MAUP), which affects the interpretation of the results by the model, as different spatial division methods and scale choices have a significant impact on the results of the analyses ([Clark](#page-13-0) and Scott 2014). Equally important, another concern is that the choice of multiple spatial scales can provide information on the built environment at different levels and scales. This helps to gain a comprehensive understanding of the factors influencing the occurrence of traffic accidents (Nigg et al. [2022](#page-13-0)). The study by Jun et al. [\(2015\)](#page-13-0) states that the regular grid scale division ranges from 300 m to 1000 m, reducing the impact of MAUP. Therefore, to reduce the effect of MAUP and to capture multi-level information, we refer to the studies of [Su](#page-13-0) et al. [\(2022\)](#page-13-0) and Liu et al. [\(2023\)](#page-13-0) and choose 300 m, 500 m, and 800 m as the spatial analysis units for this study.
+
+# *3.3. Measuring the variables*
+
+# *3.3.1. Measurement of the dependent variable: crash risk of transportation hubs*
+
+Risk (*R*) is defined as the likelihood and potential impact of hazardous events, which can be understood as the product of the likelihood (*L*) of an event occurring and the consequences (*C*), denoted as *R* = *L* × *C* ([Youli](#page-13-0) et al. 2018). Therefore, based on the risk theory, in this study, the crash risk (*CR*) of the buffer zone around the transportation hub is defined by the crash likelihood (*CL*) and crash severity (*CS*), which can be expressed as *CR* = *CL* × *CS* ([Merlin](#page-13-0) et al. 2020, Wu et al. 2023). Further, considering the accessibility of the crash dataset, we adjusted the use of the absolute number of crash occurrences to denote the likelihood of occurrence (i.e., the higher the number of crashes occurring, the higher the likelihood of a crash occurring as well), for model development and result analysis. We have considered three different crash types and propose the quantitative crash risk assessment model for the buffer zone of a transportation hub, which was integrated based on the number of crashes occurring around transportation hubs. The equation is shown below:
+
+$$CR_i = \alpha P_i + \beta I_i + \gamma F_i \tag{1}$$
+
+In Eq. (1), *CRi* denotes the crash risk equivalent of the buffer *i*. *Pi* is the number of "Property Damage Crash" crash types in buffer *i*; *Ii* is the number of "Injury Crash" accident types in buffer *i*; and *Fi* is the number of "Fatal Crash" accident types in buffer *i*. *α*, *β* and *γ* are severity weights for property damage crash, injury crash, and fatal crash, respectively. Due to the limitations of our crash dataset, which did not include information on property damage (in US dollars), injuries, and fatalities per crash, we relied on the study conducted by Wu et al. [\(2023\)](#page-13-0) as a reference. They utilized crash data from Shenzhen and employed structural equation modeling to develop a measurement model that establishes the relationship between school commuting crash risk and various measures, including the number of fatalities, serious injuries, slight injuries, and direct property damages. We acknowledge that the severity weights determined by Wu et al. [\(2023\)](#page-13-0) differ from this study in the exact coefficient values. Nevertheless, Wu et al. [\(2023\)](#page-13-0) confirmed the coefficients of fatal crashes were higher than those for injury crashes, followed by property damage crashes. Therefore, considering their findings, we adopted the estimated coefficients for *α* = 0*.*22*, γ* = 0*.*52 from Wu et al. [\(2023\)](#page-13-0) in this study. It should be noted that they differentiate between injury crash types, including slight and serious injuries, and we averaged the coefficients for both, so *β* = 0*.*33.
+
+# *3.3.2. Measurement of independent variables: metrics of the built environment of transportation hubs*
+
+The most commonly employed framework for describing the urban built environment is the 5D conceptual model proposed by [Ewing](#page-13-0) and [Cervero](#page-13-0) (2010), specifically density, diversity, design, destination accessibility, and distance. Based on this framework, the availability of the dataset and the characteristics of the POIs data were combined to select 13 metrics describing the built environment surrounding the transportation hub from three dimensions: land use, transport system, and urban design in this study. The specific details of these metrics are presented in [Table](#page-4-0) 1, which includes the names, symbols, and measurement methods.
+
+Due to space constraints, we highlight the descriptive statistics of the built environment of the total sample (N = 208) of transportation hubs at three spatial scales: 300 m, 500 m, and 800 m, as shown in [Tables](#page-4-0) 2-4, respectively. The differences in the built environment of transportation hubs at different spatial scales can be observed by comparison. As the buffer ring expands, the indicator values for many density types decrease. However, the BSD did not change within the different buffer zones, which is because the expansion of the buffer zone did not change the accessibility measurements. Moreover, multicollinearity diagnostics
+
+<sup>1</sup> Data from: <https://www.census.gov/>.
+
+<sup>2</sup> Data from: <https://zh.wikipedia.org/wiki/Maryland>.
+
+<sup>3</sup> Data from: [https://mdot.maryland.gov/pages/home.aspx#](https://mdot.maryland.gov/pages/home.aspx%23).
+
+<sup>4</sup> OSM website: <https://www.openstreetmap.org/>.
+
+<sup>5</sup> [https://opendata.maryland.gov/.](https://opendata.maryland.gov/)
+
+<span id="page-3-0"></span>![](_page_3_Figure_2.jpeg)
+
+**Fig. 1.** Geographic distribution of transportation hubs.
+
+![](_page_3_Figure_4.jpeg)
+
+**Fig. 2.** Geographic distribution of crash data.
+
+were performed by calculating the Variance Inflation Factor (VIF). The results showed no multicollinearity (Tolerances *>* 0.1, VIF *<* 5) was observed in the selected 13 variables, allowing us to proceed to the following analysis step. Itshould be noted that large orders of magnitude between different variables can lead to intrinsic anisotropy in nature, further blurring the relationship between the independent and the dependent variables. In this regard, we performed the next step of the model analysis with logarithmic changes in 2 high-level variables: population density and road intersection density.
+
+# *3.4. Measuring the method*
+
+The three models were used for the modeling analysis include OLS, GWR, and MGWR in this study.
+
+OLS is a global model that does not consider spatial effects, and its mathematical expression is shown in Eq. (2):
+
+$$y_i = \alpha_0 + \sum_m \alpha_m x_{im} + \varepsilon_i \tag{2}$$
+
+In Eq. (2), *yi* is the dependent variable, which is the CR of transportation hub *i*(*i* = 1*,* 2*,* ⋯*,* 208); *xim* denotes the independent variable is the built environment variable *m* for transportation hub *i*, which represents the built environment variables of transportation hub *i*; *α*0*, α<sup>m</sup>* are the regression coefficients; *ε<sup>i</sup>* is the random error in Eq. (2). In this study, OLS serves a dual purpose. It not only captures the global impact of the respective variables on the dependent variable, but also plays a crucial role in assessing the presence of multicollinearity in these variables.
+
+In this regard, we introduced GWR, whose regression coefficients vary with geographic location, allowing for a more spatial interpretation, and whose mathematical expression is shown in Eq. (3):
+
+$$y_i = \beta_0(\sigma_i, \tau_i) + \sum_{r=1} \beta_r(\sigma_i, \tau_i) x_{ir} + \epsilon_i$$
+(3)
+
+In Eq. (3), *yi* has the same meaning as Eq. (2); (*σi, τi*) is the longitude and latitude of transportation hub *i*; *β*0(*σi, τi*) is the constant term of transportation hub *i*;*βr*(*σi, τi*) is the coefficient of the built environment variable *r* for transportation hub *i*; *xir* is the built environment variable *r* for transportation hub *i*; ∊*<sup>i</sup>* represents uncaptured variation in random error and spatial effects.
+
+As we have illustrated in the literature review, in comparison to the GWR, the MGWR regresses each independent variable using its optimal bandwidth, personalizing the spatial scale at which each spatial process operates, resulting in a spatial process model that is closer to the real one and has more explanatory power. The mathematical formulation of the
+
+<span id="page-4-0"></span>**Table 1** Definitions and measurements of variables used to describe the built environment.
+
+| Variable (Symbol)                             | Measurement (Unit)                                                              |
+|-----------------------------------------------|---------------------------------------------------------------------------------|
+| Enterprise office POI density<br>(EO)         | Number of enterprise office POIs in the buffer<br>zone /area (#/km2<br>)        |
+| Leisure and entertainment POI<br>density (LE) | Number of leisure and entertainment POIs in the<br>buffer zone/area (#/km2<br>) |
+| Tourism POI density (TOUR)                    | Number of tourism POIs in the buffer zone/area<br>(#/km2<br>)                   |
+| Life service POI density (LS)                 | Number of life service POIs in the buffer zone/<br>area (#/km2<br>)             |
+| Shopping center POI density<br>(SC)           | Number of shopping center POIs in the buffer<br>zone/area (#/km2<br>)           |
+| Dining POI density (DIN)                      | Number of dining POIs in the buffer zone/area<br>(#/km2<br>)                    |
+| Bus stop accessibility (BSA)                  | Distance from the hubs to the nearest bus stop in<br>the buffer zone (km)       |
+| Bus stop density (BSD)                        | Number of bus stops in the buffer zone /area<br>(#/km2<br>)                     |
+| Traffic signal density (TSD)                  | Number of traffic signals in the buffer zone/area<br>(#/km2<br>)                |
+| Road intersection density (RID)               | Number of road intersections in the buffer zone/<br>area (#/km2<br>)            |
+| Road network density (RND)                    | Sum of road network lengths in the buffer zone/<br>area (km/km2<br>)            |
+| Land using mixing degree<br>(LUM)             | ∑N<br>2<br>(Xij/Xi)<br>j=1                                                      |
+| Population density (PD)                       | Number of residents in the buffer zone / areas<br>(#/km2<br>)                   |
+|                                               |                                                                                 |
+
+**Note:** 1. The "area" refers to the size of different spatial unit; 2. The Herfindahl-Hirschman Index (HHI) is used for "land use mixing degree"; the smaller the HHI value, the higher the mixing, and vice versa. *Xi* is the number of total POIs in the buffer *i*; *Xij* is the number of total POIs in category *j* in the buffer *i*.
+
+MGWR model is shown in Eq. (4):
+
+$$y_i = \beta_0(\sigma_i, \tau_i) + \sum_{j=1}^n \beta_{bwj}(\sigma_i, \tau_i)x_{ij} + \epsilon_i$$
+(4)
+
+In Eq. (4), *bwj* represents the bandwidth used for the regression coefficients of the variable *j*; *βbwj*(*σi, τi*) denotes the regression coefficient of the variable *j* of transportation hub *i* under the bandwidth *bwj*; *xij* is the built environment variable *j* of transportation hub *i*. Refer to Eq. [\(3\)](#page-3-0) for the meaning of the other terms.
+
+#### **4. Results**
+
+### *4.1. Spatial dependence and heterogeneity analysis of CR in transportation hubs*
+
+According to Eq. [\(1\),](#page-2-0) the CR values of each transportation hub were calculated under various buffers. The spatial distribution of these values is displayed in [Fig.](#page-6-0) 3. The transportation hubs with higher CR are predominantly located in downtown Baltimore, Montgomery, and Prince George's areas immediately adjacent to Washington, DC. And some of the more remote transportation hubs have lower CR. However, we combine this with [Fig.](#page-3-0) 1 to show that the CR of railway station hubs located in urban areas is higher than that of other hubs. Note that we only show the spatial distribution of CR in this section, which has no effect on the results of CR as a dependent variable.
+
+To explore the spatial dependence of CR at transportation hubs, we calculated the Moran's index under different buffers, as shown in [Table](#page-7-0) 5. The 300 m buffer zone was better at catching the spatial dependence of CR of transportation hubs when hub diversity was not considered (all transportation hubs, *N* = 208). Whereas, as the buffer zone expands, the value of Moran's index becomes smaller, implying that the spatial dependence of the CR of the transportation hub decreases, and all statistical indicators pass the significance test (*P <* 0*.*050, *Z <* 1*.*923). The total sample demonstrated good spatial dependence within any buffer zone. This further shows that it is reasonable to utilize 300 m, 500 m, and 800 m buffer zones as the spatial unit of study to investigate the impact mechanisms of CR at transportation hubs.
+
+In addition, we calculated Moran's index separately for railway station hubs and airport hubs under different buffer ranges. The Moran's index for ferry and bus stations could not be calculated due to insufficient sample size (*N <* 30). Moran's index of CR for train stations and airports is positive in any buffer and passes the significance test (*P <* 0*.*050, *Z <* 1*.*378). It shows the favorable spatial dependence of different transportation hubs CR within the 300 m, 500 m, and 800 m buffer zones. The variability of factors influencing the CR of different transportation hubs can be further explored in subsequent studies.
+
+Lastly, the Theil Index was introduced to materialize the spatial heterogeneity of the CR. Lower values of the Theil Index indicate minimal heterogeneity among buffers. [Fig.](#page-8-0) 4 depicts the results of the Theil index, which illustrates the spatial heterogeneity of CR. The airport CR exhibits the highest Theil index among buffers of the same size, while the Theil index of the total sample of CR is the lowest, suggesting that CR heterogeneity in space is most obvious in airport hubs. Comparing buffers across different spatial scales, transportation hub CR within the 300 m buffer demonstrates a higher Theil index, indicating more significant heterogeneity than the other buffers.
+
+**Table 2** Results of descriptive statistics and multicollinearity diagnostics for built environment variables in 300 m buffer zone.
+
+| Variables        | Obs | Mean     | Std.Dev. | Min   | Max       | Multi-collinearity diagnosis |       |
+|------------------|-----|----------|----------|-------|-----------|------------------------------|-------|
+|                  |     |          |          |       |           | Tolerances                   | VIF   |
+| PD (#/km2<br>)   | 208 | 1355.483 | 2119.259 | 0     | 11840.010 | 0.508                        | 1.967 |
+| LUM (no unit)    | 208 | 0.765    | 0.308    | 0.200 | 1         | 0.391                        | 2.556 |
+| RND (km/km2<br>) | 208 | 20.284   | 18.465   | 0     | 91.282    | 0.263                        | 3.809 |
+| TSD (#/km2<br>)  | 208 | 8.605    | 15.741   | 0     | 74.283    | 0.382                        | 2.620 |
+| RID (#/km2<br>)  | 208 | 87.403   | 102.398  | 0     | 291.703   | 0.256                        | 3.902 |
+| BSD (#/km2<br>)  | 208 | 4.685    | 5.644    | 0     | 18.061    | 0.339                        | 2.949 |
+| BSA (km/km2<br>) | 208 | 7.028    | 10.989   | 0     | 62.872    | 0.538                        | 1.857 |
+| EO (#/km2<br>)   | 208 | 0.901    | 2.299    | 0     | 17.687    | 0.600                        | 1.667 |
+| LE (#/km2<br>)   | 208 | 8.980    | 21.817   | 0     | 134.418   | 0.493                        | 2.030 |
+| TOUR (#/km2<br>) | 208 | 2.329    | 7.909    | 0     | 95.507    | 0.778                        | 1.285 |
+| LS (#/km2<br>)   | 208 | 8.571    | 18.593   | 0     | 99.045    | 0.288                        | 3.467 |
+| SC (#/km2<br>)   | 208 | 3.639    | 11.373   | 0     | 106.119   | 0.405                        | 2.470 |
+| DIN (#/km2<br>)  | 208 | 7.363    | 20.988   | 0     | 187.478   | 0.358                        | 2.793 |
+
+**Table 3** Results of descriptive statistics and multicollinearity diagnostics for built environment variables in 500 m buffer zone.
+
+| Variables        | Obs | Mean    | Std.Dev. | Min   | Max       | Multi-collinearity diagnosis |       |
+|------------------|-----|---------|----------|-------|-----------|------------------------------|-------|
+|                  |     |         |          |       |           | Tolerances                   | VIF   |
+| PD (#/km2<br>)   | 208 | 963.298 | 1465.740 | 0     | 10112.711 | 0.301                        | 3.325 |
+| LUM (no unit)    | 208 | 0.703   | 0.318    | 0.220 | 1         | 0.390                        | 2.565 |
+| RND (km/km2<br>) | 208 | 13.360  | 9.370    | 0     | 51.697    | 0.532                        | 1.878 |
+| TSD (#/km2<br>)  | 208 | 3.097   | 7.012    | 0     | 39.471    | 0.230                        | 4.339 |
+| RID (#/km2<br>)  | 208 | 61.739  | 80.926   | 0     | 449.738   | 0.518                        | 1.930 |
+| BSD (#/km2<br>)  | 208 | 4.083   | 8.721    | 0     | 64.936    | 0.333                        | 3.005 |
+| BSA (km/km2<br>) | 208 | 7.028   | 10.989   | 0     | 62.872    | 0.675                        | 1.483 |
+| EO (#/km2<br>)   | 208 | 0.710   | 1.274    | 0     | 7.639     | 0.471                        | 2.121 |
+| LE (#/km2<br>)   | 208 | 5.447   | 12.283   | 0     | 71.301    | 0.367                        | 2.726 |
+| TOUR (#/km2<br>) | 208 | 1.138   | 3.461    | 0     | 40.743    | 0.714                        | 1.401 |
+| LS (#/km2<br>)   | 208 | 5.754   | 13.015   | 0     | 104.406   | 0.242                        | 4.131 |
+| SC (#/km2<br>)   | 208 | 2.062   | 5.165    | 0     | 35.650    | 0.302                        | 3.306 |
+| DIN (#/km2<br>)  | 208 | 4.688   | 12.544   | 0     | 87.853    | 0.233                        | 4.292 |
+
+**Table 4** Results of descriptive statistics and multicollinearity diagnostics for built environment variables in 800 m buffer zone.
+
+| Variables        | Obs | Mean     | Std.Dev. | Min   | Max      | Multi-collinearity diagnosis |       |
+|------------------|-----|----------|----------|-------|----------|------------------------------|-------|
+|                  |     |          |          |       |          | Tolerances                   | VIF   |
+| PD (#/km2<br>)   | 208 | 864.7242 | 1213.952 | 0     | 5918.940 | 0.324                        | 3.807 |
+| LUM (no unit)    | 208 | 0.664    | 0.326    | 0.189 | 1        | 0.374                        | 2.674 |
+| RND (km/km2<br>) | 208 | 6.401    | 4.744    | 0     | 31.592   | 0.584                        | 1.712 |
+| TSD (#/km2<br>)  | 208 | 0.774    | 1.830    | 0     | 10.504   | 0.348                        | 2.875 |
+| RID (#/km2<br>)  | 208 | 49.618   | 67.324   | 0     | 264.337  | 0.803                        | 1.245 |
+| BSD (#/km2<br>)  | 208 | 1.020    | 2.128    | 0     | 12.414   | 0.479                        | 2.089 |
+| BSA (km/km2<br>) | 208 | 7.027    | 10.989   | 0     | 62.872   | 0.733                        | 1.364 |
+| EO (#/km2<br>)   | 208 | 0.511    | 0.822    | 0     | 5.411    | 0.481                        | 2.078 |
+| LE (#/km2<br>)   | 208 | 2.561    | 6.103    | 0     | 39.470   | 0.247                        | 4.048 |
+| TOUR (#/km2<br>) | 208 | 0.515    | 1.335    | 0     | 12.095   | 0.534                        | 1.873 |
+| LS (#/km2<br>)   | 208 | 2.561    | 6.104    | 0     | 39.470   | 0.158                        | 6.342 |
+| SC (#/km2<br>)   | 208 | 0.809    | 1.918    | 0     | 13.687   | 0.270                        | 3.699 |
+| DIN (#/km2<br>)  | 208 | 2.049    | 4.904    | 0     | 31.831   | 0.245                        | 4.080 |
+
+# *4.2. Modeling analysis*
+
+This study designed two types, totaling 21 modeling scenarios (1 × 3 × 3 + 2 × 3 × 3), as shown in [Fig.](#page-8-0) 5. Type I does not differentiate between transportation hubs, and the results are based on total sample regression. Type II distinguishes between transportation hubs, and as previously elaborated, bus stations and ferries have too few sample sizes to be used for developing separate models. Therefore, the Type II scenarios were used to compare the differences in the built environment of different transportation hubs regarding their CR impacts.
+
+Nine scenarios (1 × 3 × 3) are constructed to investigate the impact of the built environment on the CR of transportation hubs, and the scenarios are designed as shown in [Fig.](#page-8-0) 5, Type I. We compared the evaluation indicators of the 9 model scenarios, as shown in [Table](#page-8-0) 6, to find the optimal explanatory model.
+
+The AIC and AICc metrics favor lower values, indicating superior model performance. For the R2 and Adj R2 metrics, values closer to 1 indicate better model fit. Within the 300-meter buffer zone, the AIC for OLS is 376.915, the GWR is 365.005, and the MGWR is 335. The MGWR model demonstrates the lowest AIC value, signifying its superior performance in terms of explanatory variable selection and model fitting within the 300 m buffer zone. Similarly, within the 500 m and 800 m buffer zones, OLS and GWR underperformed MGWR based on the AIC metrics, consistently attaining the lowest values. Moreover, for both R<sup>2</sup> and Adj R2 metrics, the MGWR model has higher values relative to the other models in any buffer. This indicates that the MGWR model possesses advantages in terms of explanatory variable selection and model fitting across different buffer zone ranges and is better able to account for the correlation between the built environment around transportation hubs and CR at any spatial scale.
+
+Further, we compare the bandwidth and standard deviation (STD) of
+
+two local models, GWR and MGWR. For the MGWR model, we sorted the bandwidths of the independent variables in ascending order. The bandwidths and STD of the estimated parameters for the GWR and MGWR models for the three buffers of 300 m, 500 m, and 800 m are shown in [Figs.](#page-9-0) 6 - 8. Generally, we observe that the STD in the MGWR model decreases as the bandwidth increases. This is likely because as the bandwidth increases, the number of neighboring observations the model considers increases, which can provide more spatial information to estimate the parameters, reducing the estimation uncertainty due to sample finiteness and thus reducing the standard error. However, the STD of the GWR model varies irregularly because GWR uses a fixed bandwidth for estimation, which may result in an uneven density and distribution of observations at different locations. Hence, the standard errors may also vary irregularly.
+
+# *4.3. Mechanisms of CR impacts from the built environment of transportation hubs*
+
+From the analyses in Section 4.2.1, MGWR provides optimal results under different buffers compared to the other two methods. Further, we compare the estimated coefficients of the external built environment of transportation hubs on CR based on MGWR under different buffers, as shown in [Table](#page-10-0) 7.
+
+The findings reveal considerable variability in the key variables across different buffer distances, and their impact on CR exhibits significant variations in magnitude and significance. Within the 300 m buffer, we identified 5 key variables: population density, traffic signal density, bus stop density, enterprise office POI density, and dining POI density. As we expanded the buffer to 500 m, the number of critical variables increased to 6, encompassing population density, land use mix, road network density, traffic signal density, bus stop density, and dining
+
+<span id="page-6-0"></span>![](_page_6_Figure_2.jpeg)
+
+![](_page_6_Figure_4.jpeg)
+
+**Fig. 3.** Spatial distribution of CR under different buffers zone.
+
+POI density. Further expanding the buffer to 800 m resulted in identifying 7 key variables: population density, traffic signal density, road intersection density, bus stop density, enterprise office POI density, life service POI density, and dining POI density.
+
+A spatial presentation comprises these four variables: population density, traffic signal density, bus stop density, and enterprise office POI density. All four variables have significant survival correlations and a large degree of influence on the CR of the transportation hub, which offsets the effect of the MAUP. Population density increases external CR in the hubs of Baltimore's urban center and the northeastern Maryland region, and the same trend is observed for the other three variables, as shown in [Fig.](#page-10-0) 9.
+
+The result of the optimal model based on the 800 m buffer is analyzed as an example. Specifically, four key variables—population density (Cofe = 0*.*418*,* p *<* 0*.*01), traffic signal density (Cofe = 0*.*250*,* p *<* 0*.*01), bus stop density (Cofe = 0*.*348*,* p *<* 0*.*01), and Enterprise
+
+office POI density (Cofe = 0*.*170*,* p *<* 0*.*01) have a significant positive effect on the CR of the transportation hubs at the 1 % confidence level. This association may be attributed to the notion that higher residential population density, traffic signal density, bus stop density, and enterprise office POI density indicate a bustling transportation hub area, signifying increased traffic flow, elevated vehicle densities, and heightened points of conflict among diverse traffic flow, thereby amplifying crash risks.
+
+Conversely, three key variables—road intersection density (Cofe = − 0*.*083*,* p *<* 0*.*05), life service POI density (Cofe = − 0*.*163*,* p *<* 0*.*1), and dining POI density (Cofe = − 0*.*114*,* p *<* 0*.*1), exhibit a negative influence on the CR of transportation hubs at the 5 % to 10 % confidence levels, albeit with a moderated impact. Greater road intersection densities typically indicate a more resilient road network in the vicinity, enhancing evacuation efficiency and reducing dependence on individual intersections, hence bolstering network reliability. Higher life service
+
+<span id="page-7-0"></span>![](_page_7_Figure_2.jpeg)
+
+**Fig. 3.** (*continued*).
+
+**Table 5** Spatial dependence of CR of transportation hubs.
+
+| Categories              | Area | Global<br>Moran's I | P<br>value | Z<br>score | S.D.  |
+|-------------------------|------|---------------------|------------|------------|-------|
+| All transportation hubs | 300  | 0.852               | 0.000      | 5.359      | 0.026 |
+| (N = 208)               | m    |                     |            |            |       |
+|                         | 500  | 0.664               | 0.000      | 2.429      | 0.025 |
+|                         | m    |                     |            |            |       |
+|                         | 800  | 0.347               | 0.003      | 1.923      | 0.026 |
+|                         | m    |                     |            |            |       |
+| Railway station (N =    | 300  | 0.575               | 0.002      | 2.777      | 0.038 |
+| 110)                    | m    |                     |            |            |       |
+|                         | 500  | 0.351               | 0.030      | 3.723      | 0.039 |
+|                         | m    |                     |            |            |       |
+|                         | 800  | 0.281               | 0.048      | 1.387      | 0.038 |
+|                         | m    |                     |            |            |       |
+| Airport (N = 69)        | 300  | 0.314               | 0.001      | 2.184      | 0.033 |
+|                         | m    |                     |            |            |       |
+|                         | 500  | 0.223               | 0.047      | 2.039      | 0.012 |
+|                         | m    |                     |            |            |       |
+|                         | 800  | 0.186               | 0.004      | 2.248      | 0.057 |
+|                         |      |                     |            |            |       |
+|                         | m    |                     |            |            |       |
+
+POI densities and Dining POI densities may reflect a more balanced urban planning approach, diminishing the necessity for extended travel distances to access services, thereby alleviating traffic congestion and unnecessary vehicular movements, consequently lowering crash probabilities.
+
+[Table](#page-11-0) 8 presents the estimated bandwidth results of the MGWR model within the 800 m buffer, revealing intriguing insights. The analysis highlights a substantial variance in the optimal bandwidths across the seven key variables. Notably, the impact scale of traffic signal density is 69, representing only 33.17 % of the total sample size, indicating a relatively lower effect than the other variables. Moreover, the effect scales for the remaining five crucial factors—residential population density, bus stop density, corporate office Point of Interest (POI) density, life service POI density, and food service POI density—are relatively close, constituting 61.06 %, 59.62 %, 60.58 %, 61.54 %, and 62.02 % of the total sample size, respectively. Noteworthy is the impact scale of road intersections at 117, accounting for 56.25 % of the total sample, underscoring its significance in the model. Remarkably, the bandwidth of the intercept is observed to be 207, nearing the global scale, which is a crucial observation in the context of the analysis.
+
+*4.4. Analysis of impact mechanisms of different types of transportation hubs*
+
+Similarly, this subsection presents the results for "Railway Station − MGWR-800 m" and the results for "Airport − MGWR-800 m". The CR impacts of different types of transportation hub's built environments are also compared. The results indicate substantial variations in the CR influence mechanisms among different types of transportation hubs, as detailed in [Table](#page-11-0) 9 and 10.
+
+At the land use level, a noteworthy positive effect of population density surrounding railway stations on CR was observed at the 1 % significance level (Cofe = 0*.*396*,*p *<* 0*.*01), contrasting with the lack of significant impact of population density near airports on CR. Regarding the transport system level, the road network density encircling airports significantly positively affected CR (Cofe = 0*.*114*,*p *<* 0*.*01), indicating a direct relationship between higher road network density and increased crash likelihood. Conversely, the road network density surrounding railway stations did not display a significant effect on CR; instead, the traffic signal density near railway stations (Cofe = 0*.*401*,*p *<* 0*.*01) and road intersection density (Cofe = − 0*.*158*,*p *<* 0*.*05) demonstrated significant impacts on CR. Notably, these effects were not observed for airports. In terms of urban design, the life service POI density surround airports exhibited a significant positive effect on CR (Cofe = 0*.*400*,* p *<* 0*.*05). At the same time, the same factor surrounding railway stations did not yield a significant impact on CR. This disparity may arise from the urban setting differences between airport cities, usually situated in remote or suburban areas, and railway station, typically located in city centers, leading to distinct urban design features impacting CR. These observations are further exemplified in [Fig.](#page-12-0) 10, illustrating how higher bus stops and corporate office densities elevate CR surround railway stations predominantly positioned in Baltimore City's center. Conversely, bus stops exert a more pronounced influence on airfield in Maryland's southern region.
+
+Moreover, the effect scales of key variables with the same impact effect for neighboring CR were also compared. Notably, the effect scale of bus stop density in railway station is 98, representing 89.09 % of the total sample, whereas the effect scale of bus stop density in airport is 35, accounting for 50.72 % of the total sample. Similarly, both enterprise office POI density and dining POI density in railway station are recorded at 108, closely aligning with the global scale of 110, indicating relatively stable coefficients across space. Conversely, the enterprise office POI
+
+<span id="page-8-0"></span>![](_page_8_Figure_2.jpeg)
+
+**Fig. 4.** Theil index in different buffer zones at different transportation hubs.
+
+![](_page_8_Figure_4.jpeg)
+
+**Fig. 5.** Model development scenarios.
+
+density and dining POI density in airport stand at 30 and 68, respectively, showcasing significant spatial heterogeneity. Furthermore, the effect scale of the intercept for the railway station is calculated at 106, nearing the full sample, whereas the intercept effect scale for the airport is 54, equivalent to 78.26 % of the total dataset.
+
+## *4.5. Robustness analysis*
+
+we conducted robustness analyses of the core findings. We tested the robustness of the results by performing independent regression analyses for each of the three types of crashes (property damage crashes, injury crashes, and fatal crashes), using the number of occurrences within the 800 m buffer zone as the dependent variable, and comparing the results with those of the benchmark group model. The regression results are shown in [Table](#page-12-0) 11.
+
+The results indicate that the benchmark models are generally consistent with the direction and significance of the independent variables in the other groups. For example, Ln (PD) and TSD are positive and significant in all models and are indicated as significant at the 5 % level. Although there is a notable difference in the magnitude of significance, this does not change the core findings. This result certainly validates the robustness of the findings of this study. Therefore, the weighted combination of this study with reference to Wu et al. [\(2023\)](#page-13-0) can reflection of the general trend of the impact of the built environment around transportation hubs on crash risk.
+
+#### **5. Discussion and policy recommendations**
+
+This study reveals the relationship between the external built environment of transportation hubs and CR (Crash Risk). Despite the variability in CR performance across different buffer zones, common spatial patterns emerge. Specifically, hubs with higher CR are predominantly in the Baltimore urban center and two cities adjacent to Washington, DC. This phenomenon may be attributed to increased traffic flow in urban centers and the presence of diverse transportation users [\(Asadi](#page-13-0) et al. [2022,](#page-13-0) Yoon and Lee, 2021). Analysis of model evaluation metrics revealed that the 800 m scale captured more spatial dependencies than the 300 m and 500 m scales, likely due to the broader geographical coverage of larger buffers, yielding more informative data points [\(Liu](#page-13-0)
+
+**Table 6** Indicators of optimal model estimation results of CR impact mechanisms under different buffer zones.
+
+| Model indicators |         | 300-meters buffer |         |         | 500-meters buffer |         |         | 800-meters buffer |         |  |
+|------------------|---------|-------------------|---------|---------|-------------------|---------|---------|-------------------|---------|--|
+|                  | OLS     | GWR               | MGWR    | OLS     | GWR               | MGWR    | OLS     | GWR               | MGWR    |  |
+| AIC              | 376.915 | 365.005           | 335.000 | 237.107 | 240.583           | 229.062 | 306.136 | 229.613           | 218.426 |  |
+| AICc             | 381.415 | 376.789           | 340.952 | 241.607 | 245.741           | 234.294 | 310.636 | 261.496           | 232.339 |  |
+| R2               | 0.687   | 0.750             | 0.765   | 0.840   | 0.850             | 0.857   | 0.777   | 0.890             | 0.880   |  |
+| Adj R2           | 0.666   | 0.707             | 0.737   | 0.829   | 0.833             | 0.841   | 0.762   | 0.857             | 0.857   |  |
+
+<span id="page-9-0"></span>![](_page_9_Figure_2.jpeg)
+
+**Fig. 6.** Bandwidth and STD for GWR and MGWR model parameter estimation in 300 m buffer.
+
+![](_page_9_Figure_4.jpeg)
+
+**Fig. 7.** Bandwidth and STD for GWR and MGWR model parameter estimation in 500 m buffer.
+
+![](_page_9_Figure_6.jpeg)
+
+**Fig. 8.** Bandwidth and STD for GWR and MGWR model parameter estimation in 800 m buffer.
+
+<span id="page-10-0"></span>**Table 7** Estimated coefficients of external built environment of transportation hubs on CR under different buffers based on MGWR.
+
+| Variables | 300-meters buffer |       | 500-meters buffer |       | 800-meters buffer |       |
+|-----------|-------------------|-------|-------------------|-------|-------------------|-------|
+|           | Cofe.             | S.E.  | Cofe.             | S.E.  | Cofe.             | S.E.  |
+| Ln (PD)   | 0.095**           | 0.056 | 0.155***          | 0.052 | 0.418***          | 0.059 |
+| LUM       | − 0.030           | 0.064 | − 0.103**         | 0.046 | − 0.005           | 0.056 |
+| RND       | 0.086             | 0.078 | 0.075*            | 0.039 | 0.056             | 0.045 |
+| TSD       | 0.644***          | 0.065 | 0.433***          | 0.060 | 0.250***          | 0.058 |
+| Ln (RID)  | − 0.032           | 0.079 | 0.029             | 0.040 | − 0.083**         | 0.038 |
+| BSD       | 0.133**           | 0.069 | 0.319***          | 0.050 | 0.348***          | 0.049 |
+| BSA       | − 0.002           | 0.055 | 0.008             | 0.035 | − 0.019           | 0.039 |
+| EO        | 0.096*            | 0.052 | 0.022             | 0.042 | 0.170***          | 0.049 |
+| LE        | 0.003             | 0.057 | − 0.055           | 0.047 | 0.091             | 0.068 |
+| TOUR      | 0.020             | 0.046 | − 0.002           | 0.034 | 0.011             | 0.046 |
+| LS        | − 0.105           | 0.075 | − 0.063           | 0.058 | − 0.163*          | 0.085 |
+| SC        | 0.045             | 0.063 | − 0.053           | 0.052 | − 0.003           | 0.065 |
+| DIN       | 0.129*            | 0.067 | 0.104*            | 0.059 | − 0.114*          | 0.069 |
+| Intercept | − 0.019**         | 0.040 | − 0.011**         | 0.029 | − 0.071**         | 0.034 |
+
+Note: 1. Cofe. stands for coefficient, S.E. stands for standard error; 2. \*\*\*p *<* 0.01; \*\*p *<* 0.05; \*p *<* 0.1.
+
+et al. [2023\)](#page-13-0). This contrasts with the findings of Wu et al. [\(2023\)](#page-13-0) exploring the school perimeter built environment and CR, in which they found that small buffer zones as a spatial unit of analysis were more explanatory of the school perimeter built environment and CR. Moreover, regarding method performance comparison, it is evident that Multiscale Geographically Weighted Regression (MGWR) exhibits superior explanatory power, consistent with previous research findings (An et al. [2022,](#page-13-0) Kuo et al. 2024).
+
+This study provides insights into spatial dependence and CR heterogeneity within transportation hubs. Based on these findings, specific and practicable policy recommendations are proposed for different spatial buffer scale (300 m, 500 m, and 800 m) and hub types (railway stations and airports). These recommendations emphasize adjustments to peripheral land use, infrastructure improvements, and urban planning strategies tailored to the unique characteristics of each spatial scale and hub types.
+
+The 300 m buffer zone presents the strongest spatial dependence on
+
+CR in densely populated urban areas near the transportation hubs. This study found a significant positive correlation between the density of traffic signalization equipment and the reduction in CR at this scale. Therefore, increasing the traffic signal density in and around railway stations should be considered on a case-by-case basis to regulate traffic flow and minimize conflict points between vehicles and pedestrians. Furthermore, the study revealed that areas with a poor land use mix have a higher risk of crashes, leading to the recommendation that highrisk land uses (e.g., heavy commercial or industrial areas) within 300 m should be restricted and replaced with pedestrian-friendly mixed-use developments. Regarding pedestrian infrastructure, new developments in the vicinity of hubs should incorporate traffic calming measures, such as raised crossings, speed humps, and clear signage, to reduce vehicle speeds and enhance pedestrian safety.
+
+The 500 m buffer zone demonstrates a balanced combination of CR determinants, with significant contributions from factorslike population density, road network density, and bus stop density. It was observed that higher road network density is associated with lower CR. Consequently, the road network density within 500 m of the hub was increased to enhance connectivity and reduce congestion. Specific measures include providing additional secondary roads and alternate routes to alleviate congestion at major intersections. To improve transit accessibility, bus stop density in high pedestrian areas traffic should be increased while ensuring that the design minimizes CR by relocating bus stops away from intersections and incorporating shelters, lighting, and marked crossing areas. Regarding land use development, a mix of residential and commercial land uses will be encouraged within this buffer zone to shorten commuting distances and alleviate traffic congestion.
+
+The 800 m buffer zone illustrates a broader spatial pattern, emphasizing the significance of built environment factors, such as population density, corporate office density, and road intersection density. Given that corporate office POI density positively affects CR, it is recommended that corporate office developments be centrally located in areas with well-developed transport networks to minimize reliance on private vehicles. For instance, zoning corporate parks in locations with good access to public transport can achieve this goal. Regarding amenity facilities, increasing the density of life service POI (e.g., grocery stores,
+
+![](_page_10_Figure_11.jpeg)
+
+**Fig. 9.** Spatial distribution of estimated coefficients for four variables.
+
+<span id="page-11-0"></span>**Table 8** Results of MGWR model estimation of bandwidth in 800 m buffers.
+
+| Variables | Bandwidth | Cofe.     | STD   | Min     | Median  | Max     |
+|-----------|-----------|-----------|-------|---------|---------|---------|
+| Ln (PD)   | 127.000   | 0.418***  | 0.100 | 0.248   | 0.356   | 0.514   |
+| LUM       | 204.000   | − 0.005   | 0.007 | 0.020   | 0.038   | 0.049   |
+| RND       | 121.000   | 0.056     | 0.111 | − 0.082 | 0.190   | 0.367   |
+| TSD       | 69.000    | 0.250***  | 0.137 | − 0.170 | 0.124   | 0.417   |
+| Ln (RID)  | 117.000   | − 0.083** | 0.099 | − 0.289 | 0.089   | 0.035   |
+| BSD       | 124.000   | 0.348***  | 0.093 | 0.006   | 0.226   | 0.287   |
+| BSA       | 207.000   | − 0.019   | 0.007 | − 0.073 | − 0.057 | − 0.043 |
+| EO        | 126.000   | 0.170***  | 0.085 | 0.026   | 0.171   | 0.256   |
+| LE        | 107.000   | 0.091     | 0.181 | − 0.218 | − 0.104 | 0.206   |
+| TOUR      | 191.000   | 0.011     | 0.014 | 0.030   | 0.077   | 0.099   |
+| LS        | 128.000   | − 0.163*  | 0.090 | − 0.030 | 0.164   | 0.227   |
+| SC        | 207.000   | − 0.003   | 0.001 | − 0.007 | − 0.003 | 0.000   |
+| DIN       | 129.000   | − 0.114*  | 0.062 | − 0.230 | − 0.147 | − 0.057 |
+| Intercept | 207.000   | − 0.071** | 0.003 | − 0.088 | − 0.070 | − 0.066 |
+
+Note: 1. Cofe. stands for coefficient, STD stands for standard deviation; 2. \*\*\*p *<* 0.01; \*\*p *<* 0.05; \*p *<* 0.1.
+
+**Table 9** Estimation results for railway station-MGWR-800 m.
+
+| Variables | Bandwidth | Cofe.     | STD   | Min     | Median  | Max     |
+|-----------|-----------|-----------|-------|---------|---------|---------|
+| Ln (PD)   | 82.000    | 0.396***  | 0.322 | 0.163   | 0.822   | 0.850   |
+| LUM       | 80.000    | − 0.026   | 0.086 | − 0.158 | 0.016   | 0.147   |
+| RND       | 92.000    | 0.071     | 0.105 | − 0.058 | 0.142   | 0.356   |
+| TSD       | 107.000   | 0.401***  | 0.005 | 0.004   | 0.011   | 0.040   |
+| Ln (RID)  | 108.000   | − 0.158** | 0.011 | − 0.179 | − 0.179 | − 0.111 |
+| BSD       | 83.000    | 0.245***  | 0.101 | − 0.081 | 0.134   | 0.317   |
+| BSA       | 98.000    | − 0.075   | 0.373 | − 1.019 | − 0.916 | − 0.112 |
+| EO        | 108.000   | 0.159**   | 0.008 | 0.021   | 0.059   | 0.065   |
+| LE        | 74.000    | 0.145     | 0.167 | − 0.111 | 0.213   | 0.275   |
+| TOUR      | 108.000   | 0.046     | 0.008 | 0.043   | 0.074   | 0.081   |
+| LS        | 96.000    | − 0.143   | 0.062 | − 0.101 | 0.036   | 0.103   |
+| SC        | 108.000   | − 0.063   | 0.003 | − 0.061 | − 0.044 | − 0.043 |
+| DIN       | 108.000   | − 0.204*  | 0.003 | 0.088   | 0.106   | 0.109   |
+| Intercept | 106.000   | 0.010**   | 0.043 | − 0.242 | 0.022   | 0.027   |
+
+Note: 1. Cofe. stands for coefficient, STD stands for standard deviation; 2. \*\*\*p *<* 0.01; \*\*p *<* 0.05; \*p *<* 0.1.
+
+**Table 10** Estimation results for airport-MGWR-800 m.
+
+| Variables | Bandwidth | Cofe.     | STD   | Min     | Median  | Max     |
+|-----------|-----------|-----------|-------|---------|---------|---------|
+| Ln (PD)   | 35.000    | 0.001     | 0.137 | − 0.268 | − 0.202 | 0.152   |
+| LUM       | 53.000    | 0.078     | 0.026 | 0.000   | 0.072   | 0.087   |
+| RND       | 56.000    | 0.114***  | 0.060 | 0.039   | 0.068   | 0.213   |
+| TSD       | 53.000    | − 0.011   | 0.039 | − 0.055 | 0.090   | 0.095   |
+| Ln (RID)  | 50.000    | 0.026     | 0.025 | − 0.059 | − 0.015 | 0.039   |
+| BSD       | 54.000    | 0.463***  | 0.025 | 0.076   | 0.173   | 0.190   |
+| BSA       | 32.000    | − 0.025   | 0.065 | − 0.253 | − 0.076 | 0.046   |
+| EO        | 30.000    | 0.231***  | 0.059 | 0.047   | 0.140   | 0.291   |
+| LE        | 68.000    | 0.116     | 0.001 | 0.074   | 0.077   | 0.077   |
+| TOUR      | 39.000    | 0.086     | 0.192 | − 0.053 | 0.451   | 0.459   |
+| LS        | 68.000    | 0.400**   | 0.001 | 0.811   | 0.813   | 0.814   |
+| SC        | 53.000    | 0.094     | 0.027 | 0.137   | 0.139   | 0.237   |
+| DIN       | 68.000    | − 0.328** | 0.001 | − 0.649 | − 0.647 | − 0.646 |
+| Intercept | 54.000    | − 0.016** | 0.014 | − 0.031 | − 0.020 | 0.023   |
+
+Note: 1. Cofe. stands for coefficient, STD stands for standard deviation; 2. \*\*\*p *<* 0.01; \*\*p *<* 0.05; \*p *<* 0.1.
+
+and medical centers) can help decrease unnecessary trips and traffic congestion, thereby reducing CR. Urban design policies should focus on encouraging the development of essential services in proximity to residential areas.
+
+The study identified significant differences in CR determinants between railway stations and airports, highlighting the need for tailored recommendations for these two transportation hubs. For railway stations, it is recommended to increase the bus stops density in the vicinity to enhance "last mile" connectivity and create an integrated transportation system that connects bus stops directly to station entrances, as this significantly reduces CR. Additionally, improving traffic signal density is essential to manage the high volume of pedestrian and vehicular traffic around the railway stations. For airports, increasing the road network density surrounding airports can reduce congestion and improve evacuation efficiency. This can be achieved by constructing additional access roads and providing high-capacity roundabouts at key intersections. Furthermore, to address the impact of amenity service density in the vicinity of airports on CR, a zoning policy should be implemented to support balanced urban planning and prevent overconcentration of service facilities around airports.
+
+The limitations of this study should be acknowledged. Firstly, severity weights for crash types were adopted from Wu et al. [\(2023\)](#page-13-0) without localized calibration. While these coefficients are consistent with established crash risk theories and have been validated in previous
+
+<span id="page-12-0"></span>![](_page_12_Figure_2.jpeg)
+
+**Fig. 10.** Spatial distribution of estimated coefficients for BSD and EO: railway stations and airports.
+
+**Table 11** Comparison of results for different groups with core explanatory variables using MGWR model within 800 m buffer.
+
+| Groups                | Ln (PD)  | TSD      | Ln (RID)  | BSD      | EO       | LS        | DIN       |
+|-----------------------|----------|----------|-----------|----------|----------|-----------|-----------|
+| Benchmark group       | 0.418*** | 0.250*** | − 0.083** | 0.348*** | 0.170*** | − 0.163*  | − 0.114*  |
+| Property Damage Crash | 0.383*** | 0.216*** | − 0.064** | 0.369*** | 0.176*** | − 0.176** | − 0.086   |
+| Injury Crash          | 0.463*** | 0.293*** | − 0.042*  | 0.314*** | 0.155**  | − 0.098** | − 0.155*  |
+| Fatal Crash           | 0.059**  | 0.185*** | 0.048     | 0.165**  | 0.364*   | − 0.056   | − 0.076** |
+
+<sup>\*\*\*</sup>p *<* 0.01; \*\*p *<* 0.05; \*p *<* 0.1.
+
+studies, the application of these coefficients in different geographic environments may result in nuances. In addition, the absolute number of crashes was used as a proxy for crash risk without considering exposure factors such as traffic volume, trips. While absolute crash counts offer valuable insights into the spatial relationship between the built environment and crash occurrences, they may not fully capture the underlying risk dynamics. Despite these constraints, this study undeniably contributes to the theoretical understanding of the interplay between the built environment and traffic safety, shedding light on strategies to enhance the external safety of transportation hubs through optimized built environment planning.
+
+# **6. Conclusion**
+
+This study reveals the relationship between the built environment around transportation hubs and crash risk, with the following main findings: (1) The 800 m buffer spatial scale demonstrates a stronger explanatory power in capturing various features of the built environment that influence crash risk around transportation hubs. (2) In the explanatory model, MGWR considers a flexible bandwidth compared to the other two methods, making it more suitable for analyzing the spatial heterogeneity of the dependent variable. (3) There are heterogeneous effects in the built environment of different types of transportation hubs regarding their impacts on surrounding crash risk, but it is not appropriate to attract too high a population density and build too high a road density around all transportation hubs.
+
+Several potential directions for future research can be identified: (1)
+
+When data become available in the future, localized weights for severity should be calibrated as well as exposure indicators incorporated to provide a more accurate assessment of crash risk. (2) While this study examined multiple spatial scales, exploring different time windows could offer valuable insights. (3) Employing appropriate machine learning techniques to explore the relationship between the external traffic environment of transportation hubs and crash risk may yield intriguing findings.
+
+#### **CRediT authorship contribution statement**
+
+**Chuanyao Li:** Writing – original draft, Supervision, Project administration, Methodology, Investigation, Funding acquisition. **Li Chen:** Writing – original draft, Software, Methodology, Conceptualization.
+
+# **Declaration of competing interest**
+
+The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
+
+# **Acknowledgements**
+
+This study was supported by grants from the National Natural Science Foundation of China (72271248), the Natural Science Foundation of Hunan Province, China (2024JJ5449), the Scientific Research Program of the Education Department of Hunan Province (24B0002), the <span id="page-13-0"></span>Chinese Academy of Engineering Major Projects of Academy-Local Cooperation (2024-DFZD-57), the National Natural Science Foundation of China (72288101), the Higher-end Think-Tank Project of Central South University (2022znzk07), and the Fundamental Research Funds for the Central Universities of Central South University (Grant No.1053320240405).
+
+# **Data availability**
+
+The authors do not have permission to share data.
+
+# **References**
+
+- Alam, M.D.J., [Pinchemel,](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0005) A., Habib, M.A., Caetano, M., 2023. Airport evacuation under panic conditions: a [microsimulation](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0005) modeling applied at ottawa international airport. [Simulation-Transactions](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0005) of the Society for Modeling and Simulation [International](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0005) 99 (12), 1213–1224.
+- Ali, Y., [Washington,](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0010) S., Haque, M.M., 2023. Estimating real-time crash risk at signalized
+- [intersections:](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0010) a bayesian generalized extreme value approach. Saf. Sci. 164. Alkheder, S., 2023. Studying the effect of built [environment](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0015) on traffic safety with random
+- parameter and [generalized](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0015) ordered logit models. Urban Clim. 47, 101388. An, R., Wu, Z., Tong, Z., Qin, S., Zhu, Y., Liu, Y., 2022. How the built [environment](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0020) promotes public transportation in wuhan: a multiscale [geographically](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0020) weighted
+- [regression](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0020) analysis. Travel Behav. Soc. 29, 186–199. Asadi, M., Ulak, M.B., Geurs, K.T., Weijermars, W., Schepers, P., 2022. A [comprehensive](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0025) analysis of the [relationships](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0025) between the built environment and traffic safety in the dutch urban areas. Accid. Anal. Prev. 172, [106683](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0025).
+- Bi, H., Li, A., Zhu, H., Ye, Z., 2023. Bicycle safety outside the crosswalks: [Investigating](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0030) cyclists' risky [street-crossing](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0030) behavior and its relationship with built environment. J. Transp. Geogr. 108, [103551.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0030)
+- Chang, I., Park, H., Hong, E., Lee, J., Kwon, N., 2022. [Predicting](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0035) effects of built environment on fatal pedestrian accidents at [location-specific](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0035) level: application of [xgboost](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0035) and shap. Accid. Anal. Prev. 166, 106545.
+- Clark, A., Scott, D., 2014. [Understanding](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0040) the impact of the modifiable areal unit problem on the relationship between active travel and the built [environment.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0040) Urban Stud. 51 (2), 284–[299.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0040)
+- Costa, M., Lima Azevedo, C., Siebert, F.W., Marques, M., Moura, F., 2024. [Unraveling](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0045) the relation between cycling accidents and built [environment](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0045) typologies: capturing spatial [heterogeneity](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0045) through a latent class discrete outcome model. Accid. Anal. Prev. 200, [107533](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0045).
+- Duan, H., Cao, Q., Wang, L., Gu, X., Ashrafi, K., 2024. Exploring the [relationships](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0050) between 3d urban landscape patterns and pm2.5 pollution using the [multiscale](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0050) geographic weighted regression model. Progress in Physical [Geography-Earth](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0050) and
+- Ewing, R., Cervero, R., 2010. Travel and the built [environment.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0055) J. Am. Plann. Assoc. 76 (3), 265–[294.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0055)
+- Fotheringham, A.S., Yang, W., Kang, W., 2017. Multiscale [geographically](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0060) weighted [regression](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0060) (mgwr). Ann. Am. Assoc. Geogr. 107 (6), 1247–1265.
+- G´ alvez-P´erez, D., Guirao, B., Ortuno, ˜ A., 2023. Analysis of the elderly [pedestrian](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0065) injury severity in urban traffic accidents in spain using machine learning [techniques.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0065) Transp. Res. [Procedia](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0065) 71, 6–13.
+- Gao, C., Li, S., Sun, M., Zhao, X., Liu, D., 2024. Exploring the [relationship](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0070) between urban vibrancy and built [environment](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0070) using multi-source data: Case study in munich. [Remote](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0070) Sens. (Basel) 16 (6).
+- Jin, J., Huang, H., Zhou, R., Chen, J., Liu, P., 2024. Analysis of dynamic [determinants](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0075) of vehicles involved in crash [affecting](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0075) severity based on in-depth crash data. Traffic Inj. [Prev.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0075)
+- Jun, M.-J., Choi, K., Jeong, J.-E., Kwon, K.-H., Kim, H.-J., 2015. Land use [characteristics](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0080) of subway [catchment](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0080) areas and their influence on subway ridership in seoul. J. [Transp.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0080) Geogr. 48, 30–40.
+- Kuo, P.-F., Hsu, W.-T., Putra, I.G.B., [Sulistyah,](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0085) U.D., 2024. The proposed model for analyzing off-street parking [dynamics:](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0085) a case study of taipei city. Transp. Res. A Policy Pract. 180, [103965.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0085)
+- Lee, S., Yoon, J., Woo, A., 2020. Does elderly safety matter? [Associations](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0090) between built [environments](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0090) and pedestrian crashes in seoul, korea. Accid. Anal. Prev. 144,
+- Liu, W., Li, D., Meng, Y., Guo, C., 2024. The [relationship](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0095) between emotional perception and high-density built [environment](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0095) based on social media data: evidence from spatial [analyses](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0095) in wuhan. Land 13 (3).
+- Liu, X., Chen, X., Tian, M., De Vos, J., 2023. Effects of buffer size on [associations](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0100) between the built environment and metro ridership: a machine [learning-based](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0100) sensitive [analysis.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0100) J. Transp. Geogr. 113, 103730.
+
+- Lyu, T., Wang, Y., Ji, S., Feng, T., Wu, Z., 2023. A [multiscale](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0105) spatial analysis of taxi [ridership.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0105) J. Transp. Geogr. 113, 103718.
+- Ma, R., Huang, A., Cui, H., Yu, R., Peng, X., 2024. Spatial heterogeneity analysis on distribution of intra-city public electric vehicle charging points based on multi-scale geographically weighted regression. Travel Behaviour and Society 35.
+- Macedo, F.E., Raaphorst, K.M.C., [Bevelander,](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0115) K.E., Van Der Krabben, E., 2023. The influence of the built [environment](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0115) on active school travel in the netherlands: a mode choice analysis. Multimodal [Transportation](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0115) 2 (4), 100103.
+- Merlin, L.A., Guerra, E., [Dumbaugh,](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0120) E., 2020. Crash risk, crash exposure, and the built [environment:](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0120) a conceptual review. Accid. Anal. Prev. 134, 105244.
+- Mva, M., 2023. Crash data dashboard-maryland takes a data-driven approach to reach our goal of zero roadway fatalities and serious injuries.
+- Nigg, C., Niessner, C., Burchartz, A., Woll, A., [Schipperijn,](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0130) J., 2022. The geospatial and conceptual [configuration](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0130) of the natural environment impacts the association with health outcomes and behavior in children and [adolescents.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0130) Int. J. Health Geogr. 21
+- Salarian, A.H., [Mashhadizadeh,](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0135) A., Bagheri, M., 2020. Simulating passenger evacuation in railway station under fire [emergency](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0135) using safe zone approach. Transp. Res. Rec. [2674](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0135) (11), 806–812.
+- [Shiwakoti,](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0140) N., Wang, H., Jiang, H., Wang, L., 2020. A 'role-rule' model to examine [passengers](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0140)' likely behaviour and their perceived ability to evacuate safely from airport in an emergency [evacuation.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0140) Saf. Sci. 124.
+- Soltani, A., Roohani [Qadikolaei,](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0145) M., 2024. Space-time analysis of accident frequency and the role of built [environment](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0145) in mitigation. Transp. Policy 150, 189–205.
+- Su, L., Zhou, S., Kwan, M.-P., Chai, Y., Zhang, X., 2022. The impact of [immediate](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0150) urban
+- [environments](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0150) on people's momentary happiness. Urban Stud. 59 (1), 140–160. Tang, X., Bi, R., Wang, Z., 2023. Spatial analysis of [moving-vehicle](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0155) crashes and fixedobject crashes based on multi-scale [geographically](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0155) weighted regression. Accid. Anal. [Prev.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0155) 189.
+- Umair, M., Rana, I.A., Lodhi, R.H., 2022. The [impact](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0160) of urban design and the built [environment](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0160) on road traffic crashes: a case study of rawalpindi, pakistan. Case Studies on [Transport](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0160) Policy 10 (1), 417–426.
+- Wang, Z., Gong, X., Zhang, Y., Liu, S., Chen, N., 2023. Multi-scale [geographically](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0165) weighted elasticity [regression](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0165) model to explore the elastic effects of the built environment on ride-hailing ridership. [Sustainability](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0165) 15 (6).
+- Wu, Y., Hu, X., Ji, X., Wu, K., 2023. Exploring associations between built [environment](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0170) and crash risk of children in school [commuting.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0170) Accid. Anal. Prev. 193, 107287.
+- Xiao, D., Ding, H., Sze, N.N., Zheng, N., 2024. [Investigating](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0175) built environment and traffic flow impact on crash [frequency](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0175) in urban road networks. Accid. Anal. Prev. 201, [107561](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0175).
+- Xie, C.-Z., Tang, T.-Q., Hu, P.-C., Huang, H.-J., 2022. A civil aircraft passenger [deplaning](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0180) model [considering](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0180) patients with severe acute airborne disease. Journal of [Transportation](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0180) Safety & Security 14 (6), 1063–1084.
+- Xie, W., Liu, W., Xiao, Y., Dai, Y., Dong, J., Liu, R., 2023. [Mrsa-net:](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0185) Anomaly detection of metro stations based on residual attention network with path [aggregation](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0185) and efficient [representation.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0185) IEEE Sens. J. 23 (22), 28340–28354.
+- Xin, C., Guo, R., Wang, Z., Lu, Q., Lin, P.-S., 2017. The effects of [neighborhood](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0190) [characteristics](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0190) and the built environment on pedestrian injury severity: a random parameters generalized ordered probability model with [heterogeneity](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0190) in means and [variances.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0190) Anal. Methods Accid. Res 16, 117–132.
+- Yasmin, S., Eluru, N., Wang, L., Abdel-Aty, M.A., 2018. A joint [framework](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0195) for static and [real-time](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0195) crash risk analysis. Anal. Methods Accid. Res 18, 45–56.
+- Yin, C., Zhang, Z., Zhang, X., Chen, J., Tao, X., Yang, L., 2023. Hub seaport [multimodal](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0200) freight transport network design: perspective of regional integration [development.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0200) Ocean Coast. [Manag.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0200) 242.
+- Yoon, J., Lee, S., 2021. [Spatio-temporal](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0205) patterns in pedestrian crashes and their [determining](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0205) factors: application of a space-time cube analysis model. Accid. Anal. Prev. 161, [106291](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0205).
+- Youli, Y., Yingjian, P., Xiaoxia, L., Airan, Z., 2018. Research on safety risk [management](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0210) of civil [construction](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0210) projects based on risk matrix method. IOP Conf. Ser.: Mater. Sci. Eng. 392 (6), [062080.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0210)
+- Zhao, J., Guo, F., Zhang, H., Dong, J., 2024. Mechanisms of [non-stationary](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0215) influence of urban form on the diurnal thermal [environment](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0215) based on machine learning and mgwr [analysis.](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0215) Sustain. Cities Soc. 101.
+- Zhu, J., Cao, J., Saxena, D., Jiang, S., Ferradi, H., 2023. [Blockchain-empowered](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0220) federated learning: [challenges,](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0220) solutions, and future directions. ACM Comput. Surv. 55 (11).
+- Zhu, Y., Diao, M., Li, J.J., 2021. [Examining](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0225) indoor air pollution in a large-scale integrated transportation hub in shanghai. [Transportation](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0225) Research Part D-Transport & [Environment](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0225) 97.
+- Zou, R., Yu, H., Yang, H., Chen, C., Zhang, G., 2024. Severity [analyses](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0230) of urban two[vehicle-involved](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0230) rear-end crashes characterized by different configurations using mixed logit models. Journal of [Transportation](http://refhub.elsevier.com/S0001-4575(25)00165-4/h0230) Engineering Part A-Systems 150 (2).

Zotero/001_artiklid/Road Safety Indicators/lordapplicationconwaymaxwellpoisson2008.md

@@ -0,0 +1,478 @@
+---
+category: literaturenote
+citekey: lordapplicationconwaymaxwellpoisson2008
+title: Application of the Conway–Maxwell–Poisson generalized linear model for analyzing motor vehicle crashes
+authors: "Lord, Dominique; Guikema, Seth D.; Geedipally, Srinivas Reddy"
+year: 2008
+date: 2008-00-00 2008
+doi: 10.1016/j.aap.2007.12.003
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2007.12.003"
+zotero_key: 6EV2T927
+zotero_storage: ZZE6MXSD
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Lord, Dominique"
+status: converted
+---
+![](_page_0_Picture_0.jpeg)
+
+Accident Analysis and Prevention 40 (2008) 1123–1134
+
+![](_page_0_Picture_2.jpeg)
+
+# Application of the Conway–Maxwell–Poisson generalized linear model for analyzing motor vehicle crashes
+
+Dominique Lord <sup>∗</sup>, Seth D. Guikema 1, Srinivas Reddy Geedipally <sup>2</sup>
+
+*Zachry Department of Civil Engineering, Texas A&M University, 3136 TAMU, College Station, TX 77843-3136, USA* Received 7 September 2007; received in revised form 30 November 2007; accepted 6 December 2007
+
+#### **Abstract**
+
+This paper documents the application of the Conway–Maxwell–Poisson (COM-Poisson) generalized linear model (GLM) for modeling motor vehicle crashes. The COM-Poisson distribution, originally developed in 1962, has recently been re-introduced by statisticians for analyzing count data subjected to over- and under-dispersion. This innovative distribution is an extension of the Poisson distribution. The objectives of this study were to evaluate the application of the COM-Poisson GLM for analyzing motor vehicle crashes and compare the results with the traditional negative binomial (NB) model. The comparison analysis was carried out using the most common functional forms employed by transportation safety analysts, which link crashes to the entering flows at intersections or on segments. To accomplish the objectives of the study, several NB and COM-Poisson GLMs were developed and compared using two datasets. The first dataset contained crash data collected at signalized four-legged intersections in Toronto, Ont. The second dataset included data collected for rural four-lane divided and undivided highways in Texas. Several methods were used to assess the statistical fit and predictive performance of the models. The results of this study show that COM-Poisson GLMs perform as well as NB models in terms of GOF statistics and predictive performance. Given the fact the COM-Poisson distribution can also handle under-dispersed data (while the NB distribution cannot or has difficulties converging), which have sometimes been observed in crash databases, the COM-Poisson GLM offers a better alternative over the NB model for modeling motor vehicle crashes, especially given the important limitations recently documented in the safety literature about the latter type of model. © 2007 Elsevier Ltd. All rights reserved.
+
+*Keywords:* Statistical models; Conway–Maxwell–Poisson distribution; Negative binomial; Regression models; Bayesian models
+
+# **1. Introduction**
+
+There has been considerable research conducted on the development of statistical models for analyzing crashes on highway facilities ([Abbess et al., 1981; Hauer et al., 1988; Kulmala, 1995;](#page-10-0) [Poch and Mannering, 1996; Lord, 2000; Miaou and Lord, 2003;](#page-10-0) [Lord et al., 2005a; Miaou and Song, 2005; Xie et al., 2007\).](#page-10-0) The most common probabilistic structure of the models used by transportation safety analysts for modeling motor vehicle crashes are the traditional Poisson and Poisson-gamma (or negative binomial or NB) distributions. Since crash data have often been shown to exhibit over-dispersion (see [Lord et al., 2005b\),](#page-10-0) meaning that the variance is greater than the mean, Poissongamma models are usually preferred over Poisson regression
+
+Over the last few years, several studies have documented important limitations associated with Poisson and NB models. For instance, it has been shown that when the sample mean value becomes small, traditional methods used to assess the goodnessof-fit (GOF) of generalized linear models (GLMs) estimated using the maximum likelihood estimating (MLE) method (both for Poisson and NB) can be highly unreliable and provide a biased estimate of the fit [\(Maycock and Hall, 1984; Maher and](#page-10-0) [Summersgill, 1996; Wood, 2002; Lord, 2006\).](#page-10-0) Recent studies have also shown that the inverse dispersion parameter of NB models, φ = 1/α, can be significantly mis-estimated when the
+
+models. On occasion, crash data have sometimes shown characteristics of under-dispersion, especially in cases where the sample mean is very low ([Oh et al., 2006\).](#page-11-0) The Poisson-gamma distribution offers a simple way to accommodate the overdispersion, since the final equation has a closed form and the mathematics to manipulate the relationship between the mean and the variance structures are relatively simple [\(Hauer, 1997\).](#page-10-0)
+
+<sup>∗</sup> Corresponding author. Tel.: +1 979 458 3949; fax: +1 979 845 6481. *E-mail addresses:* [d-lord@tamu.edu](mailto:d-lord@tamu.edu) (D. Lord), [sguikema@civil.tamu.edu](mailto:sguikema@civil.tamu.edu) (S.D. Guikema), [srinivas8@tamu.edu](mailto:srinivas8@tamu.edu) (S.R. Geedipally).
+
+<sup>1</sup> Tel.: +1 979 458 3586; fax: +1 979 845 6554.
+
+<sup>2</sup> Tel.: +1 979 862 3446; fax: +1 979 845 6481.
+
+<span id="page-1-0"></span>sample size becomes small and the sample mean value is low. This characteristic was observed both for MLE ([Clark and Perry,](#page-10-0) [1989; Piegorsch, 1990; Lord, 2006; Lloyd-Smith, 2007\)](#page-10-0) and full Bayesian (FB) NB models ([Airoldi et al., 2006; Lord and](#page-10-0) [Miranda-Moreno, 2008\).](#page-10-0) A mis-estimated inverse dispersion parameter can affect analysis tools commonly used in highway safety analyses, such as the application of the empirical Bayes (EB) estimate for identifying hazardous locations ([Hauer, 1997;](#page-10-0) [Miranda-Moreno and Fu, 2007\)](#page-10-0) and building confidence intervals for evaluating and screening highway projects ([Wood, 2005;](#page-11-0) [Geedipally and Lord, 2008\).](#page-11-0) NB models have also been shown to be unable to handle data with extremely low mean values, which often produces sample data with many zeros. Given these important limitations, there is a need to evaluate whether other count data models could be used for modeling motor vehicle crashes.
+
+The objectives of this study are to evaluate the application of the Conway–Maxwell–Poisson (COM-Poisson) GLM for analyzing motor vehicle crashes and compare the results with those produced from the NB model. The COM-Poisson distribution has very recently been re-introduced by statisticians for modeling count data that are characterized by either over- or under-dispersion ([Shmueli et al., 2005; Kadane et al.,](#page-11-0) [2006; Guikema and Coffelt, 2008\).](#page-11-0) This distribution was first introduced in 1962 ([Conway and Maxwell, 1962\),](#page-10-0) but has been evaluated in the context of a GLM model only once [\(Guikema](#page-10-0) [and Coffelt, 2008\).](#page-10-0) Consequently, nobody has so far examined how the COM-Poisson GLM could be used for modeling crash data using common functional forms linking crash data to traffic flow variables (often referred to as general annual average daily traffic or AADT models). This type of functional form is the most popular type of models developed and used by transportation safety analysts ([Hauer, 1997; Persaud et al., 2001\).](#page-10-0) They are often preferred over models that include several covariates because they can be easily re-calibrated when they are developed in one jurisdiction and applied to another [\(Persaud et al.,](#page-11-0) [2002; Lord and Bonneson, 2005\).](#page-11-0) In fact, this type of model will be the kind of model used for estimating the safety performance of rural and urban highways as well as for intersections in the forthcoming highway safety manual (HSM) [\(Hughes et](#page-10-0) [al., 2005\).](#page-10-0) Although such models will suffer from an omitted variables bias (because many non-flow related factors are known to affect the frequency of crashes), the empirical assessment carried out in this work still provides valuable insight into the potential applicability of COM-Poisson GLM for modeling crash data.
+
+To accomplish the objectives of the study, NB and COM-Poisson GLMs were developed and compared using two datasets. The first dataset contained crash data collected at four-legged signalized intersections in Toronto, Ont. The second dataset included data collected for rural four-lane divided and undivided highways in Texas. The study will show that COM-Poisson GLMs perform as well as NB models when AADT is used as the only covariate. The COM-Poisson GLM offers potential for safety analysis and modeling crash data, since it can also handle under-dispersed data (which the NB GLM cannot or has difficulties converging, see below) and datasets that contain intermingled over- and under-dispersed counts (for dual-link models only, since the dispersion characteristic is captured using the covariate-dependent shape parameter).
+
+This paper is divided into six sections. Section 2 describes the parameterization characteristics of the COM-Poisson distribution and its GLM framework. Section [3](#page-3-0) describes the methodology used for estimating and comparing the models. Section [4](#page-4-0) presents the summary statistics of the two datasets. Section [5](#page-4-0) summarizes the results of the comparison analysis. Section [6](#page-9-0) provides a discussion about the application of COM-Poisson GLMs in highway safety, as well as ideas for further research. Section [7](#page-9-0) presents a summary of the analysis carried out in this work.
+
+# **2. Background**
+
+This section describes the characteristics of the COM-Poisson distribution and the GLM for modeling crash data. The first part covers the parameterization of the distribution. The second part describes its extension for modeling count data subjected to over- and under-dispersion.
+
+#### *2.1. Parameterization*
+
+The COM-Poisson distribution is a generalization of the Poisson distribution and was first introduced by [Conway and](#page-10-0) [Maxwell \(1962\)](#page-10-0) for modeling queues and service rates. [Shmueli](#page-11-0) [et al. \(2005\)](#page-11-0) further elucidated the statistical properties of the COM-Poisson distribution using the formulation given by [Conway and Maxwell \(1962\),](#page-10-0) and [Kadane et al. \(2006\)](#page-10-0) developed the conjugate distributions for the parameters of the COM-Poisson distribution. Its probability mass function (PMF) can be given by
+
+$$P(Y = y) = \frac{1}{Z(\lambda, \nu)} \frac{\lambda^{y}}{(y!)^{\nu}} \tag{1}$$
+
+$$Z(\lambda, \nu) = \sum_{n=0}^{\infty} \frac{\lambda^n}{(n!)^{\nu}}$$
+ (2)
+
+where, *Y* is a discrete count; λ is a centering parameter that is approximately the mean of the observations in many cases; and, ν is defined as the shape parameter of the COM-Poisson distribution.
+
+The COM-Poisson can model both under-dispersed (ν > 1) and over-dispersed (ν < 1) data, and several common PMFs are special cases of the COM with the original formulation. Specifically, setting ν = 0 yields the geometric distribution; λ < 1 and ν→∞ yields the Bernoulli distribution in the limit; and ν = 1 yields the Poisson distribution. This flexibility greatly expands the types of problems for which the COM-Poisson distribution can be used to model count data.
+
+With the original formulation, the first two central moments of the COM-Poisson distribution are given by
+
+$$E[Y] = \frac{\partial \log Z}{\partial \log \lambda} \tag{3}$$
+
+<span id="page-2-0"></span>
+$$var[Y] = \frac{\partial^2 \log Z}{\partial \log^2 \lambda} \tag{4}$$
+
+The COM-Poisson distribution does not have closed-form expressions for its moments in terms of the parameters λ and ν. However, the mean can be approximated through a few different approaches, including (i) using the mode, (ii) including only the first few terms of *Z* when ν is large, (iii) bounding *E*[*Y*] when ν is small, and (iv) using an asymptotic expression for *Z* in Eq. [\(1\).](#page-1-0) [Shmueli et al. \(2005\)](#page-11-0) used the last approach to derive the approximation:
+
+$$E[Y] \approx \lambda^{1/\nu} + \frac{1}{2\nu} - \frac{1}{2} \tag{5}$$
+
+Using the same approximation for *Z* as in [Shmueli et al.](#page-11-0) [\(2005\),](#page-11-0) the variance can be defined as
+
+$$var[Y] \approx \frac{1}{\nu} \lambda^{1/\nu} \tag{6}$$
+
+Care should be taken in using these approximations. In particular, they may not be accurate for ν > 1 or λ1/<sup>ν</sup> < 10 [\(Shmueli](#page-11-0) [et al., 2005\).](#page-11-0)
+
+Despite its flexibility and attractiveness, the COM-Poisson has limitations in its usefulness as a basis for a GLM, as documented in [Guikema and Coffelt \(2008\). I](#page-10-0)n particular, neither λ nor ν provide a clear centering parameter. While λ is approximately the mean when ν is close to one, it differs substantially from the mean for small ν. Given that ν would be expected to be small for over-dispersed data, this would make a COM model based on the original COM formulation difficult to interpret and use for over-dispersed data.
+
+To circumvent this problem, [Guikema and Coffelt \(2008\)](#page-10-0) proposed a re-parameterization of the COM-Poisson distribution to provide a clear centering parameter. This new formulation of the COM-Poisson is summarized:
+
+$$P(Y = y) = \frac{1}{S(\mu, \nu)} \left(\frac{\mu^y}{y!}\right)^{\nu} \tag{7}$$
+
+$$S(\mu, \nu) = \sum_{n=0}^{\infty} \left(\frac{\mu^n}{n!}\right)^{\nu} \tag{8}$$
+
+By substituting μ= λ1/<sup>ν</sup> in Eqs. (4), (5), and (41) of [Shmueli](#page-11-0) [et al. \(2005\),](#page-11-0) the mean and variance of *Y* are given in terms of the new formulation as
+
+$$E[Y] = \frac{1}{\nu} \frac{\partial \log S}{\partial \log \mu},\tag{9}$$
+
+and
+
+$$V[Y] = \frac{1}{\nu^2} \frac{\partial^2 \log S}{\partial \log^2 \mu} \tag{10}$$
+
+with asymptotic approximations *E*[*Y*] ≈ μ+ 1/2ν − 1/2 and var(*Y*) ≈ μ/ν, respectively. It is important to note that these approximations are especially accurate once μ> 10. With this new parameterization, the integral part of μ is now the mode leavingμas a reasonable approximation of the mean. The substitution μ= λ1/<sup>ν</sup> also allows ν to keep its role as a shape parameter. That is, if ν < 1, the variance is greater than the mean while ν > 1 leads to under-dispersion.
+
+This new formulation provides a good basis for developing a COM-Poisson GLM. The clear centering parameter provides a basis on which the centering link function can be built, allowing ease of interpretation across a wide range of values of the shape parameter. Furthermore, the shape parameter ν provides a basis for using a second link function to allow the amount of over-dispersion or under-dispersion to vary across measurements (similar to the varying dispersion parameter of the NB model, as discussed by [Miaou and Lord,](#page-10-0) [2003\).](#page-10-0)
+
+#### *2.2. Generalized linear model*
+
+[Guikema and Coffelt \(2008\)](#page-10-0) developed a COM-Poisson GLM framework for modeling discrete count data. Eqs. (11)–(14) describe this modeling framework. The framework is in effect a dual-link GLM in which both the mean and the variance depend on covariates. In Eqs. (11)–(14), *Y* is the count random variable being modeled, *xi* and *zj* are covariates, and there are assumed to be *p* covariates used in the centering link function and *q* covariates used in the shape link function. The sets of parameters used in the two link functions are not necessarily identical:
+
+$$P(Y=y) = \frac{1}{S(\mu,\nu)} \left(\frac{\mu^y}{y!}\right)^{\nu} \tag{11}$$
+
+$$S(\mu, \nu) = \sum_{n=0}^{\infty} \left(\frac{\mu^n}{n!}\right)^{\nu}$$
+ (12)
+
+$$\ln(\mu) = \beta_0 + \sum_{i=1}^p \beta_i x_i \tag{13}$$
+
+$$\ln(\nu) = \gamma_0 + \sum_{j=1}^{q} \gamma_j z_j \tag{14}$$
+
+The GLM described above is highly flexible and readily interpreted. It can model under-dispersed data sets, over-dispersed data sets, and data sets that contain intermingled under-dispersed and over-dispersed counts (for dual-link models only, since the dispersion characteristic is captured using the covariatedependent shape parameter). The variance is allowed to depend on the covariate values, which can be important if high (or low) values of some covariates tend to be variance decreasing while high (or low) values of other covariates tend to be variance increasing. The parameters have a direct link to either the mean or the variance, providing insight into the behavior and driving factors in the problem, and the mean and variance of the predicted counts are readily approximated based on the covariate values and regression parameter estimates. In this paper, the research team modified the formulation of [Guikema and Coffelt](#page-10-0) [\(2008\)](#page-10-0) by removing the second link and estimating a single shape parameter ν. This makes the model more directly comparable to the standard single-link NB model. In future work, the potential benefit of adding the second link function, the <span id="page-3-0"></span>shape link, could be explored in the context of modeling traffic crashes, as [Guikema and Coffelt \(2008\)](#page-10-0) have done in other contexts.
+
+The parameter estimation for the COM-Poisson GLM presented above is challenging. The likelihood equation for the COM-Poisson GLM is complex, making analytical and numerical maximum likelihood estimation difficult. Thus, Bayesian estimation provides an attractive alternative for estimating the coefficients of the model (e.g., Eqs. [\(13\)](#page-2-0) and [\(14\)\).](#page-2-0) The next section describes the methodology used for estimating and comparing COM-Poisson GLMs and NB models.
+
+#### **3. Methodology**
+
+This section describes the methodology used for estimating and comparing the two types of model. For each dataset, COM-Poisson GLMs and NB models were initially estimated using the entire dataset. Then, five samples, which consisted of 80% of the original data, were randomly extracted. The models were developed using the subsets and were then applied to the remaining 20% to evaluate their predictive performance.
+
+For demonstration purposes, the functional forms used for the models were the following (see Eq. [\(13\)](#page-2-0) above):
+
+Toronto intersection data:
+
+$$\mu_i = \beta_0 F_{\text{Maj},i}^{\beta_1} F_{\text{Min},i}^{\beta_2} \tag{15}$$
+
+Texas segment data:
+
+$$\mu_j = \beta_0 L_j F_j^{\beta_1} \tag{16}$$
+
+where, μ*i* = the mean number of crashes per year for intersection *i*; μ*<sup>j</sup>* = the mean number of crashes per year for segment *j*; *F*Maj *<sup>i</sup>* = entering flow for the major approach (average annual daily traffic or AADT) for intersection *i*; *F*Min *<sup>i</sup>* = entering flow for the minor approach (average annual daily traffic or AADT) for intersection *i*; *Fj* = flow traveling on segment *j* (average annual daily traffic or AADT) and time period *t*; *Lj* = length in miles for segment *j*; and β0, β1, β<sup>2</sup> = estimated coefficients.
+
+The functional forms described above are very frequently used by transportation safety analysts. Although they are not considered the most adequate functional form for flow-only models (see [Miaou and Lord, 2003\),](#page-10-0) since they under-perform near the boundary conditions (at least for intersections), they are still relevant for this study, as they are considered established functional forms in the highway safety literature.
+
+Several methods were used for estimating the GOF and predictive performance of the models. The methods used in this research include the following.
+
+#### *3.1. Deviance information criterion (DIC)*
+
+The DIC is defined as
+
+$$DIC = \bar{D} + p_D \tag{17}$$
+
+where <sup>D</sup>¯ = −2 log<sup>L</sup> represents the posterior mean of the deviance of the unstandardized model where *L* is the mean of the model log likelihood; pD <sup>=</sup> <sup>D</sup>¯ <sup>−</sup> <sup>D</sup>¯ (y|¯ θ) represents the penalty for the number of effective model parameters where <sup>D</sup>(y|¯ <sup>θ</sup>) is the point estimate of deviance for the posterior means ¯ θ.
+
+## *3.2. Mean absolute deviance (MAD)*
+
+MAD provides a measure of the average mis-prediction of the model ([Oh et al., 2003\).](#page-10-0) It is computed using the following equation:
+
+mean absolute deviance (MAD) = 
+$$\frac{1}{n} \sum_{i=1}^{n} |\hat{y}_i - y_i|$$
+ (18)
+
+# *3.3. Mean-squared predictive error (MSPE)*
+
+MSPE is typically used to assess the error associated with a validation or external data set ([Oh et al., 2003\).](#page-10-0) It can be computed using the
+
+mean absolute predictive error (MSPE) = 
+$$\frac{1}{n} \sum_{i=1}^{n} (\hat{y}_i - y_i)^2 (19)$$
+
+The coefficients of the COM-Poisson GLMs and FB NB models were estimated using the software WinBUGS [\(Spiegelhalter](#page-11-0) [et al., 2003\)](#page-11-0) (*note*: to distinguish between the Bayesian and the MLE methods, FB NB refers to a model estimated using a Bayesian approach while MLE NB refers to a model estimated using the Frequentist approach). Vague or non-informative hyper-priors were utilized for the COM-Poisson GLMs and FB NB (described below). A total of three Markov chains were used in the model estimation process with 35,000 iterations per chain and the thinning parameter was set to 1. The first 20,000 iterations (burn-in samples) were discarded. Thus, the remaining 15,000 iterations were used for estimating the coefficients. The Gelman–Rubin (G–R) convergence statistic was used to verify that the simulation runs converged properly. In the analysis, the research team ensured that G–R statistic was less than 1.1. For comparison, [Mitra and Washington \(2007\)](#page-10-0) suggested that convergence was achieved when the G–R statistic was less than 1.2.
+
+Given the limitations described above and to confirm the results of the FB NB models, MLE NB models were also estimated using SAS [\(SAS Institute Inc., 2002\).](#page-11-0) These models were not used for the comparison analysis, but to ensure the FB NB models were stable. It should be pointed out that the PMF of the NB models estimated in this work was the following:
+
+$$f(y_i; \phi, \mu_i) = \begin{pmatrix} y_i + \phi - 1 \\ \phi - 1 \end{pmatrix} \left( \frac{\phi}{\mu_i + \phi} \right)^{\phi} \left( \frac{\mu_i}{\mu_i + \phi} \right)^{y_i}$$
+(20)
+
+where, *yi* = response variable for observation *i*; μ*<sup>i</sup>* = mean response for observation *i*; and, φ = inverse dispersion parameter of the Poisson-gamma distribution.
+
+The next section describes the characteristics of the two datasets.
+
+# <span id="page-4-0"></span>**4. Data description**
+
+This section describes the characteristics of the two datasets. The first part summarizes the characteristics of the Toronto data. The second part presents the summary statistics for the Texas data.
+
+#### *4.1. Toronto data*
+
+The first dataset contained crash data collected in 1995 at four-legged signalized intersections located in Toronto, Ont. The data have previously been used for several research projects and have been found to be of relatively good quality ([Lord, 2000;](#page-10-0) [Miaou and Lord, 2003; Miranda-Moreno and Fu, 2007\).](#page-10-0) In total, 868 signalized intersections were used in this dataset. Table 1 presents the summary statistics for the full dataset and the five samples (both for the fitted data and predicted data).
+
+# *4.2. Texas*
+
+The second dataset contained crash data collected at fourlane rural undivided and divided segments in Texas. To increase the sample size, divided and undivided segments were merged together. Since the study objective was not related to examining the safety performance of segments per se, putting the data together was deemed adequate. The data were provided by the Texas Department of Public Safety (DPS) and the Texas Department of Transportation (TxDOT) and were used for the project NCHRP 17–29 (methodology for estimating the safety performance of multilane rural highways). The final database included 3220 segments (≥0.1 mile) and 5 years of crash data. [Table 2](#page-5-0) presents the summary statistics for the full dataset and the five samples (both for fitted data and predicted data).
+
+#### **5. Modeling results**
+
+This section presents the modeling results for the COM-Poisson GLMs as well as for the FB and MLE NB models and is divided into three parts. The first part explains the modeling results for the Toronto data. The second part provides details about the modeling results for the Texas data. The last part documents the marginal analysis used for examining the regression coefficients.
+
+# *5.1. Toronto data*
+
+[Table 3](#page-5-0) summarizes the results of the COM-Poisson GLMs for the Toronto data. This table shows that the coefficients for the flow parameters are below one, which indicates that the crash risk increases at a decreasing rate as traffic flow increases. It should be pointed out that the 95% marginal posterior credible intervals for each of the coefficients did not include the origin.
+
+Table 1 Summary statistics for the Toronto data
+
+|            | Fitting data |       |                    |       |      | Predicting data |                     |       |  |
+|------------|--------------|-------|--------------------|-------|------|-----------------|---------------------|-------|--|
+|            | Min.         | Max.  | Average            | Total | Min. | Max.            | Average             | Total |  |
+| Full dataa |              |       |                    |       |      |                 |                     |       |  |
+| Crashes    | 0            | 54    | 11.56 (10.02)      | 10030 | –    | –               | –                   | –     |  |
+| Major AADT | 5469         | 72178 | 28044.81 (10660.4) | –     | –    | –               | –                   | –     |  |
+| Minor AADT | 53           | 42644 | 11010.18 (8599.40) | –     | –    | –               | –                   | –     |  |
+| Sample 1b  |              |       |                    |       |      |                 |                     |       |  |
+| Crashes    | 0            | 54    | 11.49 (9.94)       | 7974  | 0    | 51              | 11.82 (10.33)       | 2056  |  |
+| Major AADT | 5469         | 72178 | 28097.36 (10656.9) | –     | 9622 | 67214           | 27835.21 (10702.5)  | –     |  |
+| Minor AADT | 71           | 41288 | 10904.03 (8421.3)  | –     | 53   | 42644           | 11433.55 (9289.40)  | –     |  |
+| Sample 2   |              |       |                    |       |      |                 |                     |       |  |
+| Crashes    | 0            | 54    | 11.50 (9.96)       | 7983  | 0    | 48              | 11.76 (10.28)       | 2047  |  |
+| Major AADT | 5469         | 67214 | 27946.05 (10490.5) | –     | 7361 | 72178           | 28438.69 (11335.8)  | –     |  |
+| Minor AADT | 53           | 42644 | 10862.04 (8532.00) | –     | 877  | 41029           | 11601.03 (8863.55)  | –     |  |
+| Sample 3   |              |       |                    |       |      |                 |                     |       |  |
+| Crashes    | 0            | 53    | 11.67 (10.07)      | 8099  | 0    | 54              | 11.10 (9.84)        | 1931  |  |
+| Major AADT | 5469         | 72178 | 28115.98 (10824.1) | –     | 5967 | 56623           | 27760.95 (10005.6)  | –     |  |
+| Minor AADT | 71           | 42644 | 11169.03 (8678.03) | –     | 53   | 36002           | 10376.61 (8272.24)  | –     |  |
+| Sample 4   |              |       |                    |       |      |                 |                     |       |  |
+| Crashes    | 0            | 54    | 11.63 (10.02)      | 8074  | 0    | 50              | 11.24 (10.02)       | 1956  |  |
+| Major AADT | 5469         | 68594 | 28072.28 (10652.6) | –     | 7361 | 72178           | 27935.25 (10721.7)  | –     |  |
+| Minor AADT | 465          | 42644 | 11119.84 (8749.1)  | –     | 53   | 41288           | 10572.82 (7983.41)  | –     |  |
+| Sample 5   |              |       |                    |       |      |                 |                     |       |  |
+| Crashes    | 0            | 54    | 11.69 (10.08)      | 8113  | 0    | 44              | 11.02 (9.79)        | 1917  |  |
+| Major AADT | 5469         | 72178 | 28103.95 (10641.9) | –     | 5967 | 56697           | 27808.91 (10761.52) | –     |  |
+| Minor AADT | 71           | 42644 | 11104.99 (8712.5)  | –     | 53   | 34934           | 10632.03 (8145.70)  | –     |  |
+
+<sup>a</sup> 868 observations.
+
+<sup>b</sup> 694 observations for the fitted data and 174 observations for the predicted data.
+
+<span id="page-5-0"></span>Table 2 Summary statistics for the Texas dataset (5 years)
+
+|                | Fitting data |       |                   |         | Predicting data |       |                   |         |
+|----------------|--------------|-------|-------------------|---------|-----------------|-------|-------------------|---------|
+|                | Min.         | Max.  | Average           | Total   | Min.            | Max.  | Average           | Total   |
+| Full dataa     |              |       |                   |         |                 |       |                   |         |
+| Crashes        | 0            | 108   | 4.89 (8.45)       | 15753   | –               | –     | –                 | –       |
+| AADT           | 42           | 89264 | 8639.27 (6606.57) | –       | –               | –     | –                 | –       |
+| Length (miles) | 0.1          | 11.21 | 0.80 (1.02)       | 2576.18 | –               | –     | –                 | –       |
+| Sample 1b      |              |       |                   |         |                 |       |                   |         |
+| Crashes        | 0            | 108   | 4.94 (8.75)       | 12722   | 0               | 52    | 4.71 (7.16)       | 3031    |
+| AADT           | 42           | 89264 | 8704.98 (6715.86) | –       | 420             | 52294 | 8376.39 (6147.98) | –       |
+| Length (miles) | 0.1          | 11.21 | 0.80 (1.02)       | 2054.91 | 0.1             | 8.319 | 0.81 (1.04)       | 521.269 |
+| Sample 2       |              |       |                   |         |                 |       |                   |         |
+| Crashes        | 0            | 108   | 4.93 (8.72)       | 12703   | 0               | 48    | 4.74 (7.31)       | 3050    |
+| AADT           | 42           | 89264 | 8699.58 (6681.18) | –       | 266             | 52294 | 8398.03 (6298.54) | –       |
+| Length (miles) | 0.1          | 11.21 | 0.80 (1.03)       | 2063.87 | 0.1             | 8.517 | 0.80 (0.96)       | 512.313 |
+| Sample 3       |              |       |                   |         |                 |       |                   |         |
+| Crashes        | 0            | 108   | 4.91 (8.21)       | 12655   | 0               | 97    | 4.81 (9.38)       | 3098    |
+| AADT           | 158          | 89264 | 8645.19 (6685.83) | –       | 42              | 53714 | 8615.58 (6284.50) | –       |
+| Length (miles) | 0.1          | 8.548 | 0.81 (1.02)       | 2076.86 | 0.1             | 11.21 | 0.78 (1.01)       | 499.318 |
+| Sample 4       |              |       |                   |         |                 |       |                   |         |
+| Crashes        | 0            | 108   | 4.94 (8.44)       | 12731   | 0               | 97    | 4.69 (8.53)       | 3022    |
+| AADT           | 42           | 89264 | 8646.10 (6619.57) | –       | 158             | 53714 | 8611.93 (6559.38) | –       |
+| Length (miles) | 0.1          | 11.21 | 0.80 (1.01)       | 2073.67 | 0.1             | 8.517 | 0.78 (1.04)       | 502.51  |
+| Sample 5       |              |       |                   |         |                 |       |                   |         |
+| Crashes        | 0            | 108   | 5.00 (8.69)       | 12891   | 0               | 53    | 4.44 (7.43)       | 2862    |
+| AADT           | 42           | 89264 | 8700.86 (6712.74) | –       | 264             | 52294 | 8392.89 (6162.47) | –       |
+| Length (miles) | 0.1          | 8.548 | 0.81 (1.02)       | 2091.04 | 0.1             | 11.21 | 0.75 (1.02)       | 485.143 |
+
+<sup>a</sup> 3220 observations.
+
+Table 3 Modeling results for the COM-Poisson GLMs using the Toronto data
+
+| Estimatesa | Full data        | Sample 1         | Sample 2         | Sample 3         | Sample 4         | Sample 5         | Average |
+|------------|------------------|------------------|------------------|------------------|------------------|------------------|---------|
+| ln(β0)     | −11.53 (0.4159)b | −11.7589 (0.742) | −11.7252 (0.560) | −11.2626 (0.487) | −11.2033 (0.729) | −11.1643 (0.709) | –       |
+| β1         | 0.6350 (0.04742) | 0.6527 (0.076)   | 0.6641 (0.055)   | 0.6078 (0.049)   | 0.6071 (0.072)   | 0.5949 (0.066)   | –       |
+| β2         | 0.7950 (0.03101) | 0.7999 (0.029)   | 0.7854 (0.029)   | 0.7960 (0.031)   | 0.7912 (0.030)   | 0.8010 (0.027)   | –       |
+| ν          | 0.3408 (0.02083) | 0.3333 (0.023)   | 0.3454 (0.023)   | 0.3359 (0.023)   | 0.3497 (0.024)   | 0.3499 (0.024)   | –       |
+| DIC        | 4953.7           | 3974.34          | 3953.69          | 3981.33          | 3953.66          | 3956.85          | –       |
+| MADfit     | 4.129            | 4.141            | 4.075            | 4.156            | 4.132            | 4.074            | 4.118   |
+| MSPEfit    | 33.664           | 34.433           | 33.102           | 34.108           | 33.508           | 33.176           | 33.665  |
+| MADpred    | –                | 4.082            | 4.3003           | 4.034            | 4.106            | 4.316            | 4.168   |
+| MSPEpred   | –                | 30.529           | 34.695           | 32.339           | 34.059           | 34.663           | 33.257  |
+
+<sup>a</sup> The coefficient estimates are based on the mode (posterior value) (see Section [6\).](#page-9-0)
+
+Table 4 Modeling results for the FB NB models using the Toronto data
+
+| Estimates | Full data       | Sample 1        | Sample 2        | Sample 3        | Sample 4        | Sample 5       | Average |
+|-----------|-----------------|-----------------|-----------------|-----------------|-----------------|----------------|---------|
+| ln(β0)    | −10.11 (0.4794) | −9.862 (0.4018) | −10.48 (0.4359) | −9.799 (0.5509) | −10.14 (0.5256) | −9.87 (0.5272) | –       |
+| β1        | 0.6071 (0.046)  | 0.5788 (0.039)  | 0.6459 (0.044)  | 0.5775 (0.052)  | 0.6057 (0.051)  | 0.5787 (0.055) | –       |
+| β2        | 0.6852 (0.021)  | 0.6903 (0.022)  | 0.6840 (0.025)  | 0.6848 (0.023)  | 0.6902 (0.024)  | 0.6918 (0.023) | –       |
+| φ         | 7.120 (0.619)   | 6.898 (0.669)   | 7.256 (0.721)   | 7.045 (0.687)   | 7.388 (0.734)   | 7.567 (0.756)  | –       |
+| DIC       | 4777.59         | 3821.9          | 3817.8          | 3836.35         | 3811.16         | 3824.74        | –       |
+| MADfit    | 4.141           | 4.174           | 4.094           | 4.168           | 4.145           | 4.096          | 4.136   |
+| MSPEfit   | 32.742          | 33.503          | 32.104          | 33.271          | 32.527          | 32.354         | 32.750  |
+| MADpred   | –               | 4.024           | 4.379           | 4.058           | 4.121           | 4.346          | 4.186   |
+| MSPEpred  | –               | 29.594          | 35.091          | 30.855          | 33.331          | 33.989         | 32.572  |
+
+<sup>b</sup> 2756 observations for the fitted data and 644 observations for the predicted data.
+
+<sup>b</sup> Posterior credible standard error.
+
+Estimates Full data Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Average ln(β0) −10.2458 (0.465) −10.1664 (0.525) −10.4031 (0.520) −9.7398 (0.513) −10.2040 (0.513) −9.8473 (0.511) – β<sup>1</sup> 0.6207 (0.046) 0.6079 (0.051) 0.6393 (0.051) 0.5707 (0.051) 0.6119 (0.051) 0.5778 (0.051) – β<sup>2</sup> 0.6853 (0.0211) 0.6910 (0.0241) 0.6826 (0.024) 0.6860 (0.024) 0.6905 (0.024) 0.6903 (0.0234) – α<sup>a</sup> 0.1398 (0.0122) 0.1443 (0.014) 0.1372 (0.014) 0.1410 (0.014) 0.1349 (0.0134) 0.1315 (0.0134) – AIC 5077.3 4068.8 4052.6 4080.5 4045.2 4054.3 – MADfit 4.142 4.170 4.092 4.168 4.146 4.096 4.136 MSPEfit 32.699 33.444 32.127 33.264 32.517 32.370 32.737 MADpred – 4.026 4.374 4.062 4.117 4.348 4.185 MSPEpred – 29.547 34.973 30.898 33.271 34.002 32.538
+
+Table 5 Modeling results for the MLE NB models using the Toronto data
+
+[Table 4](#page-5-0) summarizes the results of the FB NB models for the Toronto data. This table exhibits similar characteristics as for the COM-Poisson GLMs in terms of GOF statistics and predictive performance despite the fact that the coefficients are a little bit different. Nonetheless, this difference did not affect the fit and predictive capabilities of the models. A comparison of the models' output is presented below.
+
+Table 5 summarizes the results of the MLE NB models for the Toronto data. This table shows exactly the same results as for the FB NB. This is expected since a vague prior was used for the FB NB models. The results indicate that the FB NB models are relatively stable and can, therefore, be compared with the COM-Poisson GLM.
+
+Fig. 1 compares the estimated number of crashes from the COM-Poisson and NB models for three minor AADT flows (*F*Min). The figure illustrates that the estimated values are slightly different, especially when*F*Min is equal to 500 vehicles/day, with the COM-Poisson output being always lower than the NB output. For *F*Min = 500, the maximum absolute difference is about 0.9 crash per year. At the other end of the spectrum, the maximum absolute difference is about two crashes per year for *F*Maj = 70,000 and*F*Min = 3000. Although this difference appears to be large, the relative difference is about 17%. It should be pointed out that when the posterior mean value is used for the COM-Poisson model rather than the mode (e.g., assuming μ is the predicted mean), both curves get closer for all minor flow values. For*F*Min = 3000, the absolute maximum difference becomes less than one crash per year. Thus, both models could be used for analyzing this dataset.
+
+# *5.2. Texas*
+
+[Table 6](#page-7-0) summarizes the results of the COM-Poisson models for the Texas data. Similar to the first dataset, the 95% marginal posterior credible intervals for each of the coefficients did not include the origin. In addition, the coefficients do not vary significantly between the different samples.
+
+[Table 7](#page-7-0) summarizes the results of the FB NB models for the Texas data. This table indicates that FB NB models estimate a slightly lower value for the coefficient for the traffic flow variable than for the COM-Poisson GLMs. Similar to the Toronto data, the COM-Poisson GLMs offer the same statistical performance as for FB NB models.
+
+![](_page_6_Figure_11.jpeg)
+
+Fig. 1. Estimated values for the Toronto DATA. NB and COM-Poisson models: (a) minor AADT = 500 vehicles/day; (b) minor AADT = 3000 vehicles/day; and (c) minor AADT = 5000 vehicles/day.
+
+<sup>a</sup> <sup>α</sup> <sup>=</sup> <sup>1</sup>/φ.
+
+<span id="page-7-0"></span>Table 6 Modeling results for the COM-Poisson GLMs using the Texas data
+
+| Estimatesa | Full data       | Sample 1       | Sample 2       | Sample 3       | Sample 4       | Sample 5       | Average |
+|------------|-----------------|----------------|----------------|----------------|----------------|----------------|---------|
+| ln(β0)     | −8.235 (0.242)b | −8.442 (0.267) | −8.333 (0.284) | −7.877 (0.223) | −8.155 (0.234) | −8.338 (0.249) | –       |
+| β1         | 1.081 (0.025)   | 1.102 (0.028)  | 1.089 (0.030)  | 1.044 (0.023)  | 1.074 (0.025)  | 1.092 (0.026)  | –       |
+| ν          | 0.3608 (0.012)  | 0.3504 (0.014) | 0.3465 (0.013) | 0.3699 (0.014) | 0.3701 (0.015) | 0.3650 (0.013) | –       |
+| DIC        | 13325.6         | 10688.8        | 10711.2        | 10673.9        | 10652.6        | 10710.6        | –       |
+| MADfit     | 2.385           | 2.433          | 2.435          | 2.369          | 2.371          | 2.415          | 2.401   |
+| MSPEfit    | 21.985          | 24.297         | 23.708         | 18.970         | 20.050         | 22.938         | 21.991  |
+| MADpred    | –               | 2.240          | 2.242          | 2.388          | 2.413          | 2.283          | 2.313   |
+| MSPEpred   | –               | 14.462         | 16.650         | 31.748         | 28.745         | 18.835         | 22.088  |
+
+<sup>a</sup> The coefficient estimates are based on the mode (posterior value) (see Section [6\).](#page-9-0)
+
+Table 7 Modeling results for the FB NB models using the Texas data
+
+| Estimates | Full data      | Sample 1       | Sample 2       | Sample 3       | Sample 4       | Sample 5       | Average |
+|-----------|----------------|----------------|----------------|----------------|----------------|----------------|---------|
+| ln(β0)    | −6.512 (0.227) | −6.701 (0.264) | −6.488 (0.211) | −6.356 (0.221) | −6.449 (0.209) | −6.618 (0.204) | –       |
+| β1        | 0.9206 (0.025) | 0.9403 (0.029) | 0.9178 (0.023) | 0.9046 (0.024) | 0.9134 (0.023) | 0.9320 (0.022) | –       |
+| φ         | 3.229 (0.177)  | 3.155 (0.189)  | 3.073 (0.184)  | 3.235 (0.198)  | 3.253 (0.198)  | 3.328 (0.200)  | –       |
+| DIC       | 12408.7        | 9882.13        | 9908.99        | 9988.68        | 9964.02        | 9983.94        | –       |
+| MADfit    | 2.437          | 2.466          | 2.508          | 2.430          | 2.424          | 2.453          | 2.453   |
+| MSPEfit   | 20.699         | 22.758         | 22.487         | 18.070         | 18.284         | 21.651         | 20.658  |
+| MADpred   | –              | 2.297          | 2.143          | 2.509          | 2.464          | 2.378          | 2.358   |
+| MSPEpred  | –              | 12.929         | 13.307         | 31.162         | 29.854         | 17.369         | 20.924  |
+
+Table 8 Modeling results for the MLE NB models using the Texas data
+
+| Estimates | Full data       | Sample 1        | Sample 2        | Sample 3        | Sample 4        | Sample 5        | Average |
+|-----------|-----------------|-----------------|-----------------|-----------------|-----------------|-----------------|---------|
+| ln(β0)    | −6.5605 (0.199) | −6.6293 (0.224) | −6.4570 (0.224) | −6.4266 (0.222) | −6.4652 (0.220) | −6.6163 (0.220) | –       |
+| β1        | 0.9260 (0.022)  | 0.9324 (0.025)  | 0.9143 (0.025)  | 0.9125 (0.025)  | 0.9151 (0.024)  | 0.9318 (0.024)  | –       |
+| αa        | 0.3095 (0.017)  | 0.3172 (0.019)  | 0.3255 (0.020)  | 0.3094 (0.019)  | 0.3075 (0.019)  | 0.3009 (0.018)  | –       |
+| AIC       | 13375           | 10674           | 10724           | 10773           | 10741           | 10749           | –       |
+| MADfit    | 2.440           | 2.463           | 2.506           | 2.434           | 2.424           | 2.453           | 2.453   |
+| MSPEfit   | 20.766          | 22.654          | 22.439          | 18.190          | 18.289          | 21.647          | 20.664  |
+| MADpred   | –               | 2.293           | 2.141           | 2.510           | 2.463           | 2.378           | 2.357   |
+| MSPEpred  | –               | 12.856          | 13.282          | 31.146          | 29.861          | 17.365          | 20.902  |
+
+<sup>a</sup> <sup>α</sup> <sup>=</sup> <sup>1</sup>/φ.
+
+Table 8 summarizes the results of the MLE NB models for the Texas data. This table shows exactly the same results as for the FB NB.
+
+Fig. 2 shows the comparison results between the estimated number of crashes per mile per 5-year of the COM-Poisson and
+
+![](_page_7_Figure_13.jpeg)
+
+Fig. 2. Estimated values for the Texas data: NB and COM-Poisson models.
+
+NB models for the Texas data (full dataset). This figure illustrates that both estimates are indeed very close.
+
+# *5.3. Marginal effects*
+
+An important issue in developing or using a regression model is the interpretation of the coefficients. Computing the marginal value of a particular variable can provide valuable information about how the regression coefficient related to that variable influence the expected mean value. For this exercise, the calculations of the marginal effect are slightly more complicated for the COM-Poisson distribution than for the NB distribution, which is usually straightforward. This is attributed to the fact that the parameter μ for the COM-Poisson is a centering parameter, as opposed to the expected mean value typically found in NB models.
+
+The relative marginal effect of a particular variable or covariate *Xi* can be estimated using the following equation [\(Cameron](#page-10-0)
+
+<sup>b</sup> Posterior credible standard error.
+
+![](_page_8_Figure_2.jpeg)
+
+Fig. 3. Marginal effect of the traffic flows for the Toronto model. (a) Marginal effect of the major flow: (i) marginal effect of major flow with COM-Poisson and (ii) marginal effect of major flow with NB. (b) Marginal effect of the minor flow: (i) marginal effect of minor flow with COM-Poisson and (ii) marginal effect of minor flow with NB.
+
+[and Trivedi, 1998\):](#page-10-0)
+
+$$\frac{1}{E[Y|X_i]} \frac{\partial E[Y|X_i]}{\partial X_i} \tag{21}$$
+
+In estimating the relative marginal effect of the variables, the mean approximation:
+
+$$E[Y] \approx \mu + 1/2\nu - 1/2$$
+ (22)
+
+can be used for the COM-Poisson models.
+
+As seen in Fig. 3a, the relative marginal effect of the major flow for the COM-Poisson model depends on the major and minor entering flows. This figure shows that the relative marginal effect on the expected mean for a unit increase in major flow remains nearly constant for lower minor flow volumes (e.g., 100 vehicles/day) and the rate of curvature increases with the increase in minor flows. On the other hand, the relative marginal effect of the major flow for the NB model is only independent of the minor flow. With a unit increase in major flow, the relative marginal effect on the expected mean value decreases. This decrease is smaller for higher major flows. Similar results can be seen for both COM-Poisson and NB models for the marginal effect related to the minor flow (Fig. 3b).
+
+Fig. 4 shows that the relative marginal effect on the expected mean value decreases with a unit increase in flow for both COM-Poisson and NB models. In this figure, the *y*-axis is formatted under the logarithmic scale. In Fig. 4, it can be seen that the marginal effect of traffic flow is higher for the NB model than for the COM-Poisson GLM. For the NB model, there is a sharp decrease in the marginal effect at lower flows and the curve decreases slightly for flows above 10,000 vehicles/day.
+
+![](_page_8_Figure_11.jpeg)
+
+Fig. 4. Marginal effect of traffic flow for the Texas model (*note*: *y*-axis is formatted under a logarithmic scale).
+
+## <span id="page-9-0"></span>**6. Discussion**
+
+This paper has shown that the COM-Poisson GLM offers potential for modeling motor vehicle crashes. First, the model performs as well as the NB model (FB and MLE) for the functional form that only includes traffic flow as covariates. As detailed in the modeling results, both models provided similar GOF statistics and predictive performance. [Guikema and](#page-10-0) [Coffelt \(2008\)](#page-10-0) have reported similar comparison results between the COM-Poisson GLM and the FB NB model. The models used in [Guikema and Coffelt \(2008\)](#page-10-0) included six covariates in both the centering and shape links. Hence, it is expected that COM-Poisson GLMs developed with several covariates, such as lane and shoulder widths, should work as well as the NB model.
+
+Second, although almost all crash datasets have been shown to exhibit over-dispersion (see [Lord et al., 2005b\),](#page-10-0) it has been documented that some crash datasets can sometimes experience under-dispersion [\(Oh et al., 2006\).](#page-11-0) The NB GLM could theoretically handle under-dispersion, since the dispersion parameter can be negative (var(*Y*) =μ+ (−α)μ2)). However, in this case, the mean of the Poisson is no longer gamma distributed because this latter distribution cannot have negative parameters (i.e., gamma(φ, φ)). In addition, researchers who have worked on the characterization of the NB distribution and GLM have indicated that a negative dispersion parameter could lead to a mis-specification of the pmf (when −1/(max of counts) < α) [\(Clark and Perry, 1989; Saha and Paul, 2005\).](#page-10-0) On the other hand, the COM-Poisson distribution has been shown to easily handle such datasets ([Shmueli et al., 2005; Kadane et al., 2006;](#page-11-0) [Geedipally et al., 2007; Guikema and Coffelt, 2008\).](#page-11-0) The fact that the model handles under-dispersed data makes it more useful than the NB model, which has difficulty coping with this kind of data (as described above). Although not a good analysis approach, a transportation safety analyst could theoretically not have to worry about the characteristics of the dispersion in the data, since the COM-Poisson GLM can handle both over- and under-dispersed data, and a combination of both, if the data are characterized as such.
+
+Third, as discussed above, crash data can sometimes be subjected to very low sample mean values, which create data characterized by a large number of zeros (with the hypothesis that the space and time scales have been appropriately used, see [Lord et al., 2005b\).](#page-10-0) Consequently, NB models do not perform well with such datasets since they may tend to under-predict zero values (or over-estimate non-zero count values). To overcome this problem, some researchers have suggested the use of zeroinflated Poisson and NB models ([Shankar et al., 1997\).](#page-11-0) However, other researchers have argued against the use of such models for modeling crash data, since this kind of data does not exhibit two distinct generating processes, one of which is characterized by having a long-term mean equal to zero (which is not feasible for crash data) [\(Lord et al., 2005b, 2007; Warton, 2005; Wedagama](#page-10-0) [et al., 2006; Kadane et al., 2006\).](#page-10-0) Depending upon the specification of the parameters λ and ν, the COM-Poisson model can predict more zeros than the NB model for the same mean value. Nonetheless, both models should not be used as a direct substitute to zero-inflated models (when they are warranted) (see [Kadane et al., 2006\).](#page-10-0) In short, further work is needed on this topic.
+
+Fourth, the COM-Poisson model is not significantly more difficult to implement than the FB NB model once the code for the likelihood is available. [Guikema and Coffelt \(2008\)](#page-10-0) developed the code needed implement the COM-Poisson GLM in WinBUGS, and this code will be made available through the WinBUGS developer web page ([www.winbugs](http://www.winbugs-development.org.uk/)development.org.uk/). For the models produced in this work, non-informative or vague priors were used for the regression coefficients. For the β coefficients, normal(0, 100) priors were used, for ν, a gamma(0.03, 0.1) prior was used, and for φ a gamma(0.1, 0.1) prior was used. The research team also experimented with other non-informative priors, but found that the priors did not significantly affect either GOF of the models or the posterior parameter estimates. In addition, the difference in computational times for these models was not enormous. For example, for the full Toronto data set, a run of the COM-Poisson model with 35,000 replications took about 5 h while a run of the FB NB model with 35,000 replications took between 1 and 1.5 h; the absolute difference seems large, but some simulation runs can sometimes take up to 2 or 3 days in WinBUGS to converge depending on the complexity of the model hierarchical structure. Overall, implementing the COM-Poisson model is not significantly more difficult than implementing the FB NB model once the code for the COM-Poisson model is available.
+
+Given the fact that the COM-Poisson GLM has so far only been applied once, there are many lines of research activities that could be investigated. First, similar to the work performed in this research, general AADT models should be evaluated for under-dispersed data using the COM-Poisson GLM. Second, further research should be conducted about data characterized by extremely low sample mean values (data with many zeros). Third, given the fact that crash data are often subjected to low sample mean values and small sample size, the stability of the COM-Poisson GLM should be investigated. Fourth, since the EB method is now used frequently in highway safety analyzes, an EB modeling framework should be developed for the COM-Poisson model. Fifth, methods about how to use COM-Poisson GLMs for identifying hazardous sites should be investigated. Sixth, although the analysis carried out in this research was conducted by assuming a fixed shape parameter (independent of covariates), further research should be done to examine the effects of a covariate-dependent shape parameter on COM-Poisson GLMs. Preliminary work conducted by the research team on some of these topics seems to offer positive results about using the COM-Poisson GLM for analyzing motor vehicle crashes (see, e.g., [Geedipally et al.,](#page-10-0) [2007\).](#page-10-0)
+
+#### **7. Summary and conclusions**
+
+This paper has documented the application of the COM-Poisson GLM for analyzing motor vehicle crashes. The COM-Poisson distribution, originally developed in 1962, has recently been re-introduced by statisticians for analyzing count <span id="page-10-0"></span>data subjected to either over- or under-dispersion. This innovative distribution is an extension of the Poisson distribution. The objectives of this study were to evaluate and compare the COM-Poisson GLM with the NB model commonly used for analyzing motor vehicle crashes. The comparison analysis was carried out using the most common functional forms used by transportation safety analysts, which link crashes to the entering flows at intersections or on segments. To accomplish the study objectives, several FB NB and COM-Poisson GLMs were developed using two datasets. The first dataset contained crash data collected at four-legged signalized intersections in Toronto, Ont. The second dataset included data collected for rural fourlane divided and undivided highways in Texas. Several methods were used to assess the statistical fit and predictive performance of the models.
+
+The results of this study show that COM-Poisson GLMs perform as well as FB NB models in terms of GOF statistics and predictive performance. This result is supported by another recent study on this topic (Guikema and Coffelt, 2008). Given the fact the COM-Poisson distribution can also handle underdispersed data, which have sometimes been observed in crash databases, the COM-Poisson GLM offers a better alternative over the NB model for modeling motor vehicle crashes. The advantage may be even greater if the model is found to be more stable than its counterpart for small sample sizes and low sample mean values. Finally, it is hoped that the new type of model described in this paper will open the door for better and more reliable tools for estimating the safety performance of entities located on transportation networks.
+
+#### **Acknowledgments**
+
+The authors would like to thank two anonymous reviewers for providing useful comments to improve this paper.
+
+# **References**
+
+- Abbess, C., Jarett, D., Wright, C.C., 1981. Accidents at blackspots: estimating the effectiveness of remedial treatment, with special reference to the "regression-to-mean" effect. Traffic Engineering and Control 22 (10), 535–542.
+- Airoldi, E.M., Anderson, A.G., Fienberg, S.E., Skinner, K.K., 2006. Who Wrote Ronald Reagan's Radio Addresses? Bayesian Analysis 1 (2), 289– 320.
+- Cameron, A.C., Trivedi, P.K., 1998. Regression analysis of count data. Econometric Society Monograph No. 30. Cambridge University Press, Boston.
+- Clark, S.J., Perry, J.N., 1989. Estimation of the negative binomial parameter κ by maximum quasi-likelihood. Biometrics 45, 309–316.
+- Conway, R.W., Maxwell, W.L., 1962. A queuing model with state dependent service rates. Journal of Industrial Engineering 12, 132–136.
+- Geedipally S., Guikema, S.D., Dhavala, S., Lord, D., 2007. Characterizing the Performance of a Bayesian Conway–Maxwell Poisson GLM. Working Paper, Zachry Department of Civil Engineering, Texas A&M University, College Station, TX.
+- Geedipally, S.R., Lord, D., 2008. Effects of the Varying Dispersion Parameter of Poisson-gamma models on the Estimation of Confidence Intervals of Crash Prediction models. In: Presented at the 87th Annual Meeting of the Transportation Research Board, Washington, D.C.
+- Guikema, S.D., Coffelt, J.P., 2008. A flexible count data regression model for risk analysis. Risk Analysis, in press.
+
+- Hauer, E., 1997. Observational Before–After Studies in Road Safety: Estimating the Effect of Highway and Traffic Engineering Measures on Road Safety. Elsevier Science Ltd., Oxford.
+- Hauer, E., Ng, J.C.N., Lovell, J., 1988. Estimation of safety at signalized intersections. Transportation Research Record 1185, 48–61.
+- Hughes, W., Eccles, K., Harwood, D., Potts, I., Hauer, E., 2005. Development of a Highway Safety Manual. Appendix C: Highway Safety Manual Prototype Chapter: Two-Lane Highways. NCHRP Web Document 62 (Project 17-18(4)). Washington, D.C. [\(http://onlinepubs.trb.org/onlinepubs/nchrp/nchrp](http://onlinepubs.trb.org/onlinepubs/nchrp/nchrp_w62.pdf) w62.pdf, accessed October 2007).
+- Kadane, J.B., Shmueli, G., Minka, T.P., Borle, S., Boatwright, P., 2006. Conjugate analysis of the Conway–Maxwell–Poisson distribution. Bayesian Analysis 1, 363–374.
+- Kulmala, R., 1995. Safety at Rural Three- and Four-Arm Junctions: Development and Applications of Accident Prediction Models. VTT Publications 233, Technical Research Centre of Finland, Espoo.
+- Lloyd-Smith, J.O., 2007. Maximum likelihood estimation of the negative binomial dispersion parameter for highly overdispersed data, with applications to infectious diseases. PLoS ONE 2 (2), e180 [\(http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1791715](http://www.pubmedcentral.nih.gov/articlerender.fcgi%3Fartid=1791715), accessed July 2007.
+- Lord, D., 2000. The prediction of accidents on digital networks: characteristics and issues related to the application of accident prediction models. Ph.D. Dissertation. Department of Civil Engineering, University of Toronto, Toronto, Ontario.
+- Lord, D., 2006. Modeling motor vehicle crashes using Poisson-gamma models: examining the effects of low sample mean values and small sample size on the estimation of the fixed dispersion parameter. Accident Analysis & Prevention 38 (4), 751–766.
+- Lord, D., Bonneson, J.A., 2005. Calibration of predictive models for estimating the safety of ramp design configurations. Transportation Research Record 1908, 88–95.
+- Lord, D., Manar, A., Vizioli, A., 2005a. Modeling crash-flow-density and crashflow-V/C ratio for rural and urban freeway segments. Accident Analysis & Prevention 37 (1), 185–199.
+- Lord, D., Miranda-Moreno, L.F., 2008. Effects of low sample mean values and small sample size on the estimation of the fixed dispersion parameter of Poisson-gamma models for modeling motor vehicle crashes: a Bayesian perspective. Safety Science, [doi:10.1016/j.ssci.2007.03.005](http://dx.doi.org/10.1016/j.ssci.2007.03.005), in press.
+- Lord, D., Washington, S.P., Ivan, J.N., 2005b. Poisson, Poisson-gamma and zero inflated regression models of motor vehicle crashes: balancing statistical fit and theory. Accident Analysis & Prevention 37 (1), 35–46.
+- Lord, D., Washington, S.P., Ivan, J.N., 2007. Further notes on the application of zero inflated models in highway safety. Accident Analysis & Prevention 39 (1), 53–57.
+- Maher, M.J., Summersgill, I., 1996. A Comprehensive methodology for the fitting predictive accident models. Accident Analysis & Prevention 28 (3), 281–296.
+- Maycock, G., Hall, R.D., 1984. Accidents at 4-arm roundabouts. TRRL Laboratory Report 1120. Transportation and Road Research Laboratory, Crowthorne, Bershire.
+- Miaou, S.-P., Lord, D., 2003. Modeling traffic crash-flow relationships for intersections: dispersion parameter, functional form, and Bayes versus empirical Bayes. Transportation Research Record 1840, 31–40.
+- Miaou, S.-P., Song, J.J., 2005. Bayesian ranking of sites for engineering safety improvements: decision parameter, treatability concept, statistical criterion and spatial dependence. Accident Analysis & Prevention 37 (4), 699–720.
+- Mitra, S., Washington, S., 2007. On the nature of over-dispersion in motor vehicle crash prediction models. Accident Analysis & Prevention 39 (3), 459–568.
+- Miranda-Moreno, L.F., Fu, L., 2007. Traffic safety study: empirical Bayes or full Bayes? Paper 07-1680. In: Proceedings of the 84th Annual Meeting of the Transportation Research Board, Washington, D.C.
+- Oh, J., Lyon, C., Washington, S.P., Persaud, B.N., Bared, J., 2003. Validation of the FHWA crash models for rural intersections: lessons learned. Transportation Research Record 1840, 41–49.
+
+- <span id="page-11-0"></span>Oh, J., Washington, S.P., Nam, D., 2006. Accident prediction model for railway-highway interfaces. Accident Analysis & Prevention 38 (2), 346–356.
+- Persaud, B.N., Lord, D., Palminaso, J., 2002. Issues of calibration and transferability in developing accident prediction models for urban intersections. Transportation Research Record 1784, 57–64.
+- Persaud B.N., Retting, R., Garder, P., Lord, D., 2001. Observational before-after study of U.S. roundabout conversions using the empirical Bayes method. Transportation Research Record 1751, 1–8.
+- Piegorsch, W.W., 1990. Maximum likelihood estimation for the negative binomial dispersion parameter. Biometrics 46, 863–867.
+- Poch, M., Mannering, F.L., 1996. Negative binomial analysis of intersectionaccident frequency. Journal of Transportation Engineering 122 (2), 105– 113.
+- Saha, K., Paul, S., 2005. Bias-corrected maximum likelihood estimator of the negative binomial dispersion parameter. Biometrics 61 (1), 179–185.
+- SAS Institute Inc., 2002. SAS System for Windows. Version 9. Cary, NC.
+- Shankar, V., Milton, J., Mannering, F.L., 1997. Modeling accident frequency as zero-altered probability processes: an empirical inquiry. Accident Analysis & Prevention 29 (6), 829–837.
+
+- Shmueli, G., Minka, T.P., Kadane, J.B., Borle, S., Boatwright, P., 2005. A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution. Journal of the Royal Statistical Society, Part C 54, 127–142.
+- Spiegelhalter, D.J., Thomas, A., Best, N.G., Lun, D., 2003. WinBUGS Version 1.4.1 User Manual. MRC Biostatistics Unit, Cambridge. Available from: <http://www.mrcbsu.cam.ac.uk/bugs/welcome.shtml>.
+- Warton, D.I., 2005. Many zeros does not mean zero inflation: comparing the goodness-of-fit of parametric models to multivariate abundance data. Environmetrics 16, 275–289.
+- Wedagama, D.M., Bird, R.N., Metcalf, A.V., 2006. The influence of urban landuse on non-motorised transport casualties. Accident Analysis & Prevention 38 (6), 1049–1057.
+- Wood, G.R., 2002. Generalized linear accident models and goodness of fit testing. Accident Analysis & Prevention 34 (1), 417–427.
+- Wood, G.R., 2005. Confidence and prediction intervals for generalized linear accident models. Accident Analysis & Prevention 37 (2), 267–273.
+- Xie, Y., Lord, D., Zhang, Y., 2007. Predicting motor vehicle collisions using Bayesian neural networks: an empirical analysis. Accident Analysis & Prevention 39 (5), 922–933.

Zotero/001_artiklid/Road Safety Indicators/lordnegativebinomiallindleydistribution2011.md

@@ -0,0 +1,263 @@
+---
+category: literaturenote
+citekey: lordnegativebinomiallindleydistribution2011
+title: The negative binomial–Lindley distribution as a tool for analyzing crash data characterized by a large amount of zeros
+authors: "Lord, Dominique; Geedipally, Srinivas Reddy"
+year: 2011
+date: 2011-00-00 2011
+doi: 10.1016/j.aap.2011.04.004
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2011.04.004"
+zotero_key: SG9BG7BW
+zotero_storage: GPIZJGNZ
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Lord, Dominique"
+status: converted
+---
+Contents lists available at [ScienceDirect](http://www.sciencedirect.com/science/journal/00014575)
+
+# Accident Analysis and Prevention
+
+journal homepage: [www.elsevier.com/locate/aap](http://www.elsevier.com/locate/aap)
+
+![](_page_0_Picture_5.jpeg)
+
+# The negative binomial–Lindley distribution as a tool for analyzing crash data characterized by a large amount of zeros
+
+Dominique Lord<sup>a</sup>,∗, Srinivas Reddy Geedipally <sup>b</sup>,<sup>1</sup>
+
+- <sup>a</sup> Zachry Department of Civil Engineering, Texas A&M University, 3136 TAMU, College Station, TX 77843-3136, USA
+- <sup>b</sup> Engineering Research Associate, Texas Transportation Institute, Texas A&M University, 3135 TAMU, College Station, TX 77843-3135, USA
+
+## article info
+
+Article history: Received 17 January 2011 Received in revised form 1 April 2011 Accepted 2 April 2011
+
+Key words: Count data Safety Negative binomial distribution Poisson distribution Negative binomial–Lindley distribution
+
+### abstract
+
+The modeling of crash count data is a very important topic in highway safety. As documented in the literature, given the characteristics associated with crash data, transportation safety analysts have proposed a significant number of analysis tools, statistical methods and models for analyzing such data. Among the data issues, we find the one related to crash data which have a large amount of zeros and a long or heavy tail. It has been found that using this kind of dataset could lead to erroneous results or conclusions if the wrong statistical tools or methods are used. Thus, the purpose of this paper is to introduce a new distribution, known as the negative binomial–Lindley (NB-L), which has very recently been introduced for analyzing data characterized by a large number of zeros. The NB–L offers the advantage of being able to handle this kind of datasets, while still maintaining similar characteristics as the traditional negative binomial (NB). In other words, the NB–L is a two-parameter distribution and the long-term mean is never equal to zero. To examine this distribution, simulated and observed data were used. The results show that the NB–L can provide a better statistical fit than the traditional NB for datasets that contain a large amount of zeros.
+
+© 2011 Elsevier Ltd. All rights reserved.
+
+#### **1. Introduction**
+
+The modeling of crash count data is a very important topic in highway safety. As documented in [Lord and Mannering \(2010\),](#page-4-0) given the characteristics associated with crash data, transportation safety analysts have proposed a significant number of analysis tools and models for analyzing such data. Among the data issues documented in the paper, we find the one related to crash data which have a large amount of zeros and a long or heavy tail. For such datasets, the number of sites where no crash is observed is so large that traditional statistical distributions or models, such as the Poisson and Poisson-gamma or negative binomial (NB) distributions, cannot be used efficiently. The Poisson distribution tends to underestimate the number of zeros given the mean of the data, while the NB may over-estimate zeros, but under-estimate observations with a count. This is obviously dependent upon the characteristics of the tail.
+
+The large amount of zeros observed in crash data have initially been attributed to observations or sites that can be categorized under two states: a safe state, where no crash can occur, and a non-safe state [\(Miaou, 1994; Shankar et al., 1997, 2003\).](#page-4-0) A portion of the zero counts come from the safe state, while the rest of the zero counts come from a Poisson or NB distribution. The observations classified under the first state are considered as 'added' zeros. The zero-inflated model (both used for the Poisson and NB) has been consequently proposed to analyze this kind of dataset, usually because they provide better statistical fit ([Shankar et al., 1997,](#page-4-0) [2003; Kumara and Chin, 2003\).](#page-4-0) Some researchers [\(Warton, 2005;](#page-4-0) [Lord et al., 2005, 2007\)](#page-4-0) however have raised important methodological issues about the use of such models, including the fact that the safe state has a distribution with a long-term mean equal to zero, which is theoretically impossible. [Lord et al. \(2005\)](#page-4-0) noted that the large amount of zeros can be attributed to the following factors: (1) sites with a combination of low exposure, high heterogeneity, and sites categorized as high risk; (2) analyses conducted with small time or spatial scales; (3) data with a relatively high percentage of missing or mis-reported crashes; and (4) crash models with omitted important variables. More recently, [Mayshkina and Mannering](#page-4-0) [\(2009\)](#page-4-0) have proposed a zero-state Markov switching model, which overcomes some of the criticisms discussed above, for analyzing longitudinal datasets characterized by a large number of zeros.
+
+In cases in which the characteristics of the dataset cannot or is very difficult to be changed (as it will be discussed further below), the large number of zeros could still create a lot of difficulties for properly analyzing such dataset. This could obviously lead to erroneous results or conclusions if the wrong statistical tools or
+
+<sup>∗</sup> Corresponding author. Tel.: +1 979 458 3949; fax: +1 979 845 6481. E-mail addresses: [d-lord@tamu.edu](mailto:d-lord@tamu.edu) (D. Lord), [srinivas-g@ttimail.tamu.edu](mailto:srinivas-g@ttimail.tamu.edu)
+
+<sup>(</sup>S.R. Geedipally). <sup>1</sup> Tel.: +1 979 862 1651; fax: +1 979 845 6006.
+
+<span id="page-1-0"></span>methods are used. Thus, the purpose of this paper is to introduce a new distribution that has very recently been introduced for analyzing data characterized by a large number of zeros. This mixed distribution is known as the NB–Lindley (NB–L) distribution ([Zamani and Ismail, 2010\),](#page-4-0) which as the name implies, is a mixture of the NB and the Lindley distributions [\(Lindley, 1958; Ghitany](#page-4-0) [et al., 2008\).](#page-4-0) This two-parameter distribution has interesting and sound theoretical properties in which the distribution is characterized by a single long-term mean that is never equal to zero and a single variance function, similar to the traditional NB distribution. The properties of the NB–Lindley distribution are examined using simulated and observed data and a discussion is presented about the potential use of the NB-L distribution for traffic safety analyses. It important to point out that all documented distributions, such as the Poisson-gamma, Poisson-lognormal, Poisson-Pascal or the NB–L in highway safety research are in fact used as an approximation to describe the crash process. This process is known as the Poisson trials with unequal probability of events (See [Lord et al.,](#page-4-0) [2005, f](#page-4-0)or additional details).
+
+The paper is divided into five sections. Section 2 describes the characteristics of the NB–Lindley distribution. Section 3 presents the comparison analysis between the Poisson, NB and NB–L using simulated and observed data. Section [4](#page-3-0) provides additional information for future work. Section [5](#page-4-0) summarizes the study results.
+
+#### **2. Characteristics of the negative binomial–Lindley distribution**
+
+As discussed above, the NB–L distribution is a mixture of negative binomial and Lindley distributions. This mixed distribution has a thick tail and can be used when the data contains large number of zeros.
+
+The negative binomial distribution is a mixture of Poisson and gamma distribution. The probability mass function (pmf) of the NB distribution can be given as:
+
+$$P(Y = y; \phi, p) = \frac{\Gamma(\phi + y)}{\Gamma(\phi) \times y!} (1 - p)^{\phi}(p)^{y}; \quad \phi > 0, \quad 0$$
+
+The parameter 'p' is defined as the probability of success in each trial and is given as:
+
+$$p = \frac{\mu}{\mu + \phi} \tag{2}$$
+
+where, = E(Y) = mean; and, -= inverse dispersion parameter.
+
+Then, it can be shown that the variance is ([Casella and Berger,](#page-4-0) [1990\):](#page-4-0)
+
+$$Var(Y) = \phi \frac{p}{(1-p)^2} = \frac{1}{\phi}\mu^2 + \mu$$
+ (3)
+
+Using Eqs. (2) and (3), the pmf of the NB distribution can be re-parameterized this way:
+
+$$P(Y = y; \mu, \phi) = \frac{\Gamma(\phi + y)}{\Gamma(\phi)\Gamma(y + 1)} \left(\frac{\phi}{\mu + \phi}\right)^{\phi} \left(\frac{\mu}{\mu + \phi}\right)^{y}$$
+(4)
+
+The pmf in Eq. (4) is the one normally used for crash count data. The Lindley distribution is a mixture of exponential and gamma distribution ([Lindley, 1958; Ghitany et al., 2008; Zamani and Ismail,](#page-4-0) [2010\).](#page-4-0) The pmf of the Lindley distribution can be defined as follows:
+
+$$f(X = x; \theta) = \frac{\theta^2}{\theta + 1} (1 + x) e^{-\theta x}; \quad \theta > 0, \quad x > 0$$
+ (5)
+
+A random variable Z is assumed to follow a NB–L (r,) distribution when the following conditions satisfy:
+
+Z
+$$\sim$$
+NB  $\left(r,P=1-\mathrm{e}^{-\lambda}\right)$  and  $\lambda\sim$ Lindley  $\left(\theta\right)$ 
+
+The pmf of the NB–L distribution is given as ([Zamani and Ismail,](#page-4-0) [2010\):](#page-4-0)
+
+$$P(Z=z;r,\theta) = \frac{\Gamma(r+z)}{\Gamma(r)\times z!} \frac{\theta^2}{\theta+1} \sum_{j=0}^{z} \frac{\Gamma(z+1)}{\Gamma(j+1)\times\Gamma(z+j+1)} (-1)^{j} \times \frac{\theta+r+j-1}{(\theta+r+j)^2}$$
+(6)
+
+The parameter 'r' is the shape parameter of NB–L distribution, similar to the inverse dispersion parameter '-' of the NB distribution. The parameter '', in combination with shape parameter 'r' dictates the mean and variance of the NB–L distribution.
+
+The first moment (i.e., the mean) of the NB–L (r,) is given as:
+
+$$E(Z) = r \left[ \frac{\theta^3}{(\theta + 1)(\theta - 1)^2} - 1 \right]$$
+ (7)
+
+It should be noted that E[Z] = E[Y] =
+
+The second moment of the NB–L (r,) is given as:
+
+$$E(Z^{2}) = (r + r^{2}) \left[ \frac{\theta^{2}(\theta - 1)}{(\theta + 1)(\theta - 2)^{2}} \right] - (r + 2r^{2}) \left[ \frac{\theta^{3}}{(\theta + 1)(\theta - 1)^{2}} \right] + r^{2}$$
+(8)
+
+The variance of the NB–L (r,) is calculated as:
+
+$$Var(Z) = E(Z^{2}) - (E(Z))^{2}$$
+(9)
+
+As described above, the NB–L distribution is a two-parameter distribution, which implies that the mean is never equal to zero. The NB–L is in fact an extension of the NB distribution.
+
+To estimate the parameters r and , Eqs. (7) and (8) need to be solved iteratively and both parameters should be greater than 0 (r > 0 and > 0). The parameters can also be estimated by solving the likelihood function, but the function is difficult to manipulate, since the partial derivatives contain multiple solutions. Additional work is therefore needed for finding the optimal solution among all the possible ones. More detailed information can be found in [Zamani and Ismail \(2010\).](#page-4-0)
+
+## **3. Application of the negative binomial–Lindley distribution**
+
+This section presents the comparison analysis results between the Poisson, NB and NB–L distributions using simulated and observed data.
+
+### 3.1. Simulated data
+
+The simulation protocol is the same one used by [Lord et al.](#page-4-0) [\(2005\), w](#page-4-0)hich consisted in simulating a Poisson distribution and add observations with the value zero to 'simulate' a two-state process, one of which is characterized by a long-term equal to zero. For this example, count data with 100 observations were simulated using a Poisson distribution with a mean equal to 0.50. Then, 100, 150 and 200 observations with the value zero were added to the data. The simulated data are summarized in [Fig. 1. T](#page-2-0)he original simulated data produced 57 observations with the value zero, 34 with the value 1, 8 with the value 2, and 1 with a value above 3 (5 to be exact).
+
+The Poisson, NB and NB–L distributions were fitted based on the simulated data. Using the mean and variance of the data, the parameters were estimated with Eqs.(2) and (3)for the NB distribution and Eqs.(7)–(9)for the NB–L distribution. After the parameters
+
+<span id="page-2-0"></span>![](_page_2_Figure_2.jpeg)
+
+**Fig. 1.** Simulated data: Poisson distribution with a mean equal to 0.5 (and added 0s).
+
+were estimated, the predicted probability and frequency were calculated for each count. The results are summarized in Table 1. The goodness-of-fit was assessed using the Chi-squared test and the log-likelihood value. As discussed in [Mitra and Washington \(2007\)](#page-4-0) and [Lord and Park \(2008\),](#page-4-0) it is recommended to use more than one goodness-of-fit (GOF) measure for evaluating models and distributions. The log-likelihood value was calculated in two steps. First, the likelihood was calculated for each observation using the estimated parameters and pmf. Then, the natural logarithm of the likelihood was computed. For each dataset, the sum of the loglikelihood was calculated and compared for the Poisson, NB and NB–L distributions.
+
+Table 1 shows that both the NB and the NB–L distributions provide adequate fit, but the NB–L offers an overall better statistical fit for all sample sizes with added zeros. Furthermore, the fit improves as the number or proportion of zeros increases. It should be pointed out that the fit using the log-likelihood shows that that all three distributions are very close. However, as discussed above about using different GOF measures, when a non-likelihood-based GOF criterion is used the NB–L shows a better fit. When 200 observations with zeros are added, the ratio between the two Chi-square values becomes larger than the ratio for the 150 added zeros.
+
+As discussed in [Miaou and Lord \(2003\)](#page-4-0) and [Lord et al. \(2005,](#page-4-0) [2007\), t](#page-4-0)he primary goal for analyzing distributions and regression models should not be solely based on finding the absolute best statistical fit. It is also very important to look at the data generating process, the relationship between the variables and whether the distribution or model is logically or theoretically sound. [Miaou and](#page-4-0) [Lord \(2003\)](#page-4-0) referred to the latter characteristic as "goodness-oflogic." Thus, although the NB–L offers a better statistical fit, it is still a rational distribution that can be used for characterizing crash data, given the fact that the long-term mean can never equal zero.
+
+**Table 1** Goodness-of-fit analysis – simulated results.
+
+| Count          | Observed frequency                                   | Poisson distribution | NB distribution    | NB–L distribution  |
+|----------------|------------------------------------------------------|----------------------|--------------------|--------------------|
+|                | Original data + 100 observations (% of zeros = 78.5) |                      |                    |                    |
+| 0              | 157                                                  | 151.9                | 158.4              | 158.2              |
+| 1              | 34                                                   | 41.8                 | 31.6               | 32.1               |
+| 2              | 8                                                    | 5.7                  | 7.5                | 7.2                |
+| 3+             | 1                                                    | 0.6                  | 2.5                | 2.5                |
+| Parameters     |                                                      | = 0.275              | = 0.275<br>= 0.725 | = 2.7113 r = 6.925 |
+| Chi-square     |                                                      | 2.77                 | 0.30               | 0.17               |
+| Log-likelihood |                                                      | −87.1970             | −87.1767           | −87.1757           |
+|                | Original data + 150 observations (% of zeros = 82.8) |                      |                    |                    |
+| 0              | 207                                                  | 200.6                | 208.3              | 207.7              |
+| 1              | 34                                                   | 44.1                 | 31.9               | 33.1               |
+| 2              | 8                                                    | 4.9                  | 7.3                | 6.8                |
+| 3+             | 1                                                    | 0.4                  | 2.5                | 2.4                |
+| Parameters     |                                                      | = 0·22               | = 0.220<br>= 0.506 | = 14.065 r = 2.682 |
+| Chi-square     |                                                      | 5.25                 | 0.21               | 0.03               |
+| Log-likelihood |                                                      | −108.9260            | −108.8877          | −108.8858          |
+|                | Original Data + 200 observations (% of zeros = 85.7) |                      |                    |                    |
+| 0              | 257                                                  | 249.8                | 258.2              | 257.4              |
+| 1              | 34                                                   | 45.8                 | 32.2               | 33.7               |
+| 2              | 8                                                    | 4.2                  | 7.2                | 6.5                |
+| 3+             | 1                                                    | 0.3                  | 2.5                | 2.4                |
+| Parameters     |                                                      | = 0.183              | = 0.183<br>= 0.389 | = 1.0889 r = 1.659 |
+| Chi-square     |                                                      | 7.85                 | 0.16               | 0.01               |
+| Log-likelihood |                                                      | −130.5944            | −130.5342          | −130.5320          |
+
+Bold value indicates best goodness-of-fit.
+
+<span id="page-3-0"></span>**Table 2** Single-vehicle fatal crashes on divided multilane rural highways in Texas between 1997 and 2001.
+
+| Crashes        | Observed frequency | Poisson | NB                 | NB–L               |
+|----------------|--------------------|---------|--------------------|--------------------|
+| 0              | 1532               | 1509.2  | 1534.4             | 1532.9             |
+| 1              | 162                | 198.2   | 154.7              | 158.3              |
+| 2              | 19                 | 13.0    | 25.8               | 23.7               |
+| 3              | 6                  | 0.6     | 4.9                | 4.6                |
+| 4+             | 2                  | 0.0     | 1.2                | 1.4                |
+| Parameters     |                    | = 0.131 | = 0.131<br>= 0.434 | = 15.984 r = 1.851 |
+| Chi-square     |                    | 102.99  | 2.73               | 1.68               |
+| Log-likelihood |                    | −715.1  | −696.1             | −695.6             |
+
+Bold value indicates best goodness-of-fit.
+
+**Table 3** Single-vehicle roadway departure crashes on rural two-lane horizontal curves in Texas between 2003 and 2008.
+
+| Crashes        | Observed frequency | Poisson   | NB                 | NB–L              |
+|----------------|--------------------|-----------|--------------------|-------------------|
+| 0              | 29087              | 28471.6   | 29204.8            | 29133.6           |
+| 1              | 2952               | 3918.0    | 2706.0             | 2855.5            |
+| 2              | 464                | 269.6     | 567.4              | 503.1             |
+| 3              | 108                | 12.4      | 141.1              | 120.9             |
+| 4              | 40                 | 0.4       | 37.8               | 35.9              |
+| 5              | 9                  | 0.0       | 10.6               | 13.1              |
+| 6              | 5                  | 0.0       | 3.0                | 3.3               |
+| 7              | 2                  | 0.0       | 0.9                | 3.3               |
+| 8              | 3                  | 0.0       | 0.3                | 0.0               |
+| 9              | 1                  | 0.0       | 0.1                | 0.0               |
+| 10+            | 1                  | 0.0       | 0.0                | 3.3               |
+| Parameters     |                    | = 0.138   | = 0.138<br>= 0.284 | = 9.212 r = 1.018 |
+| Chi-square     |                    | 2297.31   | 57.47              | 11.68             |
+| Log-likelihood |                    | −14,208.1 | −13,557.7          | −13,529.8         |
+
+Bold value indicates best goodness-of-fit.
+
+The next section describes the results for the observed data.
+
+### 3.2. Observed data
+
+Two datasets were used for this part of the analysis. The first one included single-vehicle fatal crashes that occurred on divided multilane rural highways between 1997 and 2001. The data were collected as a part of NCHRP 17-29 research project titled "Methodology for estimating the safety performance of multilane rural highways" [\(Lord et al., 2008\).](#page-4-0) The data contained 1721 segments that varied from 0.10 mile to 11.21 miles, with an average equal to 1.01 miles. The sample mean was equal to 0.13. About 89% of the segments had no fatal crash. As discussed above, one way to reduce the number of zeros would be to change the spatial scale by aggregating shorter segments and create a new sample that contains longer segments. However, it may not always possible to aggregate the data due to loss of homogeneity of various geometric elements among different segments. For the purpose of this example, we will leave this dataset as is.
+
+The second dataset included single-vehicle roadway departure fatal crashes that occurred on 32,672 rural two-lane horizontal curves between 2003 and 2008. The sample mean is equal to 0.14. For this dataset, about 90% of the data experienced no crash during the 5-year period. Given the proportion of zeros, one may believe that horizontal curves are very safe, although previous research indicate that they are not when compared to tangent segments ([AAHSTO, 2010\).](#page-4-0) The large number of zeros for this dataset can be explained by the sample that contains very short segments (about 90% are less than 0.3 miles). It would consequently be very difficult to change the spatial scale to reduce the number of zeros. One could perhaps increase the number of years to increase the sample size, if the data are available.
+
+Tables 2 and 3 summarize the goodness-of-fit results for the two datasets. Similar to the results shown in the previous section, the two tables show that the NB–L distribution provides a better fit.
+
+In summary, the NB–L provided a better fit than the traditional NB distribution for all simulated and observed datasets that included a large number of zeros. However, as pointed out by [Zamani and Ismail \(2010\), t](#page-4-0)he NB–L distribution works very well only when the dataset contains many observations with zero count. Based on the additional simulated data (not shown here), when the proportion of zeros is below 80%, the traditional NB distribution offers a performance that is equal to that of the NB–L. This is probably explained by the fact that the variance may not be large enough for the NB–L distribution to properly capture the variation. Since the distribution is also influenced by the length of the tail, it is suggested to evaluate both the NB and NB–L distributions when the percentage of zeros is above 70% (to be conservative) and select the one that provides the best goodness-of-fit statistic, as both are logically sound.
+
+# **4. Further work**
+
+Since this is a newly introduced distribution, there are a lot of avenues for further work, both in terms of theoretical and practical applications. This paper focused on the distribution, but most of the research work in highway safety research is related to the development and application regression or statistical models that link crashes to covariates, such as traffic flow and geometric design characteristics. This means that the data should be analyzed in the context of a generalized linear model (GLM) similar to what [Guikema and Coffelt \(2008\)](#page-4-0) and [Sellers and Shmueli \(2010\)](#page-4-0) did for the Conway-Maxwell-Poisson (COM-Poisson) model.
+
+Like the COM-Poisson distribution, the NB–L is a complex distribution and the pmf and likelihood need to be further manipulated in order to use it for modeling crash data. Unlike the NB model, the mean cannot be easily linked to the covariates when the MLE is used for estimating the coefficients (note: the centric parameter is not equal or the same as the mean, , of the NB distribution). Furthermore, this type of distribution falls under the category of <span id="page-4-0"></span>hierarchical models, where the intermediate parameters need to be integrated separately (Booth et al., 2001). Although Zamani and Ismail (2010) have provided an approach to maximize the log-likelihood, Bayesian methods may be needed for developing the GLM. This is governed by the function that uses intermediate parameters; this is in addition to the limitations described at the end of Section [2.](#page-1-0)
+
+Once the GLM is fully developed, the NB–L model should be evaluated and compared to other existing models that could potentially be used for handling data characterized by a large number of zeros. Those include random parameter (Anastasopoulos and Mannering, 2009) and zero-state Markov switching models (Mayshkina and Mannering, 2009) among others. This comparison will be the true test for the applicability of the NB–L for handling data with a lot of zeros. Finally, the NB–L model should also be evaluated for the small sample size and low sample mean problem (Lord, 2006).
+
+#### **5. Summary and conclusions**
+
+This paper has described the application of the NB–L distribution to datasets characterized by a large number of zeros and a heavy tail. Traditional statistical methods that have been proposed for analyzing such datasets have been found to suffer from important numerical and methodological problems. The newly introduced NB–L offers the advantage of being able to handle datasets with a large number of zeros, while still maintaining similar characteristics as the traditional NB. In other words, the NB–L is a two-parameter distribution and the long-term mean is never equal to zero. To examine this distribution, simulated and observed data were used. The results have shown that the NB-L always provided better statistical fit than the NB for datasets having a large number of zeros. However, since the distribution is also influenced by the length of the tail, it is suggested to evaluate both the NB and NB–L distributions and select the one that provides the best goodness-of-fit statistic. In conclusion, it is believed that this new distribution and subsequent GLM may offer a very useful tool for analyzing data characterized with a large amount of zeros.
+
+#### **Acknowledgements**
+
+The authors would like to thank Dr. Noriszura Ismail from the School of Mathematical Sciences at the Universiti Kebangsaan Malaysia and Dr. Soma S. Dhavala from the Department of Statistics at Texas A&M University for their useful advice. The paper also benefited from the input of Dr. David Warton from the School of Mathematics and Statistics at the University of New South Wales and Dr. Mike Maher from the Institute for Transport Studies at the University of Leeds as well as three anonymous reviewers.
+
+#### **References**
+
+- AAHSTO, 2010. Highway Safety Manual, 1st ed. American Association of State Highway and Transportation Officials, Washington, DC.
+- Anastasopoulos, P.C., Mannering, F.L., 2009. A note on modeling vehicle accident frequencies with random-parameters count models. Accident Analysis and Prevention 41 (1), 153–159.
+- Booth, J.G., Casella, G., Fried, H., Hobert, J.P., 2001. Negative Binomial Loglinear Mixed Models. Working Paper. Department of Statistics, University of Florida, Gainesville, FL.
+- Casella, G., Berger, R.L., 1990. Statistical Inference. Wadsworth Brooks/Cole, Pacific Grove, CA.
+- Ghitany, M.E., Atieh, B., Nadarajah, S., 2008. Lindley distribution and its application. Mathematics and Computers in Simulation 78, 39–49.
+- Guikema, S.D., Coffelt, J.P., 2008. A flexible count data regression model for risk analysis. Risk Analysis 28 (1), 213–223.
+- Kumara, S.S.P., Chin, H.C., 2003.Modeling accident occurrence at signalized tee intersections with special emphasis on excess zeros. Traffic Injury Prevention 3 (4), 53–57.
+- Lindley, D.V., 1958. Fiducial distributions and Bayes' theorem. J.R. Stat. Soc. (20), 102–107. http://www.jstor.org/stable/2983909.
+- Lord, D., 2006.Modelingmotor vehicle crashes using Poisson-gammamodels: examining the effects of low sample mean values and small sample size on the estimation of the fixed dispersion parameter. Accident Analysis & Prevention 38 (4), 751–766.
+- Lord, D., Geedipally, S.R., Persaud, B.N., Washington, S.P., van Schalkwyk, I., Ivan, J.N., Lyon, C., Jonsson, T., 2008. Methodology for Estimating the Safety Performance of Multilane Rural Highways NCHRP Web-Only Document 126. National Cooperation Highway Research Program, Washington, DC.
+- Lord, D., Mannering, F.L., 2010. The statistical analysis of crash-frequency data: a review and assessment of methodological alternatives. Transportation Research - Part A 44 (5), 291–305.
+- Lord, D., Park, P.Y-J., 2008. Investigating the effects of the fixed and varying dispersion parameters of Poisson-gamma models on empirical Bayes estimates. Accident Analysis & Prevention 40 (4), 1441–1457.
+- Lord, D., Washington, S.P., Ivan, J.N., 2005. Poisson Poisson-gamma and zero inflated regression models of motor vehicle crashes: balancing statistical fit and theory. Accident Analysis & Prevention 37 (1), 35–46.
+- Lord, D., Washington, S.P., Ivan, J.N., 2007. Further notes on the application of zero inflated models in highway safety. Accident Analysis & Prevention 39 (1), 53–57.
+- Mayshkina, N.V., Mannering, F.L., 2009. Zero-state Markov switching count-data models: an empirical assessment. Accident Analysis and Prevention Vol. 42 (1), 122–130.
+- Miaou, S.-P., 1994. The relationship between truck accidents and geometric design of road sections: Poisson versus negative binomial regressions. Accident Analysis & Prevention 26 (4), 471–482.
+- Miaou, S.P., Lord, D., 2003. Modeling traffic crash-flow relationships for intersections: dispersion parameter, functional form, and Bayes versus empirical Bayes. Transportation Research Record 1840, 31–40.
+- Mitra, S., Washington, S., 2007. On the nature of over-dispersion in motor vehicle crash prediction models. Accident Analysis & Prevention 39 (3), 459–468.
+- Shankar, V., Milton, J., Mannering, F.L., 1997. Modeling accident frequency as zero-altered probability processes: an empirical inquiry. Accident Analysis & Prevention 29 (6), 829–837.
+- Shankar, V.N., Ulfarsson, G.F., Pendyala, R.M., Nebergal, M.B., 2003. Modeling crashes involving pedestrians and motorized traffic. Safety Science 41 (7), 627– 640.
+- Sellers, K.F., Shmueli, G., 2010. A flexible regression model for count data. Annals of Applied Statistics 4, 943–961.
+- Warton, D.I., 2005. Many zeros does not mean zero inflation: comparing the goodness-of-fit of parametric models to multivariate abundance data. Environmetrics 16 (2), 275–289.
+- Zamani, H., Ismail, N., 2010. Negative binomial–Lindley distribution and its application. Journal of Mathematics and Statistics 6 (1), 4–9.

Zotero/001_artiklid/Road Safety Indicators/lordpoissonpoissongammazero2005.md

@@ -0,0 +1,418 @@
+---
+category: literaturenote
+citekey: lordpoissonpoissongammazero2005
+title: "Poisson, Poisson-gamma and zero-inflated regression models of motor vehicle crashes: balancing statistical fit and theory"
+authors: "Lord, Dominique; Washington, Simon P.; Ivan, John N."
+year: 2005
+date: 2005-00-00 2005
+doi: 10.1016/j.aap.2004.02.004
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2004.02.004"
+zotero_key: 9H6KSIDQ
+zotero_storage: GEC26ZIN
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Lord, Dominique"
+status: converted
+---
+![](_page_0_Picture_0.jpeg)
+
+Accident Analysis and Prevention 37 (2005) 35–46
+
+![](_page_0_Picture_2.jpeg)
+
+# Poisson, Poisson-gamma and zero-inflated regression models of motor vehicle crashes: balancing statistical fit and theory
+
+Dominique Lorda,∗, Simon P. Washingtonb,1, John N. Ivanc,<sup>2</sup>
+
+<sup>a</sup> *Center for Transportation Safety, Texas Transportation Institute, Texas A and M University System, 3135 TAMU, College Station, TX 77843-3135, USA* <sup>b</sup> *Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, AZ 85721-0072, USA* <sup>c</sup> *Department of Civil and Environmental Engineering, University of Connecticut, 261 Glenbrook Road, Unit 2037, Storrs, CT 06269-2037, USA*
+
+Received 14 November 2003; received in revised form 9 February 2004; accepted 11 February 2004
+
+#### **Abstract**
+
+There has been considerable research conducted over the last 20 years focused on predicting motor vehicle crashes on transportation facilities. The range of statistical models commonly applied includes binomial, Poisson, Poisson-gamma (or negative binomial), zero-inflated Poisson and negative binomial models (ZIP and ZINB), and multinomial probability models. Given the range of possible modeling approaches and the host of assumptions with each modeling approach, making an intelligent choice for modeling motor vehicle crash data is difficult. There is little discussion in the literature comparing different statistical modeling approaches, identifying which statistical models are most appropriate for modeling crash data, and providing a strong justification from basic crash principles. In the recent literature, it has been suggested that the motor vehicle crash process can successfully be modeled by assuming a dual-state data-generating process, which implies that entities (e.g., intersections, road segments, pedestrian crossings, etc.) exist in one of two states—perfectly safe and unsafe. As a result, the ZIP and ZINB are two models that have been applied to account for the preponderance of "excess" zeros frequently observed in crash count data.
+
+The objective of this study is to provide defensible guidance on how to appropriate model crash data. We first examine the motor vehicle crash process using theoretical principles and a basic understanding of the crash process. It is shown that the fundamental crash process follows a Bernoulli trial with unequal probability of independent events, also known as Poisson trials. We examine the evolution of statistical models as they apply to the motor vehicle crash process, and indicate how well they statistically approximate the crash process. We also present the theory behind dual-state process count models, and note why they have become popular for modeling crash data. A simulation experiment is then conducted to demonstrate how crash data give rise to "excess" zeros frequently observed in crash data. It is shown that the Poisson and other mixed probabilistic structures are approximations assumed for modeling the motor vehicle crash process. Furthermore, it is demonstrated that under certain (fairly common) circumstances excess zeros are observed—and that these circumstances arise from low exposure and/or inappropriate selection of time/space scales and not an underlying dual state process. In conclusion, carefully selecting the time/space scales for analysis, including an improved set of explanatory variables and/or unobserved heterogeneity effects in count regression models, or applying small-area statistical methods (observations with low exposure) represent the most defensible modeling approaches for datasets with a preponderance of zeros.
+
+© 2004 Elsevier Ltd. All rights reserved.
+
+*Keywords:* Zero-inflated models; Poisson distribution; Negative binomial distribution; Bernoulli trials; Safety performance functions; Small area analysis
+
+This paper was presented under a different title at the 83rd Annual Meeting of Transportation Research Board.
+
+<sup>∗</sup> Corresponding author. Tel.: +1 979 458 1218; fax: +1 979 845 4872.
+
+*E-mail addresses:* d-lord@tamu.edu (D. Lord), simonw@engr.arizona.edu (S.P. Washington), johnivan@engr.uconn.edu (J.N. Ivan).
+
+<sup>1</sup> Tel.: +1 520 621 4686; fax: +1 520 621 2550.
+
+<sup>2</sup> Tel.: +1 860 486 0352; fax: +1 860 486 2298.
+
+#### **1. Introduction**
+
+There has been considerable research conducted over the last 20 years on the development of safety performance functions (SPFs) (or crash prediction models) for predicting crashes on highway facilities [\(Abbess et al., 1981; Hauer](#page-10-0) [et al., 1988; Persaud and Dzbik, 1993; Kulmala, 1995; Poch](#page-10-0) [and Mannering, 1996; Lord, 2000; Ivan et al., 2000; Lyon](#page-10-0) [et al., 2003; Miaou and Lord, 2003; Oh et al., 2003](#page-10-0)). During this period, most improvements have been focused on the tools for developing predictive models, such as the application of random-effect models [\(Miaou and Lord, 2003\)](#page-11-0), the generalized estimating equations (GEE) ([Lord and Persaud,](#page-11-0) [2000\),](#page-11-0) and Markov Chain Monte Carlo (MCMC) ([Qin et al.,](#page-11-0) [2004; Miaou and Lord, 2003\)](#page-11-0) methods for modeling crash data.
+
+Regardless of the statistical tool applied, these models require the use of a probabilistic structure for describing the data generating process (dgp) (motor vehicle crashes). Traditional Poisson and Poisson-gamma (or negative binomial) processes are the most common choice, but in recent years, some researchers have applied "zero-inflated" or "zero altered" probability models, which assume that a dual-state process is responsible for generating the crash data [\(Shankar](#page-11-0) [et al., 1997, 2003; Qin et al., 2004; Kumara and Chin, 2003;](#page-11-0) [Lee and Mannering, 2002\).](#page-11-0) These models have been applied to capture the apparent "excess" zeros that commonly arise in crash data—and generally have provided improved statistical fit to data compared to Poisson and negative binomial (NB) regression models.
+
+Given the broad range of possible modeling approaches and the host of assumptions associated with each modeling approach, making an intelligent choice for modeling motor vehicle crash data is difficult. Using statistical models such as zero-inflated count models or spline functions to obtain the best statistical fit is no longer a challenge, and over-fitting data can become problematic ([Loader, 1999\).](#page-11-0) Hence, a balance must be struck between the logical underpinnings of the statistical theory and the predictive capabilities of the selected predictive models ([Miaou and Lord, 2003\).](#page-11-0) In the very least, modelers of motor vehicle crash data should carefully contemplate the implications of their choice of modeling tools.
+
+The objective of this study is to provide defensible guidance on how to appropriately model relationships between road safety and traffic exposure. After providing a brief background, the motor vehicle crash process is examined using theoretical principles and a basic understanding of the crash process. It is shown that the fundamental crash process follows a Bernoulli trial with unequal probability of independent events, also known as Poisson trials (rather than a Poisson process). The evolution of statistical models is then presented as it applies to the motor vehicle crash process, and how well various statistical models approximate the crash process is discussed. The theory behind dual-state process count models is provided, and why they have become popular for modeling crash data is noted.
+
+A simulation experiment, where crash data are generated from Bernoulli trials with unequal outcome probabilities, is conducted to demonstrate how crash data give rise to "excess" zeros. It is shown that the Poisson and other mixed probabilistic structures are simply approximations assumed for modeling the motor vehicle crash process. Furthermore, it is shown that the fairly common observance of excess zeros is a consequence of low exposure and selection of inappropriate time/space scales rather than an underlying dual-state process. In conclusion, selecting an appropriate time/space scale for analysis, including an improved set of explanatory variables and/or unobserved heterogeneity effects in count regression models, or applying small-area statistical methods (observations with low exposure) represent the most theoretically defensible modeling approaches for datasets with a preponderance of zeros.
+
+## **2. Background: theoretical principles of motor vehicle crashes**
+
+It is useful at this point to examine the crash process—and how to model it—using theoretical principles and a current understanding of the motor vehicle crash process. This process has rarely been discussed in the traffic safety literature in this manner, and is the stepping stone for arguments presented later in this paper (see also [Kononov and Janson, 2002;](#page-10-0) [Lord et al., 2004\).](#page-10-0) Examined in this section are concepts of a single-state crash process, the notion of a dual-state process, and over-dispersion. Many of the basic concepts presented are not new and can be found in textbooks on discrete distributions ([Olkin et al., 1980; Johnson and Kotz, 1969\).](#page-11-0)
+
+A crash is, in theory, the result of a Bernoulli trial. Each time a vehicle enters an intersection, a highway segment, or any other type of entity (a trial) on a given transportation network, it will either crash or not crash. For purposes of consistency a crash is termed a "success" while failure to crash is a "failure". For the Bernoulli trial, a random variable, defined as *X*, can be generated with the following probability model: if the outcome *w* is a particular event outcome (e.g. a crash), then *X* (*w*) = 1, whereas if the outcome is a failure, then *X* (*w*) = 0. Thus, the probability model becomes
+
+| X        | 1 | 0 |
+|----------|---|---|
+| P(x = X) | p | q |
+
+where *p* is the probability of success (a crash) and *q* = (1 − *p*) is the probability of failure (no crash).
+
+In general, if there are *N* independent trials (vehicles passing through an intersection, road segment, etc.) that give rise to a Bernoulli distribution, then it is natural to consider the random variable *Z* that records the number of successes out of the *N* trials. Under the assumption that all trials are characterized by the same failure process (this assumption is revisited later in the paper), the appropriate probability model that <span id="page-2-0"></span>accounts for a series of Bernoulli trials is known as the binomial distribution, and is given as:
+
+$$P(Z = n) = \binom{N}{n} p^{n} (1 - p)^{N - n}, \tag{1}$$
+
+where *n* = 0,1,2, ..., *N*. In Eq. (1), *n* is defined as the number of crashes or collisions (successes). The mean and variance of the binomial distribution are *E*(*Z*) = *Np* and VAR(*Z*) = *Np*(1−*p*) respectively.
+
+For typical motor vehicle crashes where the event has a very low probability of occurrence and a large number of trials exist (e.g. million entering vehicles, vehicle-milestraveled, etc.), it can be shown that the binomial distribution is approximated by a Poisson distribution. Under the binomial distribution with parameters *N* and *p*, let *p* = λ/*N*, so that a large sample size *N* will be offset by the diminution of *p* to produce a constant mean number of events λ for all values of *p*. Then as N → ∞, it can be shown that (see [Olkin et al.,](#page-11-0) [1980\)](#page-11-0)
+
+$$P(Z=n) = \binom{N}{n} \left(\frac{\lambda}{N}\right)^n \left(1 - \frac{\lambda}{N}\right)^{N-p} \cong \frac{\lambda^n}{n!} e^{-\lambda}$$
+ (2)
+
+where, *n* = 0,1,2, ..., *N* and λ is the mean of a Poisson distribution.
+
+The approximation illustrated in Eq. (2) works well when the mean λ and *p* are assumed to be constant. In practice however, it is not reasonable to assume that crash probabilities across drivers and across road segments (intersections, etc.) are constant. Specifically, each driver–vehicle combination is likely to have a probability *p*<sup>i</sup> that is a function of driving experience, attentiveness, mental workload, risk adversity, vision, sobriety, reaction times, vehicle characteristics, etc. Furthermore, crash probabilities are likely to vary as a function of the complexity and traffic conditions of the transportation network (road segment, intersection, etc.). All these factors and others will affect to various degrees the individual risk of a crash.
+
+These and other characteristics affecting the crash process create inconsistencies with the approximation illustrated in Eq. (2). Outcome probabilities that vary from trial to trial are known as Poisson trials (note: Poisson trials are not the summation of independent Poisson distributions; this term is used to designate Bernoulli trials with unequal probability of events). As discussed by [Feller \(1968\),](#page-10-0) count data that arise from Poisson trials do not follow a standard distribution. However, the mean and variance for these trials share similar characteristics to the binomial distribution when the number of trials *N* and the expected value *E*(*Z*) are fixed. Unfortunately, these assumptions for crash data analysis are not valid: *N* is not known with certainty—but is an estimated value—and varies for each site, in which
+
+$$E(Z|p_1,\ldots,p_n) = N\bar{p} \tag{3}$$
+
+$$VAR(Z|p_1, ..., p_n) = N\bar{p}(1 - \bar{p}) - Ns_p^2$$
+(4)
+
+where ¯p = <sup>N</sup> <sup>i</sup>=1pi/N and Ns<sup>2</sup> <sup>p</sup> <sup>=</sup> <sup>N</sup> <sup>i</sup>=1(pi − p¯) <sup>2</sup> ≥ 0. The only difference between these two distributions is the computation of the variance. [Drezner and Farnum \(1993\)](#page-10-0) and [Vellaisamy and Punnen \(2001\)](#page-11-0) provide additional information about the properties described by [Feller \(1968\).](#page-10-0)
+
+[Neldelman and Wallenius \(1986\)](#page-11-0) have suggested that the unequal outcome occurrence of independent probabilities is related to the convex functions of the variance-to-mean relationships. Hence, they indicated that the variance-to-mean ratio would be smaller or larger than 1 whether the relationship is a concave or a convex function of the mean, respectively, providing evidence to this effect. Examples of distributions that have concave relationships include the Bernoulli and hypergeometric distributions, while examples of convex variance-to-mean relationships include the negative binomial (NB), the exponential, and uniform distributions. The Poisson distribution has a relationship that is both convex and concave, which only happens when *E*(*Z*) = *Var*(*Z*) = λ.
+
+[Neldelman and Wallenius \(1986\)](#page-11-0) maintain that many phenomena observed in nature tend to exhibit convex relationships, concavity being very rare. As an example, they referred to a paper written by [Taylor \(1961\), w](#page-11-0)ho evaluated 24 studies on the sampling of biological organisms for determining population sizes. Taylor found that 23 of the 24 studies followed a NB distribution. As discussed previously, crash data have been observed with variance-to-mean ratios above 1 [\(Abbess](#page-10-0) [et al., 1981; Poch and Mannering, 1996; Hauer, 1997\).](#page-10-0)
+
+[Barbour et al. \(1992\)](#page-10-0) proposed several methods for determining if the unequal event of independent probabilities can be approximated by a Poisson process. They used the Stein–Chen method combined with coupling methods to validate these approximations. One of these procedures states that
+
+$$d_{\text{TV}}(L(Z), Po(\lambda)) \le \min\{1, \lambda^{-1}\} \sum_{i=1}^{N} p_i^2 \le \max_{1 \le i \le N} p_i$$
+ (5)
+
+where, *d*TV = the total variance distance between the two probabilities measures *L*(*Z*) and *Po*(λ); (see appendix A.1 in [Barbour et al. \(1992\)](#page-10-0) for the mathematical properties of the total variance distance); *L*(*Z*) = count data generated by unequal events of independent probabilities; *Po*(λ) = count data generated from a Poisson distribution with mean λ = *E*(*Z*).
+
+Thus, since the individual *p*<sup>i</sup> values for road crash data are almost certainly very small, Poisson approximation to the total number of crashes occurring on a given set of roads over a given time period should be excellent.
+
+However, taking together the number of crashes from different sets of roads, or from the same set of roads over different time periods, the distribution of the observed counts is often over-dispersed, in that the crash count variance VAR(*Z*) is larger than the crash count mean *E*(*Z*). Now, if *W* is any nonnegative integer valued random variable such that VAR(*W*) is much larger that *E*(*W*) = λ, Poisson approximation is unlikely to be good. For instance, it is also shown in Barbour et al. <span id="page-3-0"></span>that if,
+
+$$\varepsilon = \min\{1, \lambda\} \left( \frac{\text{VAR}(W)}{E(W)} - 1 \right) > 0 \tag{6}$$
+
+then
+
+$$d_{\text{TV}}(L(W), Po(\lambda)) \ge (\varepsilon/\delta_r^2)^{r/(r-2)}$$
+ (7)
+
+for any *r* > 2, where
+
+$$\delta_r = \min\{1, \lambda^{-1/2}\}\{E(|W - \lambda|^r)\}^{1/r}$$
+(8)
+
+and a similar lower bound is true, whatever the mean chosen for the approximating Poisson distribution. Hence, the value of ε, which is a measure of the difference between mean and variance of *W*, imposes limits on the accuracy of any Poisson approximation to the distribution of *W*. Thus, if the data are over-dispersed, the simple Poisson trials model needs reappraisal.
+
+One important limitation about the methods proposed by Barbour et al. is that the probability for each event must be known. Unfortunately, the individual crash risk (*p*i) cannot be estimated in field studies since it varies for each driver–vehicle combination and across road segments. Although [Eqs. \(5\)–\(8\)](#page-2-0) are not ideally suited for motor vehicle crash data analyses, they illustrate that Poisson and NB distributions represent approximations of the underlying motor vehicle crash process that is derived from a Bernoulli distribution with unequal event probabilities ([Neldelman and](#page-11-0) [Wallenius, 1986; Barbour et al., 1992\).](#page-11-0) Data that do not meet the equality found in Eq. (7) will show over-dispersion, representing a convex relationship. Crash data commonly exhibit this characteristic (see also [Hauer, 2001](#page-10-0) for additional information). In contrast, over-dispersion resulting from other types of processes (not based on a Bernoulli trial) can be explained by the clustering of data (neighborhood, regions, wiring boards, etc.), unaccounted temporal correlation, and model mis-specification. The reader is referred to [Gourieroux](#page-10-0) [and Visser \(1986\),](#page-10-0) [Poormeta \(1999\)](#page-11-0) and[Cameron and Trivedi](#page-10-0) [\(1998\)](#page-10-0) for additional information on the characteristics of over-dispersion for different types of data.
+
+Over-dispersion, then, arises from crash data as a result of Bernoulli trials with non-equal success probabilities. Overdispersed data are characterized by "excess" zeros (more zeros than expected under an approximate Poisson process), "excess" large outcomes (large values of *Z*), or both. Zeroinflated models, presented and discussed in the next section, are models that explicitly account for "excess" zeros. Explored later in the paper is whether the theory of zero-inflated models is consistent with the crash process.
+
+# **3. Zero-inflated models**
+
+The first concept of a zero-inflated distribution originated from the work of [Rider \(1961\)](#page-11-0) and [Cohen \(1963\), w](#page-10-0)ho examined the characteristics of mixed Poisson distributions. Mixed Poisson distributions are characterized by data that have been mixed with two Poisson distributions in the proportions α and 1−α, respectively. The probability density function (pdf) of such a mixed distribution is
+
+$$P(n) = \alpha \frac{\lambda_1^n e^{-\lambda_1}}{n!} + (1 - \alpha) \frac{\lambda_2^n e^{-\lambda_2}}{n!}$$
+ (9)
+
+where λ<sup>2</sup> > λ<sup>1</sup> (the means of the two distributions) and *n* is the observed count data (0,1,2, ..., *N*). Both [Rider \(1961\)](#page-11-0) and [Cohen \(1963\)](#page-10-0) have proposed different approaches using the method of moments for estimating the parameter α. Cohen further described an approach for estimating the parameter α with zero sample frequency. [Johnson and Kotz](#page-10-0) [\(1969\)](#page-10-0) were the first to explicitly define a modified Poisson distribution (known as Poisson with added zeros) that explicitly accounted for excess zeros in the data. The modified distribution is the following:
+
+$$P(n) = \alpha + (1 - \alpha)e^{-\lambda}; \quad n = 0$$
+  
+
+$$P(n) = (1 - \alpha)\frac{e^{-\lambda}\lambda^n}{n!}; \quad n \ge 1$$
+(10)
+
+[Johnson and Kotz \(1969\)](#page-10-0) proposed a similar procedure to the one suggested by [Cohen \(1963\)](#page-10-0) for estimating the parameter α. Based on the work of [Yoneda \(1962\),](#page-11-0) they also developed a general modified Poisson distribution that accounts for any kind of excess in the frequency of the data. Under this distribution, *n* = 0,1,2, ..., *K* are inflated counts while the rest of the distribution *K* + 1,*K* + 2, ..., *N* follows a Poisson process.
+
+The concept of the mixed Poisson distribution introduced by the previous authors has been particularly useful to describe data characterized with a preponderance of zeros. For this type of data, more zeros are observed than what would have been predicted by a normal Poisson or Poisson-gamma process. It is generally believed that data with excess zeros come from two sources or two distinct distributions, hence the apply-named dual-state process. The underlying assumption for this system is that the excess zeros solely explain the heterogeneity found in the data (if we make abstraction of the ZINB) and each observation has the same mean λ. Two different types of regression or predictive models have been proposed in the literature for handling this type of data. The first type is known as the hurdle model ([Cragg, 1971;](#page-10-0) [Mullahy, 1986\).](#page-10-0) This type of model has not been extensively applied in the statistical field and is therefore not the focus of this paper. The reader is referred to [Cragg \(1971\),](#page-10-0) [Mullahy](#page-11-0) [\(1986\)](#page-11-0) and [Schmidt and Witte \(1989\)](#page-11-0) for additional information about hurdle models.
+
+The zero-inflated count models (also called zero-altered probability or count models with added zeros) represent an alternative way to handle data with a preponderance of zeros. Since their formal introduction by [Lambert \(1992\)](#page-10-0) (who expanded the work of [Johnson and Kotz \(1969\)\)](#page-10-0), the use of these models has grown almost boundlessly and can be found in numerous fields, including traffic safety. For example, these models have been applied in the fields of <span id="page-4-0"></span>manufacturing [\(Lambert, 1992; Li et al., 1999\)](#page-10-0), economics ([Green, 1994\)](#page-10-0), epidemiology [\(Heilbron, 1994\),](#page-10-0) sociology ([Land et al., 1996\),](#page-10-0) trip distribution ([Terza and Wilson, 1990\)](#page-11-0) and political science ([Zorn, 1996\)](#page-11-0) among others.
+
+In essence, zero-inflated regression models are characterized by a dual-state process, where the observed count can either be located in a perfect state or in an imperfect state with a mean µ. This type of model is suitable for data generated from two fundamentally different states. As described in [Washington et al. \(2003\), c](#page-11-0)onsider the following example. A transportation survey asks how many times you have taken mass transit to work during the past week. An observed zero could arise in two distinct ways. First, last week a respondent may have opted to take the vanpool instead of mass transit. Alternatively, a respondent may never take transit, as a result of other commitments on the way to and from their place of employment, lack of viable transit routes, lack of proximity of transit stations, etc. Thus two states are present, one being a normal count-process state and the other being a zero-count state. The count of zeros observed across the entire population of respondents under this dual state process results in "excess" zeros not explained by a Poisson or negative binomial process. The reader is referred to [Lambert \(1992\),](#page-10-0) [Li et](#page-11-0) [al. \(1999\),](#page-11-0) and [Lord et al. \(2004\)](#page-11-0) for additional information about the pdf and the characteristics of likelihood function of zero-inflated models.
+
+#### **4. What empirical crash data tell us**
+
+This section presents the results of a critical review of the work performed on the application of zero-inflated models in traffic safety. Table 1 summarizes crash data characteristics from six published studies. The table includes the proxy for exposure to risk, statistics on crash counts, and type and
+
+Table 2 Number of crashes and crash rate for rural highways by functional class ([US](#page-11-0) DOT, 1995)
+
+| Functional class   | Fatal injury |       | Non-fatal injury |        |
+|--------------------|--------------|-------|------------------|--------|
+|                    | Number       | Ratea | Number           | Ratea  |
+| Interstate         | 2,076        | 1.01  | 41,758           | 20.37  |
+| Principal arterial | 3,452        | 1.76  | 83,599           | 42.62  |
+| Minor arterial     | 3,760        | 2.56  | 108,069          | 73.61  |
+| Major collector    | 5,400        | 2.93  | 166,276          | 90.21  |
+| Minor collector    | 1,437        | 2.86  | 55,885           | 111.89 |
+
+<sup>a</sup> Rates are per 100 million vehicles-miles.
+
+location of segments. The mean number of crashes per year is very low for each dataset. This is expected, since all the studies have a substantially high number of zeros. Where the information is available, the datasets are shown to have between 2% and 12% more zeros than what would be expected given the mean number of crashes per year and a normal Poisson distribution (column #5). Second, with the exception of the last two studies, all highway segments are classified as arterial or collector roads and are located in a rural area. As discussed in the next paragraph, these types of highways are usually classified as 'high risk'. Third, the first five datasets are shown to have very low average exposure; this characteristic is confirmed by looking at the right-hand side of Table 1. It is possible that the low exposure may explain the preponderance of zeros in the data. The prevalence of zeros in the last dataset is probably explained by another phenomenon, which is discussed below.
+
+Table 2 summarizes the crash rate by functional class for rural highways in the United States. This table clearly shows that arterial and collector rural segments are more dangerous than interstate highways. In fact, a person is about five and two times more likely, given the exposure, to be involved
+
+Table 1 Characteristics of data used for zero-inflated models
+
+| Study                    | Location                 |     | Crashes per year |       |         | Exposure (AADT) |         |         |         | Observations  |
+|--------------------------|--------------------------|-----|------------------|-------|---------|-----------------|---------|---------|---------|---------------|
+|                          |                          | Min | Max              | Mean  | Std dev | Min             | Max     | Mean    | Std dev | 0   1+        |
+| Miaou (1994)             | Rural principal arterial | 0   | 8                | 0.20  | –       | 8 × 10−4a       | 5.03a   | 0.25a   | –       | 7610  653     |
+| Shankar et al. (1997)    | Rural principal arterial | 0   | 84               | 0.29  | 1.09    | 251             | 26,415  | 4,534   | 4,255   | N/A           |
+|                          | Rural minor arterial     | 0   | 7                | 0.09  | 0.35    | 187             | 11,016  | 1,691   | 1,537   | N/A           |
+|                          | Rural collector arterial | 0   | 6                | 0.61  | 0.28    | 146             | 10,438  | 982     | 931     | N/A           |
+| Lee and Mannering (2002) | Rural principal arterial | 0b  | 9b               | 0.11b | 0.37b   | 988             | 6,522   | 2,194   | 998     | 120c          |
+| Qin et al. (2004)        | Rural (SV)               | 0   | 61               | 0.68  | –       | 240             | 40,000  | 992     | –       | 19,480/10,320 |
+|                          | Rural (SD)               | 0   | 23               | 0.15  | –       | 240             | 40,000  | 992     | –       | 26,640/3,160  |
+|                          | Rural (OD)               | 0   | 7                | 0.04  | –       | 240             | 40,000  | 992     | –       | 28,609/1,191  |
+|                          | Rural (ID)               | 0   | 23               | 0.08  | –       | 240             | 40,000  | 992     | –       | 28,068/1,732  |
+| Shankar et al. (2003)    | Suburban                 | 0   | 4                | 0.16  | 0.476   | 10,350d         | 98,875d | 25,425d | 20,475d | 440c          |
+| Kumara and Chin (2003)   | Urban intersection       | 0   | 6                | 0.29  | 0.70    | 1,093e          | 66,378e | 12,992e | 9,479e  | 2,238 542     |
+
+SV = single-vehicle, SD = same direction (multi-vehicle), OD = opposite direction (multi-vehicle) ID = intersection (multi-vehicle).
+
+- <sup>a</sup> Number of trucks in veh-miles × 106.
+- <sup>b</sup> Crashes per month.
+- <sup>c</sup> Total number of observations.
+- <sup>d</sup> AADT for vehicles (no pedestrian volumes are available for this data set).
+- <sup>e</sup> Total entering flow (the data did not include flows for each approach).
+
+in a crash on a minor collector or a principal arterial than a freeway respectively. Other researchers have confirmed the results shown in [Table 2.](#page-4-0) For instance, [Amoros et al. \(2003\)](#page-10-0) found that rural collector roads are generally about twice as dangerous as rural freeway segments for eight counties in France. [Brown and Baass \(1995\)](#page-10-0) evaluated crash rates by severity for different types of highway located within the vicinity of Montreal, Quebec. They reported that a driver is about two to three times more likely to be involved in a collision on rural principal and arterial roads than on freeways. The crashes were also found to be more severe for principal arterial roads.
+
+The characteristics illustrated in [Tables 1 and 2](#page-4-0) show interesting, if not counterintuitive results. In a general sense, it is puzzling to note that a typical rural highway that has inherently safe segments would, at the same time, belong to a group classified as the most dangerous type of highways. This observation can be explained in one of two ways: either (1) rural highways include many inherently safe segments and a few tremendously dangerous segments, which results in an overall average crash rate much higher than that of freeway segments, or (2) crashes simply do not follow a dual-state process, but a single process characterized by low exposure. Another point worth mentioning is that rural highway segments characterized by high exposure or traffic volumes have never been found to follow a dual-state process [\(Hauer and](#page-10-0) [Persaud, 1995; Harwood et al., 2000\).](#page-10-0)
+
+Another characteristic detailed in [Table 1](#page-4-0) concerns the high percentage of zeros (80%) for approaches of signalized three-legged intersections located in Singapore [\(Kumara and](#page-10-0) [Chin, 2003\).](#page-10-0) At first glance, this high percentage appears to be counterintuitive, especially since intersection-related crashes usually account for about half of all crashes occurring in urban environments [\(NHTSA, 2000\).](#page-11-0) In another study, [Lord \(2000\)](#page-11-0) found that 3-legged signalized intersections in Toronto, Ontario, experienced on average 4.8 crashes per year and only 10% of the intersections had zero crashes (all approaches combined). This observation merits further reflection: given the characteristics of each city, their effort placed on improving safety, and the same level of exposure, are three-legged signalized intersections in Toronto truly three times more dangerous than in Singapore? Perhaps variations in driver behavior and traffic signal design may explain a portion of this difference. However, given that 80% of the approaches had no crashes, the excess zeros at signalized intersections in Singapore are likely attributable to non-reported crashes and perhaps other differences to a lesser degree. [Kumara and Chin \(2003\)](#page-10-0) acknowledged that the excess zeros could be explained by non-reported crashes. Unfortunately, they did not explore this option further, since assuming this prospect would entail that crash data cannot be generated from a dual-state process (see p. 53). Other examples of misconception between an apparent "excess" zeros in crash data and non-reported crashes on rural transportation networks can be found here [\(CIMA International, 2000;](#page-10-0) [Lord](#page-11-0) [et al., 2003a & 2003b;](#page-11-0) [Lord et al., 2004\).](#page-11-0)
+
+## **5. Simulated crash data from bernoulli trials**
+
+This section presents the results of a simulation study intended to illustrate the concepts discussed in the previous section. In this exercise, a series of simulation runs was performed by replicating a Bernoulli trial with unequal probability of events—the nature of the motor vehicle crash process described previously. A sample of 100 sites was used in the simulation. The individual risk *p*<sup>i</sup> values was simulated from a combination of uniform and lognormal distributions (e.g., the risk for each site is uniformly distributed and the risk for each driver is lognormal distributed). Data were not generated from a Poisson-gamma distribution, since Poisson trials resulting from a Bernoulli trial (with low *p* and large *N*) do not follow a standard distribution. In addition, one goal of the simulation effort is to show that the Poisson-gamma model (the most commonly used) can be used for approximating the motor vehicle crash process; however, this does not imply that it is the best model available. Average individual risk in the simulation varies from about 20–111 in 200 million, as detailed in [Table 2;](#page-4-0) these values are similar to what is found elsewhere (see [Hauer and Persaud, 1995](#page-10-0) or [Brown and Baass,](#page-10-0) [1995\).](#page-10-0) For each site, the estimated mean number of events was computed such that
+
+$$\hat{\lambda} = N\bar{p} = \sum_{i=1}^{N} p_i \tag{11}$$
+
+where λˆ = the estimated mean number of events in one year; p¯ = <sup>N</sup> <sup>i</sup>=1pi; and *N* = the number of trials.
+
+The simulation was performed using GENSTAT [\(Payne,](#page-11-0) [2000\) f](#page-11-0)or the following three traffic flow conditions (AADT): 50 vehicles per day, 500 vehicles per day, and 5000 vehicles per day. Thus, the total number of vehicles in 1 year is 365 days times AADT. The simulation runs are meant to represent sites with very low exposure, low exposure, and medium exposure to risk, respectively. Four series of simulation runs were performed for each of the exposure levels, characterized by the degree of crash risk and the heterogeneity in crash risk (variability in crash risk across different sites).
+
+- (1) high heterogeneity–high risk (uniform: 0.0001 (max), 0.00000001 (min); variance for lognormal: 0.5),
+- (2) low heterogeneity–low risk (uniform: 0.000001 (max), 0.00000001 (min); variance for lognormal: 0.5),
+- (3) very low heterogeneity–medium/high risk (fixed high risk = 0.00001; variance for lognomal: 0.5 - all the variance is explained by this distribution),
+- (4) very low heterogeneity–low risk (fixed low risk = 0.0000001; variance for lognomal: 0.5–all the variance is explained by this distribution).
+
+The results of the four simulation runs are presented in [Tables 3–5.](#page-6-0) The data shown in these tables are the fitted frequency distribution estimates, as provided by GEN-STAT. Note that the simulation performed in this work is not meant to replicate the actual crash process occurring on
+
+<span id="page-6-0"></span>Table 3Simulation results for sites characterized by high heterogeneity and high risk
+
+| AA<br>DT<br>=5<br>00<br>0               |                             |                                     | AA<br>DT<br>=5<br>00                                 |                                         |                         |                                     | AA<br>DT<br>=5<br>0                                  |                                         |                         |                                     |                                                      |
+|-----------------------------------------|-----------------------------|-------------------------------------|------------------------------------------------------|-----------------------------------------|-------------------------|-------------------------------------|------------------------------------------------------|-----------------------------------------|-------------------------|-------------------------------------|------------------------------------------------------|
+| Nu<br>mb<br>er<br>of<br>she<br>cra<br>s | Ob<br>ved<br>ser            | Ex<br>ted<br>pec<br>Po<br>iss<br>on | Ex<br>ted<br>pec<br>Po<br>iss<br>on<br><br>gam<br>ma | Nu<br>mb<br>er<br>of<br>she<br>cra<br>s | Ob<br>ved<br>ser        | Ex<br>ted<br>pec<br>Po<br>iss<br>on | Ex<br>ted<br>pec<br>Po<br>iss<br>on<br><br>gam<br>ma | Nu<br>mb<br>er<br>of<br>she<br>cra<br>s | Ob<br>ved<br>ser        | Ex<br>ted<br>pec<br>Po<br>iss<br>on | Ex<br>ted<br>pec<br>Po<br>iss<br>on<br><br>gam<br>ma |
+| 0–<br>2                                 | 9                           | 0.1<br>1                            | 9.2<br>7                                             | 0                                       | 46                      | 34<br>.30                           | 43<br>.31                                            | 0                                       | 92                      | 90<br>.42                           | –                                                    |
+| 3–4                                     | 11                          | 1.2<br>5                            | 10<br>.84                                            | 1                                       | 20                      | 36<br>.70                           | 28<br>.39                                            | 1+                                      | 8                       | 9.5<br>8                            | –                                                    |
+| 5–<br>6                                 | 14                          | 5.8<br>7                            | 11<br>.93                                            | 2                                       | 20                      | 19<br>.63                           | 14<br>.98                                            |                                         |                         |                                     |                                                      |
+| 7–<br>9                                 | 10                          | 25<br>.10                           | 17<br>.07                                            | 3+                                      | 14                      | 9.3<br>6                            | 13<br>.32                                            |                                         |                         |                                     |                                                      |
+| 10–<br>11                               | 14                          | 23<br>.67                           | 9.8<br>9                                             |                                         |                         |                                     |                                                      |                                         |                         |                                     |                                                      |
+| 12–<br>14                               | 10                          | 28<br>.21                           | 12<br>.11                                            |                                         |                         |                                     |                                                      |                                         |                         |                                     |                                                      |
+| 15–<br>16                               | 10                          | 9.5<br>9                            | 6.3<br>3                                             |                                         |                         |                                     |                                                      |                                         |                         |                                     |                                                      |
+| 17–<br>22                               | 11                          | 6.0<br>8                            | 12<br>.33                                            |                                         |                         |                                     |                                                      |                                         |                         |                                     |                                                      |
+| 23<br>+                                 | 10                          | 0.1<br>3                            | 10<br>.24                                            |                                         |                         |                                     |                                                      |                                         |                         |                                     |                                                      |
+| To<br>tal                               | 10<br>0                     | 10<br>0                             | 10<br>0                                              |                                         | 10<br>0                 | 10<br>0                             | 10<br>0                                              |                                         | 10<br>0                 | 10<br>0                             | –                                                    |
+| Me<br>an                                | 11<br>.51<br>(<br>8.0<br>6) | 11<br>.16<br>(<br>0.3<br>3)         | 11<br>.48<br>(<br>0.8<br>3)                          |                                         | 1.1<br>0 (<br>1.3<br>0) | 1.0<br>6 (<br>0.1<br>0)             | 1.0<br>9 (<br>0.1<br>3)                              |                                         | 0.1<br>1 (<br>0.4<br>1) | 0.1<br>0 (<br>0.0<br>3)             | –                                                    |
+| a<br>φ                                  |                             | –                                   | 2.2<br>5 (<br>0.3<br>9)                              |                                         |                         | –                                   | 1.6<br>4 (<br>0.7<br>8)                              |                                         |                         |                                     | –                                                    |
+
+Dispersion parameter of the Poisson-gamma.
+
+Table 4Simulation results for sites characterized by low heterogeneity and low risk
+
+| AA<br>DT<br>=5<br>00<br>0               |                         |                                     | AA<br>DT<br>=5<br>00                                 |                                         |                         |                                     |                                                      | 50                                      |                  |                                     |                                                      |
+|-----------------------------------------|-------------------------|-------------------------------------|------------------------------------------------------|-----------------------------------------|-------------------------|-------------------------------------|------------------------------------------------------|-----------------------------------------|------------------|-------------------------------------|------------------------------------------------------|
+| Nu<br>mb<br>er<br>of<br>she<br>cra<br>s | Ob<br>ved<br>ser        | Ex<br>ted<br>pec<br>Po<br>iss<br>on | Ex<br>ted<br>pec<br>Po<br>iss<br>on<br><br>gam<br>ma | Nu<br>mb<br>er<br>of<br>she<br>cra<br>s | Ob<br>ved<br>ser        | Ex<br>ted<br>pec<br>Po<br>iss<br>on | Ex<br>ted<br>pec<br>Po<br>iss<br>on<br><br>gam<br>ma | Nu<br>mb<br>er<br>of<br>she<br>cra<br>s | Ob<br>ved<br>ser | Ex<br>ted<br>pec<br>Po<br>iss<br>on | Ex<br>ted<br>pec<br>Po<br>iss<br>on<br><br>gam<br>ma |
+| 0                                       | 35                      | 30<br>.66                           | 34<br>.30                                            | 0                                       | 89                      | 87<br>.73                           | 89<br>.00                                            | 0                                       | 10<br>0          | –                                   | –                                                    |
+| 1                                       | 32                      | 36<br>.25                           | 33<br>.27                                            | 1+                                      | 11                      | 12<br>.27                           | 11<br>.00                                            | 1+                                      | –                | –                                   | –                                                    |
+| 2                                       | 19                      | 21<br>.43                           | 19<br>.15                                            |                                         |                         |                                     |                                                      |                                         |                  |                                     |                                                      |
+| 3                                       | 9                       | 8.4<br>4                            | 8.5<br>0                                             |                                         |                         |                                     |                                                      |                                         |                  |                                     |                                                      |
+| 4+                                      | 5                       | 3.2<br>3                            | 4.7<br>8                                             |                                         |                         |                                     |                                                      |                                         |                  |                                     |                                                      |
+| To<br>tal                               | 10<br>0                 | 10<br>0                             | 10<br>0                                              |                                         | 10<br>0                 | 10<br>0                             | 10<br>0                                              |                                         | 10<br>0          |                                     |                                                      |
+| Me<br>an                                | 1.1<br>9 (<br>1.2<br>2) | 1.1<br>8 (<br>0.1<br>1)             | 1.1<br>8 (<br>0.1<br>2)                              |                                         | 0.1<br>4 (<br>0.4<br>4) | 0.1<br>3 (<br>0.0<br>6)             | 0.1<br>4 (<br>0.4<br>4)                              |                                         |                  |                                     |                                                      |
+| a<br>φ                                  |                         | –                                   | 5.3<br>6 (<br>4.7<br>2)                              |                                         |                         | –                                   | 0.4<br>0 (<br>0.4<br>4)                              |                                         |                  |                                     |                                                      |
+
+Dispersion parameter of the Poisson-gamma.
+
+Table 5 Simulation results for sites characterized by very low heterogeneity
+
+|                                  | AADT=5000 (high/medium risk) |                     |                           | AADT=5000 (low risk) |             |                     |                           |  |
+|----------------------------------|------------------------------|---------------------|---------------------------|----------------------|-------------|---------------------|---------------------------|--|
+| Number<br>Observed<br>of crashes |                              | Expected<br>Poisson | Expected<br>Poisson-gamma | Number<br>of crashes | Observed    | Expected<br>Poisson | Expected<br>Poisson-gamma |  |
+| 0–16                             | 9                            | 7.81                | 8.09                      | 0                    | 94          | 94                  | –                         |  |
+| 17–19                            | 15                           | 15.13               | 15.19                     | 1+                   | 6           | 6                   | –                         |  |
+| 20–21                            | 13                           | 14.95               | 14.23                     |                      |             |                     |                           |  |
+| 22                               | 14                           | 8.24                | 8.14                      |                      |             |                     |                           |  |
+| 23–24                            | 10                           | 16.27               | 16.05                     |                      |             |                     |                           |  |
+| 25                               | 10                           | 7.39                | 7.30                      |                      |             |                     |                           |  |
+| 26–27                            | 11                           | 12.21               | 12.10                     |                      |             |                     |                           |  |
+| 28–30                            | 10                           | 11.23               | 11.26                     |                      |             |                     |                           |  |
+| 31+                              | 8                            | 5.78                | 7.04                      |                      |             |                     |                           |  |
+| Total                            | 100                          | 100                 | 100                       |                      | 100         | 100                 | –                         |  |
+| Mean                             | 23.21 (4.84)                 | 23.13 (0.48)        | 23.13 (0.49)              |                      | 0.06 (0.25) | 0.06 (0.02)         | –                         |  |
+| φa                               |                              | –                   | 844.78b (4594)            |                      |             | –                   | –                         |  |
+
+<sup>a</sup> Dispersion parameter of the Poisson-gamma.
+
+transportation networks—the effects of covariates on crashes are important and not assumed, so simplifications have been made. Similarly, all sites used in this exercise are subjected to the same exposure; obviously, it is unreasonable to assume in practice that all sites will have the same exposure. Nonetheless, the simplifications described herein do not hinder conclusions made regarding the simulations.
+
+[Table 3](#page-6-0) summarizes the results of the simulation run used for simulating data with a high degree of heterogeneity and for observations that include sites categorized as high risk. This table shows that the number of events follow a Poisson distribution for sites with very low exposure, with about 92% of the sites having zero crashes. For low exposure, in contrast, the simulation generated more zeros than what would be expected from the Poisson and Poisson-gamma (NB) distributions. This result seems to agree with the results illustrated in [Tables 1 and 2,](#page-4-0) respectively; sites with a preponderance of zeros are characterized by low exposure and high risk. Finally, despite the fact the data were generated from a uniform-lognormal distribution, the Poisson-gamma (NB) distribution provides a superior statistical fit than the Poisson distribution for sites with medium exposure.
+
+[Table 4](#page-6-0) exhibits the results of the simulation run for low heterogeneity and sites with low risk. The results demonstrate that for low exposure the Poisson distribution offers good statistical fit, although some heterogeneity exists in the data. However, the Poisson-gamma distribution provides a better fit than the Poisson distribution for medium exposure. [Table 4](#page-6-0) also reveals that the only situation where a site may be inherently safe is when the exposure tends towards zero, as illustrated in the last columns. Table 5 summarizes the results for sites with very low heterogeneity. The simulation demonstrates that the Poisson distribution can be used to approximate data generated by a nearly homogeneous Bernoulli process. This outcome can clearly be seen both for high and low crash risks.
+
+## **6. Discussion**
+
+Zero-inflated models have been used in traffic safety studies for modeling crashes for various applications: singleand multi-vehicle crashes on rural two-lane roads [\(Miaou,](#page-11-0) [1994; Shankar et al., 1997; Qin et al., 2004\); r](#page-11-0)un-off-the-road crashes in rural areas [\(Lee and Mannering, 2002\);](#page-10-0) crashes occurring at intersections [\(Mitra et al., 2002; Kumara and Chin,](#page-11-0) [2003\);](#page-11-0) and pedestrian crashes taking place on urban and suburban corridors [\(Shankar et al., 2003\)](#page-11-0). In these studies, the use of the zero-inflated models has often been justified based on their superior statistical fit (almost always centered on the Vuong statistics—see [Vuong, 1989\).](#page-11-0) Given the characteristics of these models, these authors have automatically assumed that crashes must follow a dual-state process, with the exception of [Miaou \(1994\)](#page-11-0) who justified the use of the ZIP not as a dual-state process, but to account for unreported crash data.
+
+Although zero-inflated models offer improved statistical fit to crash data in many cases, it is argued in this paper that the inherent assumption of a dual state process underlying the development of these models is inconsistent with crash data. As was shown with both empirical and simulated data, excess observed zeros arise from low exposure and high-risk conditions (this explains the counts at the tail). In addition, the selection of spatial and temporal scales influences the number of observed zeros. For example, using crashes per 10th of a mile on urban arterials will result in a greater proportion of observed zeros than using crashes per mile (note: some studies have used small time or space intervals—e.g., [Lee](#page-10-0) [and Mannering, 2002\).](#page-10-0) Surely, whether a process is dual state should not rest on the selection of spatial or temporal scale of analysis (see example below). For an inherently safe state to exist, either one of the following two conditions must be present: (1) the probability *p*<sup>i</sup> values must either be 0 or so low that it approaches 0 for each vehicle that enters the site and/or
+
+<sup>b</sup> This is basically a normal Poisson process.
+
+(2) the number of vehicles *N* (or exposure) is or tends towards 0 for the given time period under study. Since it unreasonable to assume that each driver (no matter how many) will have a probability *p*<sup>i</sup> equal to 0 (a large body of research suggests that crashes are 70–90% human error), the number of vehicles or exposure must approach 0 (or be sufficiently small). This outcome is in accordance with both empirical and simulated crash data presented in this paper.
+
+The characteristic described in the previous paragraph can be expressed or conceptualized in a dual state framework such that (see [Eqs. \(9\)–\(11\)\):](#page-3-0)
+
+$$\hat{\lambda}_1 = N\bar{p}_1 = \sum_{i=1}^{N} p_i \to 0$$
+
+$$\hat{\lambda}_2 = M\bar{p}_2 = \sum_{j=1}^{M} p_j$$
+(12)
+
+where λ<sup>1</sup> is the mean of the first distribution (or state) that tends towards 0; λ<sup>2</sup> is the mean of the second distribution; *N* is the number of vehicles traveling through sites included in the first distribution; and *M* is the number of vehicles traveling through sites contained in the second distribution. If the second distribution (i.e., for λ2) follows a normal Poisson process (such as the ZIP), then every site has theoretically the same mean and the data must meet the properties of [Eq. \(5\).](#page-2-0) Furthermore, this attribute entails that each site must automatically have nearly the same exposure (or the same number of vehicles), given the characteristics of [Eq. \(5\),](#page-2-0) in order to obtain the same mean. In practice, it is unreasonable to assume that each site will be subjected to the same exposure; thus, violating the properties of [Eq.](#page-2-0) (5) and the fixed mean across different sites. If the second distribution is characterized by a variance-to-mean ratio above 1 (such as the ZINB), the inequality of [Eq. \(6\)](#page-3-0) holds and over-dispersion exists in the second dataset.
+
+As explained previously, the use of different space or time scales affects exposure, which in return influences the number of zeros present in the data (note: it will also affect the entire distribution, not just the number of zeros). To demonstrate this characteristic, a simple example is illustrated in Fig. 1. In this figure, crashes estimated from a Poisson distribution were simulated with a mean λ = 0.5 for 150 sites. This simulation could represent 1-km highway segments with similar characteristics. Then, 50 sites with 0 crash count were added randomly to this distribution to characterize a dual-state process for a total of 200 1-km segments. A Poisson-gamma distribution could not be fitted adequately to the proposed mixed distribution, confirming that too many zeros exist in the data. Once the simulation was completed, the sections were grouped into 2-km and 5-km segments, respectively. As expected, Fig. 1 shows that the number of sites with zeros diminishes drastically with longer segments. In fact, even with 2-km segments, the over-presentation of zeros disappears (i.e., more zeros than what would be expected given the new estimated mean), as shown with the fitted Poisson-gamma distribution in Fig. 1b. Although this
+
+![](_page_8_Figure_7.jpeg)
+
+![](_page_8_Figure_8.jpeg)
+
+![](_page_8_Figure_9.jpeg)
+
+Fig. 1. Frequency distribution based on a Poisson process (λ = 0.5) with added 0s for 1 km, 2 km and 5 km segments. (a) 1 km segments, (b) 2 km segments, (c) 5 km segments.
+
+example focused on the space scale, the same results would be observed when different time scales are used (e.g., crashes per 5 years versus crashes per 5 min for the same segment length, etc.).
+
+If ZIP and ZINB models provide good statistical fit but do not characterize the underlying crash process, several interesting questions arise. First, what are the consequences of applying these models? Second, what alternative models should be used instead? Answers to these two questions are now provided.
+
+If the only goal consists of finding the best statistical fit then the zero-inflated models may be appropriate, since they offer improved statistical fit compared to Poisson or Poissongamma models. However, this may not be sound practice. It can be shown that other types of models provide better fit (i.e., likelihood, MSE, etc.) than Poisson-gamma models, and some even better than ZIP or ZINB models. For instance, one could use the multilogit-Poisson for modeling multi-state processes. For this type of model, the crash process could be divided into "m" states, representing categories of perfect safety, above average safety, average safety, less than average safety, and least safe. The multinomial-Poisson process is another modeling approach that will fit the data better than Poisson-gamma models. The main characteristic of this model is that each count *Y*<sup>i</sup> follows a multinomial distribution [\(Dalal et al., 1990; Lang, 2002\).](#page-10-0) Although this type of model provides good statistical fit, it is not commonly used in practice, since the likelihood function can be very difficult to compute [\(Baker, 1994\).](#page-10-0) The ultimate statistical fit can be found using non-parametric (including spline functions) or semiparametric models [\(Ullah, 1989; Efromovich, 1999\).](#page-11-0) These models will fit the data better than any parametric model. Their advantage: there is no longer any assumption about whether the data follow any statistical distribution, both for the functional form and the error distribution. The disadvantage is that much faith is lost in agreement with theoretical knowledge of crashes and that any dataset will be grossly overfit by the model and results will not be transferable or generalizable to other locations.
+
+Usually what is sought besides good statistical fit is insight into the underlying data generating process—motor vehicle crashes in this study. At risk when using zeroinflated models is the possibility that mis-interpretations about safe and unsafe intersections/road segments/etc. will result. Someone might falsely believe that certain engineering investments—as predicted by the zero-inflated models—will lead to inherently safe locations. The resulting models will be difficult, if not impossible, to interpret in the 'real world' of crash causation. It may also lead modelers down the path of finding the best fit to data, possibly over-fitting the models. As stated in [Washington et al. \(2003\), "](#page-11-0)If there is not a compelling reason to suspect that two states might be present, the use of a zero-inflated model may be simply capturing model mis-specification that could result from factors such as unobserved effects (heterogeneity) in the data." Examples of commonly omitted variables include non-reporting of crashes, driver distractions, driver competence, etc. The set of variables commonly omitted from crash models may also contribute to explaining the 'excessive' zeros observed in motor vehicle crash data.
+
+How then, should crashes be modeled appropriately? The answer derives from revisiting the root of excess zeros, which are caused by the following four issues: (1) spatial or time scales that are too small; (2) under-or mis-reporting of crashes; (3) sites characterized by low exposure and high risk; and (4) important omitted variables describing the crash process. An obvious fix for (1) is to select appropriate time and spatial scales, which in some cases will result in more costly data collection. The remedies for (2), (3), and (4) require modeling solutions. One solution is to estimate NB and Poisson models with a term added to capture unobserved heterogeneity effects. This could account for under- or misreporting of crashes and omitted important variables in the crash process, and will yield a fit similar to zero-inflated models.
+
+In addition, there exist other statistical tools that can be used for modeling data with a preponderance of zeros commonly observed in crash data. Recall that the excess zeros are mainly attributed to low exposure (given the time-scale defined as crashes per year) and high risk. Small-area statistics (SAS) (also known as small area estimation or SAE) are such tools that can be used for data characterized by low exposure. These tools originated from researchers in survey sciences who frequently deal with small sample sizes (some even have a sample size of zero) [\(Rao, 2003\).](#page-11-0) Typically, the term "small area" is usually employed for areas where direct estimates of adequate precision cannot be produced. Thus, the goal of SAS is to use "indirect" estimators that "borrow strength" by using values from related areas and/or time periods to increase the "effective" sample size of interest. Empirical and Hierarchical Bayes methods combined with random-effect models are used to model this effect [\(Hauer,](#page-10-0) 1997; Carlin and Louis, 2000; Miaou and Lord, 2003). As opposed to zero-inflated models, SAS offer a much better and more rational modeling approach for conducting safety analyses for sites with low exposure. The reader is referred to [Ghosh et al. \(1998\),](#page-10-0) [Mukhopadhyay \(1998\)](#page-11-0) and [Rao \(2003\)](#page-11-0) for additional information on these model types.
+
+#### **7. Conclusions**
+
+The objective of this study was to provide defensible guidance on how to appropriate model crash data. The paper was motivated by the vast array of modeling choices, the lack of formal guidance for selecting appropriate statistical tools, and the peculiar nature of crash data, which has lead to lack of consensus among transportation safety modelers. Four main conclusions are drawn from this research.
+
+Crash data are best characterized as Bernoulli trials with independence among crashes and unequal crash probabilities across drivers, vehicles, roadways, and environmental conditions. Because of the small probability of a crash and the large number of trails these Bernoulli trails can be well approximated as Poisson trials.
+
+Poisson and negative binomial models serve as statistical approximations to the crash process. Poisson models serve well under nearly homogenous conditions, while negative binomial models serve better in other conditions.
+
+Crash data characterized by a preponderance of zeros is not indicative of an underlying dual-state process. One or more of four conditions lead to excess zeros in crash data: (1) sites with a combination of low exposure, high heterogeneity, and sites categorized as high risk; (2) analyses conducted with small time or spatial scales; (3) data with a relatively high percentage of missing or mis-reported crashes; and (4) crash models with omitted important variables.
+
+<span id="page-10-0"></span>Theoretically defensible solutions for modeling crash data with excess zeros include changing the spatial or time scale of analysis, including unobserved heterogeneity terms in NB and Poisson models, improving the set of explanatory variables, and applying small-area statistical methods.
+
+The intent of this paper was to foster new thinking about how to approach the modeling of crash data. A bottom-up focus on the crash process has been lacking, and it is hoped that this paper begins to fill this gap in the literature. It may be preferable to begin to develop models that consider the fundamental process of a crash and avoid striving for "best fit" models in isolation. Further work should concentrate on modeling Bernoulli trials with unequal probability of events. In the end, it may be possible to estimate the individual probability of crash risk for different time periods and driving conditions.
+
+### **Acknowledgements**
+
+The authors would like to express their greatest gratitude to Prof. A.D. Barbour for providing additional information about the Poisson approximation, reviewing our work for the mathematical content and providing suggestions to improve the clarity of the discussion for the section on the theoretical principles of motor vehicle crashes. They also thank Dr. Xiao Qin from the Maricopa Association of Governments for his assistance in obtaining additional information about the crash data on rural two-lane roads in Connecticut. This paper benefited from the input of Drs. Jake Kononov, Ezra Hauer, Per E. Garder, Fred L. Mannering, and two anony- ˚ mous referees. Their comments and suggestions were greatly appreciated.
+
+# **References**
+
+- Abbess, C., Jarett, D., Wright, C.C., 1981. Accidents at blackspots: estimating the effectiveness of remedial treatment, with special reference to the "regression-to-mean" effect. Traffic Eng.Cont. 22 (10), 535–542.
+- Amoros, E., Martin, J.L., Laumon, B., 2003. Comparison of road crashes incidence and severity between some French counties. Accid. Anal. Prev. 35 (4), 349–361.
+- Barbour, A.D., Holst, L., Janson, S., 1992. Poisson Approximation. Clarendon Press, New York, NY.
+- Baker, S.G., 1994. The multinomial-Poisson transformation. The Statistician 43 (4), 495–504.
+- Brown, B., Baass, K.A., 1995. Projet d'identification des sites dangereux sur les routes numerot ´ ees en Mont ´ er´ egie, vol. 1: Synth ´ ese. Direction ` de la sante publique. St-Hubert, Quebec (in French). ´
+- Cameron, A.C., Trivedi, P.K., 1998. Regression Analysis of Count Data. Cambridge University Press, Cambridge, UK.
+- CIMA International, 2000. Etude relative ´ a l' ` elaboration d'un plan ´ d'actions en matiere de s ` ecurit ´ e routi ´ ere en milieu interurbain au ` Burkina Faso. Final Report. Laval, Quebec, (in French). ´
+- Carlin, B.P., Louis, T.A., 2000. Bayes and Empirical Bayes Methods for Data Analysis. CRC Press, Chapman & Hall, London, UK.
+
+- Cohen, A.C., 1963. Estimation in Mixtures of Discrete Distributions. In: Proceedings of the International Symposium on Discrete Distributions, Montreal, Quebec.
+- Cragg, J.G., 1971. Some statistical models for limited dependent variables with application to the demand for durable goods. Econometrica 39, 829–844.
+- Dalal, S.R., Lee, J.C., Sabaval, D.J., 1990. Empirical bayes prediction for compound Poisson-multinomial process. Statistic Probability Lett. 9, 385–389.
+- Drezner, Z., Farnum, N., 1993. A generalized binomial distribution. Comm. Statist.: Theory Methods 22 (11), 3051–3063.
+- Efromovich, S., 1999. Nonparametric Curve Estimation: Methods, Theory and Applications. Springer, New York, NY.
+- Feller, W., 1968. An Introduction to Probability Theory and its Application, 3rd Ed. John Wiley, New York, NY.
+- Ghosh, M., Natarajan, K., Stroud, T.W.F., Carlin, B.P., 1998. Generalized linear models for small area estimation. J. Amer. Statistical Assoc. 93, 273–282.
+- Gourieroux, C.A., Visser, M., 1986. A count data model with unobserved heterogeneity. J. Econometrics 79, 247–268.
+- Green, W., 1994. Accounting for Excess Zeros and Sample Selection in Poisson and Negative Binomial Regression Models. Working paper EC-94-10, Department of Economics, Stern School of Business, New York University, New York, NY.
+- Harwood, D.W., Council, F.M., Hauer, E., Hughes, W.E., Vogt, A., 2000. Prediction of the Expected Safety Performance of Rural Two-Lane Roads. FHWA-RD-99-207. U.S. Department of Transportation, Washington, DC.
+- Hauer, E., 1997. Observational Before-After Studies in Road Safety: Estimating the Effect of Highway and Traffic Engineering Measures on Road Safety. Elsevier Science Ltd., Oxford.
+- Hauer, E., 2001. Overdispersion in modelling accidents on road sections and in empirical bayes estimation. Accid. Anal. Prev. 33 (6), 799–808.
+- Hauer, E., Ng, J.C.N., Lovell, J., 1988. Estimation of safety at signalized intersections. Transp. Res. Record 1185, 48–61.
+- Hauer, E., Persaud, B.P., 1995. Safety Analysis of Roadway Geometric and Ancillary Features. Research Report for the Transportation Association of Canada, Ottawa, Canada.
+- Heilbron, D.C., 1994. Zero-altered and other regression models for count data with added zeros. J. Biometrics 5, 531–547.
+- Ivan, J.N., Wang, C., Bernardo, N.R., 2000. Explaining two-lane highway crash rates using land use and hourly exposure. Accid. Anal. Prev. 32 (6), 787–795.
+- Johnson, N.L., Kotz, S., 1969. Discrete Distributions: Distributions in Statistics. Wiley, New York, NY.
+- Kononov, J., Janson, B.N., 2002. Diagnostic methodology for detection of safety problems at intersections. Transp. Res. Record 1784, 51– 56.
+- Kulmala, R., 1995. Safety at Rural Three- and Four-Arm Junctions: Development and Applications of Accident Prediction Models. VTT Publications, Technical Research Centre of Finland, Espoo.
+- Kumara, S.S.P., Chin, H.C., 2003. Modeling accident occurrence at signalized tee intersections with special emphasis on excess zeros. Traffic Injury Prev. 3 (4), 53–57.
+- Lambert, D., 1992. Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics 34 (1), 1–14.
+- Land, K.C., McCall, P.L., Nagin, D.S., 1996. A comparison of Poisson, negative binomial, semiparametric mixed Poisson regression models, with empirical applications to criminal careers data. Sociologic. Methods Res. 24 (4), 387–442.
+- Lang, J.B., 2002. Multinomial-Poisson homogeneous and homogeneous linear predictor models: a brief description. Online html document: http://www.stat.uiowa.edu/∼[jblang/mph.fitting/mph.fit.documentation](http://www.stat.uiowa.edu/~jblang/mph.fitting/mph.fit.documentation.htm). [htm](http://www.stat.uiowa.edu/~jblang/mph.fitting/mph.fit.documentation.htm). Accessed on 11 February 2004.
+- Lee, J., Mannering, F.L., 2002. Impact of roadside features on the frequency and severity of run-off-road accidents: an empirical analysis. Accid. Anal. Prev. 34 (2), 349–361.
+
+- <span id="page-11-0"></span>Li, C.-C., Lu, J.-C., Park, J., Kim, K., Brinkley, P.A., Peterson, J.P., 1999. Multivariate zero-inflated Poisson models and their applications. Technometrics 41 (1), 29–38.
+- Loader, C., 1999. Local Regression and Likelihood. Springer-Verlag, New York, NY.
+- Lord, D., 2000. The prediction of accidents on digital networks: characteristics and issues related to the application of accident prediction models. Ph.D. Dissertation, Department of Civil Engineering, University of Toronto, Toronto.
+- Lord, D., Abdou, H.A., N'Zue, A., Dionne, G., Laberge-Nadeau, C., ´ 2003a. Traffic safety diagnostic and application of countermeasures for rural roads in Burkina Faso. Transp. Res. Record 1846, 39–43.
+- Lord, D., Abdou, H.A., N'Zue, A., Dionne, G., Laberge-Nadeau, C., ´ 2003b. Investigating Sites Located on Rural Two-lane Roads in Burkina Faso for Safety Improvement. Working paper, Texas Transportation Institute, College Station, TX.
+- Lord, D., Persaud, B.N., 2000. Accident prediction models with and without trend: application of the generalized estimating equations procedure. Transp. Res. Record 1717, 102–108.
+- Lord, D., Washington, S.P., Ivan, J.N., 2004. Statistical Challenges with Modeling Motor Vehicle Crashes: Understanding the Implications of Alternative Approaches. Research Report CTS-04 01, Texas Transportation Institute, College Station, Texas. (available at [http://tti.tamu.edu/product/catalog/reports/CTS-04](http://tti.tamu.edu/product/catalog/reports/cts-04_01.pdf) 01.pdf) (also presented at the 83rd Annual Meeting of the Transportation Research Board, Washington, D.C.).
+- Lyon, C., Oh, J., Persaud, B.N., Washington, S.P., Bared, J., 2003. Empirical investigation of the IHSDM accident prediction algorithm for rural intersections. Transp. Res. Record 1840, 78–86.
+- Miaou, S.-P., 1994. The relationship between truck accidents and geometric design of road sections: Poisson versus negative binomial regressions. Accid. Anal. Prev. 26 (4), 471–482.
+- Miaou, S.-P., Lord, D., 2003. Modeling traffic crash-flow relationships for Intersections: dispersion parameter, functional form, and bayes versus empirical bayes. Transp. Res. Record 1840, 31–40.
+- Mitra, S., Chin, H.C., Quddus, M.A., 2002. Study of intersection accident by maneuvre type. Transp. Res. Record 1784, 43–50.
+- Mukhopadhyay, P., 1998. Small Area Estimation in Survey Sampling. Nasora Publishing House, New Delhi.
+- Mullahy, J., 1986. Specification and testing of some modified count data models. J. Econometrics 33, 341–365.
+- Neldelman, J., Wallenius, T., 1986. Bernoulli trials, Poisson trials, surprising variances, and Jensen's inequality. Amer. Statistician 40 (4), 286–289.
+- NHTSA, 2000. Traffic Safety Facts: 1999. U.S. Department of Transportation, Washington, D.C.
+- Payne, R.W. (Ed.), 2000. Lawes Agricultural Trust. Rothamsted Experimental Station, Oxford, U.K.
+
+- Oh, J., Lyon, C., Washington, S.P., Persaud, B.N., Bared, J., 2003. Validation of the FHWA crash models for rural intersections: lessons learned. Transp. Res. Record 1840, 41–49.
+- Olkin, I., Gleser, L.J., Derman, C., 1980. Probability Models and Applications. MacMillan Publishing Co., Inc., New York, NY.
+- Persaud, B.N., Dzbik, L., 1993. Accident prediction models for freeways. Transp. Res. Record 1401, 55–60.
+- Poch, M., Mannering, F.L., 1996. Negative binomial analysis of intersection-accident frequency. J. Transp. Eng. 122 (2), 105–113.
+- Poormeta, K., 1999. On the modelling of overdispersion of counts. Statistica Neerlandica 53 (1), 5–20.
+- Rao, J.N.K., 2003. Small Area Estimation. Wiley, Hoboken, N.J.
+- Rider, P.R., 1961. Estimating the Parameters of Mixed Poisson, Binomial and Weibull Distributions by Method of Moments. Bulletin de l'Institut International de Statistiques 38, Part 2.
+- Qin, X., Ivan, J.N., Ravishankar, N., 2004. Selecting exposure measures in crash rate prediction for two-lane highway segments. Accid. Anal. Prev. 36 (2), 183–191.
+- Shankar, V., Milton, J., Mannering, F.L., 1997. Modeling accident frequency as zero-altered probability processes: an empirical inquiry. Accid. Anal. Prev. 29 (6), 829–837.
+- Shankar, V.N., Ulfarsson, G.F., Pendyala, R.M., Nebergal, M.B., 2003. Modeling crashes involving pedestrians and motorized traffic. Safety Sci. 41 (7), 627–640.
+- Schmidt, P., Witte, A., 1989. Predicting criminal recidivism using splitpopulation survival time models. J. Econometrics 40, 141–159.
+- Taylor, L.R., 1961. Aggregation, variance, and the mean. Nature 189, 732–735.
+- Terza, J.V., Wilson, P., 1990. Analyzing frequencies of several types of events: a mixed multinomial-Poisson approach. Rev. Econ. Statist. 72 (2), 108–115.
+- Ullah, A., 1989. Semiparametric and nonparametric econometrics. Physica-Verlag, New York, NY.
+- US DOT, 1995. Highway Safety Performance 1992: Fatal and Injury Accident Rates on Public Roads in the United States. Federal Highway Administration, Washington, D.C.
+- Vellaisamy, P., Punnen, A.P., 2001. On the nature of the binomial distribution. J. Appl. Probability 38 (1), 36–44.
+- Vuong, Q.H., 1989. Likelihood ratio tests for model selection and nonnested hypothesis. Econometrica 57, 307–333.
+- Washington, S.P., Karlaftis, M., Mannering, F.L., 2003. Statistical and Econometric Methods for Transportation Data Analysis. Chapman and Hall, Boca Raton.
+- Zorn, C.J.W., 1996. Evaluating Zero-Inflated and Hurdle Poisson Specifications. Working paper. Department of Political Science, Ohio State University, Columbus, OH.
+- Yoneda, K., 1962. Estimations in some modified Poisson distributions. Yokohama Mathematical J. 10, 73–96.

Zotero/001_artiklid/Road Safety Indicators/maneinfluencenetworktrafficsignal2018.md

@@ -0,0 +1,585 @@
+---
+category: literaturenote
+citekey: maneinfluencenetworktrafficsignal2018
+title: "Influence of on-network, traffic, signal, demographic, and land use characteristics by area type on red light violation crashes"
+authors: "Mane, Ajinkya S.; Pulugurtha, Srinivas S."
+year: 2018
+date: 2018-00-00 2018
+doi: 10.1016/j.aap.2018.08.006
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2018.08.006"
+zotero_key: AGGVCW9I
+zotero_storage: CE7GGGE5
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Mane, Ajinkya S."
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+Contents lists available at [ScienceDirect](http://www.sciencedirect.com/science/journal/00014575)
+
+# Accident Analysis and Prevention
+
+journal homepage: [www.elsevier.com/locate/aap](https://www.elsevier.com/locate/aap)
+
+![](_page_0_Picture_5.jpeg)
+
+# Influence of on-network, traffic, signal, demographic, and land use characteristics by area type on red light violation crashes
+
+![](_page_0_Picture_7.jpeg)
+
+Ajinkya S. Mane[a](#page-0-0) , Srinivas S. Pulugurtha[b](#page-0-1),[⁎](#page-0-2)
+
+- <span id="page-0-0"></span><sup>a</sup> Student of Infrastructure & Environmental Systems (INES) Ph.D. Program, The University of North Carolina at Charlotte, 9201 University City Boulevard, Charlotte, NC 28223-0001, USA
+- <span id="page-0-1"></span><sup>b</sup> Professor of Civil & Environmental Engineering Department, Director of IDEAS Center, The University of North Carolina at Charlotte, 9201 University City Boulevard, Charlotte, NC 28223-0001, USA
+
+#### ARTICLE INFO
+
+Keywords: Red light violation Crash Area type Demographic Land use Traffic
+
+#### ABSTRACT
+
+The focus of this research paper is on extraction of predictor variables pertaining to on-network, traffic, signal, demographic, and land use characteristics, by area type, and examining their influence on the number of red light violation crashes. Data for the city of Charlotte, North Carolina was extracted and used for analysis. Three different sets of signalized intersections were selected in the three different area types - Central Business District (CBD), urban, and suburban areas. Each set is comprised of sixty signalized intersections (total 180 signalized intersections). The number of red light violation crashes from January 2010 to December 2014, within the vicinity of each selected signalized intersection, was considered as the dependent variable to develop crash estimation models for each area type. The crash estimation models by area type were compared with the crash estimation model developed considering all the 180 signalized intersections together. Different predictor variables were found to be significant at a 95% confidence level in three different areas. Log-link model with Negative Binomial distribution was observed to best fit the data used in this research. Findings indicate that enforcement, either manually or using red light running cameras (RLCs), at signalized intersections with high traffic volume in the CBD area; at signalized intersections with high traffic volume, high all-red clearance time, near high density of horizontal mixed non-residential and open space/recreational type land uses in urban area; at signalized intersections with high traffic volume, speed limit on the major approach, the number of lanes on the minor approach, and all-red clearance time and areas surrounded with horizontal mixed non-residential and retail type land use in suburban areas, would lead to a reduction in the number of red light violation crashes.
+
+### 1. Introduction
+
+An estimated 165,000 people were injured and 800 people were killed, annually, in crashes due to red light running in the United States ([Insurance Institute for Highway Safety \(IIHS, 2007](#page-12-0)). [Retting and](#page-12-1) [Williams \(1996\)](#page-12-1) reported two red light runners per hour in, total, 234 h of data collection. Their study concluded that the red light runners are drivers under the age of thirty years, with poor driving history, less likely to use seat belts, and drove small and older cars. [Retting et al.](#page-12-2) [\(1998\)](#page-12-2) reported red light violations at the rate of 5.2 runners/hour and 1.3 runners/hour for two different sites. [Porter and England \(2000\)](#page-12-3) stated that unbuckled and non-Caucasian drivers are more likely to run and violate red light.
+
+Most of the past studies on red light violation have analyzed the safety effect of red light running cameras (RLCs). [Maisey \(1981\),](#page-12-4) [South](#page-12-5) [et al. \(1998\),](#page-12-5) [Kamyab et al. \(2002\)](#page-12-6), [Garber et al. \(2007\),](#page-12-7) [Tay and de](#page-12-8) [Barros \(2009\),](#page-12-8) [\(2011\),](#page-12-9) [Pulugurtha and Otturu \(2014\)](#page-12-10) investigated red light running violations and/or the effectiveness of RLCs in reducing red light running violations or crashes. [Retting and Kyrychenko \(2002\)](#page-12-11) found that injury crashes reduced by 29%, while right-angle crashes reduced by 32% after the implementation of RLCs. On the contrary, [Maisey \(1981\),](#page-12-4) [South et al. \(1998\)](#page-12-5) reported an increase in rear-end collisions after the implementation of RLCs. [Weldegiorgis and Jha](#page-12-12) [\(2009\)](#page-12-12) stated that factors such as driver behavior, approach speed, and intersection geometry contribute to red light violations.
+
+Studies in the past concluded that younger drivers are prone to traffic violations while older drivers are prone to crashes at intersections ([McGwin and Brown, 1999;](#page-12-13) [Chapman et al., 2014](#page-12-14)). Similarly, [Shell et al. \(2015\)](#page-12-15) found that teens taking driver education are less likely to be involved in crashes or traffic violations in the first two years of driving. [Alver et al. \(2014\)](#page-12-16) studied the relationship between demographic characteristics, socio-economic characteristics, traffic rule
+
+E-mail addresses: [amane@uncc.edu](mailto:amane@uncc.edu) (A.S. Mane), [sspulugurtha@uncc.edu](mailto:sspulugurtha@uncc.edu) (S.S. Pulugurtha).
+
+<span id="page-0-2"></span><sup>⁎</sup> Corresponding author.
+
+<span id="page-1-0"></span>Table 1 Summary of Past Research on Factors Attributed to Red Light Violation Behavior and Crashes.
+
+| Category             | Study                                                                        | Contributing Factors                                                                                                                                                                                                                          |
+|----------------------|------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
+| Behavior             | Van der Horst et al. (1986)                                                  | Yellow change time                                                                                                                                                                                                                            |
+|                      | Baguley (1988)                                                               | High traffic volume                                                                                                                                                                                                                           |
+|                      | Retting and Williams (1996)                                                  | Profiling of violators: young drivers, drivers of small and older cars, drivers with poorer driving record, and<br>drivers who are less likely to use seat belts                                                                              |
+|                      | Kamyab et al. (2000)                                                         | High traffic volume                                                                                                                                                                                                                           |
+|                      | Porter and England (2000)                                                    | Large intersections, and high traffic volume, time period (3 pm to 6 pm), safety belt use, and ethnicity                                                                                                                                      |
+|                      | Schattler et al. (2003)                                                      | All-red clearance and yellow change time                                                                                                                                                                                                      |
+|                      | Kulanthayan et al. (2007)                                                    | Day (weekday / weekend), camera enforcement, type of vehicle (four-wheel / two-wheel vehicles), cycle<br>length, and type of traffic signal                                                                                                   |
+|                      | Zimmerman (2007)                                                             | Dilemma zone, additional time for trucks at a signalized intersection                                                                                                                                                                         |
+|                      | Gates and Noyce (2010)                                                       | Dilemma zone, vehicle type, time of the day and platooning                                                                                                                                                                                    |
+| Crashes              | Mohamedshah et al. (2000)                                                    | Number of lanes on both streets, left- and right-turn lanes, entering/crossing traffic flow, traffic control type<br>(fully actuated, semi actuated, and pre-timed), speed limit (a potential variable but was not present in the<br>dataset) |
+| Behavior and crashes | Bonneson et al. (2002)<br>Datta et al. (2000), Schattler and Datta<br>(2004) | Traffic volume of the subject approach, traffic volume of the cross-street, and red light running rate<br>All-red clearance and yellow change time                                                                                            |
+
+violations, and crashes among young drivers. Four different traffic violations, red light violation, seat-belt violation, speed limit violation, and driving under the influence of alcohol or drugs, were considered in their study. Their results indicate that crash rate increased to 38.3% for drivers who received at least one traffic citation.
+
+Studies in the past have also reported factors contributing to red light violation behavior and red light violation crashes. [Table 1](#page-1-0) summarizes factors contributing to red light running behavior and red light violation crashes from selected past research studies.
+
+[Van der Horst et al. \(1986\)](#page-12-17) found that extending yellow change time from three to four seconds would decrease red light runners (car drivers) by 50%. [Datta et al. \(2000\)](#page-12-18) found that introducing a well-designed all-red clearance time following the yellow change time would significantly reduce red light violations and right-angle crashes at urban signalized intersections. [Schattler and Datta \(2004\)](#page-12-19) concluded that at urban intersections, all-red clearance time would decrease the probability of right-angled crashes even after red light violation. A similar result was observed in the [Schattler et al. \(2003\)](#page-12-20) study. [Zimmerman](#page-12-21) [\(2007\)](#page-12-21) observed that if yellow change time was extended by 1.5 s, it would improve intersection safety by reducing the number of trucks by 47% in the dilemma zone. [Gates and Noyce \(2010\)](#page-12-22) observed that tractor-trailer and single-unit trucks are more likely to be involved in red light running compared to passenger car. Also, the number of red light violations are observed to be higher during peak hours compared to off-peak hours.
+
+#### 1.1. Limitations of past research and need for this research
+
+Most of the past studies have investigated the effectiveness of RLCs, while a few researchers focused on identifying on-network (the number of lanes, width of intersection, speed limit, etc.), traffic, and signal (phasing and timing plan) characteristics that influence red light violations and associated crashes. The spacing between intersections and width of intersections in Central Business District (CBD) area is comparatively lower than the spacing between intersections and width of intersections in urban and suburban area. Likewise, the speed limit, network density, traffic volume, and signal timings also differ by area type. While on-network, traffic, and signal characteristics were observed to have an influence on the number of red light violation crashes, the influence of these characteristics, by area type, on the number of red light violation crashes were not explored much in the past.
+
+Different land use types such as office buildings, commercial centers, industrial areas, and recreational areas attract people from different parts of the city. High-rise office buildings are typically located in the CBD area but not in the suburban area. On the other hand, singlefamily residential areas prevail more in the suburban area. High-rise commercial centers and mixed land uses are typically located in the CBD area, while horizontal mixed use or commercial centers are typically located in urban and suburban areas. The demographic characteristics as well as the number of trips generated and attracted by each land use type vary by area type. Therefore, traffic volume and congestion within the vicinity of each land use differ by area type. As stated in Kamyab [et al. \(2000\)](#page-12-23), an increase in traffic volume and congestion influence drivers' aggressive behavior and hence, the number of traffic violations. In other words, the demographic and land use characteristics vary by area type and could also influence the number of red light violation crashes.
+
+Literature review indicates that majority of past studies on red light violation were conducted considering urban signalized intersections. As on-network, traffic, signal, demographic, and land use characteristics not only vary by area type but influence the number of red light violation crashes, examining the relationship between the number of red light violation crashes and predictor variables such as on-network, traffic, signal, demographic, and land use characteristics, by area type, helps proactively identify where they could potentially happen for enforcement or implementation of remedial measures (implementation of manual or automated enforcement; signal timing changes; etc.). The findings from such a research are vital to identify, prioritize, and allocate limited available transportation funds, either manually or using RLCs.
+
+#### 2. Methodology
+
+The city of Charlotte, North Carolina was considered as the study area to gather data and conduct analysis. The methodology includes the following steps.
+
+- 1 Identify study signalized intersections by area type
+- 2 Identify predictor variables contributing to red light violation crashes
+- 3 Generate buffers and conduct geospatial analysis
+  - a Extract the number of red light violation crashes
+  - b Extract on-network, traffic, and signal characteristics
+  - c Extract demographic characteristics
+  - d Extract land use characteristics
+- 4 Check for multicollinearity between predictor variables
+- 5 Develop crash estimation models by area type
+- 6 Validate the models
+
+Each step is explained next in detail.
+
+#### 2.1. Identify study signalized intersections by area type
+
+In this research, the city of Charlotte was divided into three different areas: CBD area, urban area, and suburban area. Freeways (I-277 and I-77), which surround the Charlotte downtown/uptown area were considered as boundary lines for the CBD area. The CBD area primarily comprises of high-rise office buildings, apartments/condominiums, businesses, and related commercial activity. Intersections are closely spaced, road widths and speed limits are relatively lower, and network density is higher in the CBD area. The outer freeway (I-485; beltway) was considered as the extreme boundary line for the suburban area. The suburban area primarily comprises of residential areas, institutions/ schools, commercial centers, and recreational areas. The area between the CBD area and the suburban area was considered as the urban area. The density of land uses in the urban area is higher than in the suburban area, but lower than in the CBD area. Likewise, spacing between intersections and number of lane miles is higher in the urban area than in the CBD area, but lower than in the suburban area. Road widths and speed limits are comparable in both, urban and suburban areas.
+
+<span id="page-2-0"></span>Sixty signalized intersections in each area type (total 180 signalized intersections) were randomly selected to conduct analysis and develop crash estimation models. These signalized intersections were ensured to be spatially distributed in the respective areas ([Fig. 1](#page-2-0)). In addition, crash estimation models were developed by considering all the selected 180 signalized intersections.
+
+#### 2.2. Identify predictor variables contributing to red light violation crashes
+
+As stated previously, the on-network, traffic, signal, demographic, and land use characteristics vary by area type. On-network characteristics are those that exist on the road network. They include speed limit, the number of lanes, the number of approaches, presence of median, road width, etc. These factors may influence the number of red light violation crashes. In this research, the speed limit and the number of lanes were considered as the predictor variables.
+
+Traffic volume, yellow change time, and all-red clearance time are traffic and signal characteristics that may influence the number of red light violation crashes. These characteristics were also considered as predictor variables.
+
+The traffic volume and the number of crashes (in general) could increase with an increase in population and the number of households. Moreover, different land use patterns could attract people from
+
+![](_page_2_Figure_10.jpeg)
+
+Fig. 1. Selected signalized intersections in the City of Charlotte, North Carolina.
+
+<span id="page-3-0"></span>Table 2 Predictor Variables.
+
+| On-network, traffic and signal characteristics                  |                                                                                                                                                                                                                                                                                |
+|-----------------------------------------------------------------|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
+| Traffic Volume (TV)                                             | Average traffic volume per year, including all the approaches                                                                                                                                                                                                                  |
+| Speed Limit                                                     | Speed limit was categorized as the speed limit on the major approach (SLMA) and the speed Limit on the minor approach<br>(SLMI)                                                                                                                                                |
+| # of Lanes                                                      | # of lanes was categorized as the number of lanes on the major approach (NLMA) and the number of lanes on the minor<br>approach (NLMI)                                                                                                                                         |
+| # of Approaches (NAPR)                                          | Total number of approaches at the signalized intersection                                                                                                                                                                                                                      |
+| Average Yellow Change Time (YCT)                                | Time required for driver to comfortably decelerate to stop or to clear the signalized intersection.                                                                                                                                                                            |
+| All-Red Clearance Time (AR)                                     | Time required to clear the vehicles in the signalized intersection.                                                                                                                                                                                                            |
+| Land use characteristics (square feet)                          |                                                                                                                                                                                                                                                                                |
+| Civic/Institutional (Civic)                                     | Major cultural, educational, medical, government, athletic institutions                                                                                                                                                                                                        |
+| Horizontal Mixed Use Residential & Non-Residential<br>(HMRNR)   | Typical block consists of multiple buildings. Each building is used as single-use buildings. Residential and compatible non<br>residential uses are placed in the separate buildings.                                                                                          |
+| Horizontal Mixed Use Non-Residential (HMNR)<br>Industrial (IND) | Typical block consists of single-use buildings, only for non-residential uses such as commercial, institutional, etc.<br>Areas with manufacturing, processing and assembling of parts, distribution centers and transportation terminals;<br>specialized industrial operations |
+| Large lots of residential (LLoR)                                | Large area utilized for residential                                                                                                                                                                                                                                            |
+| Multi-family (MF)                                               | Areas with a variety of housing types; 12–43 dwelling units per acre                                                                                                                                                                                                           |
+| Office (OFF)                                                    | Areas conducive to establishment and operation of offices, institutions, and commercial activities not involving in sale                                                                                                                                                       |
+| Open Space/Recreational (OS)                                    | Area utilized for recreational purposes such as parks                                                                                                                                                                                                                          |
+| Parking (Park)                                                  | Area utilized for parking purpose                                                                                                                                                                                                                                              |
+| Retail (Retail)                                                 | Area utilized for retail shops                                                                                                                                                                                                                                                 |
+| Single-family – Attached (SFA)                                  | Area with primarily single-family housing where houses have one common wall with the adjacent house                                                                                                                                                                            |
+| Single-family – Detached (SFD)                                  | Independent residential area for single-family. Houses are not connected.                                                                                                                                                                                                      |
+| Transportation (Trans)                                          | Area utilized for transportation purpose such as air transport, water transport, and rail transport                                                                                                                                                                            |
+| Utility (Utility)                                               | Areas utilized to provide the services for everyone in a community                                                                                                                                                                                                             |
+|                                                                 |                                                                                                                                                                                                                                                                                |
+
+Vertical Mixed Use (VMU) Combines residential and compatible non-residential uses within the same building. Typically, non-residential uses are provided on the lower floors and residential uses on the upper floors
+
+Warehouse/Distribution (WH) Area utilized for manufacturing and wholesale trade/distribution process
+
+different parts of the city. As these demographic and land use characteristics within the vicinity of a signalized intersections may influence the number of red light violation crashes, they were also considered as predictor variables.
+
+Total Household Units (HU) Total number of household units
+
+Population (PL) Total population
+
+The selected predictor variables for analysis and model development in this research are summarized in [Table 2](#page-3-0).
+
+# 2.3. Generate buffers and conduct geospatial analysis
+
+Demographic characteristics
+
+Buffers were generated around each signalized intersection using "buffer" feature in ArcGIS. Further, "intersect" feature in ArcGIS was used to extract data by overlaying the generated buffers on individual geospatial layers pertaining to crashes, on-network, traffic, demographic, and land use characteristics. The width of the buffer to capture data is different for the number of red light violation crashes, on-network, traffic, demographic, and land use characteristics.
+
+# 2.3.1. Extract the number of red light violation crashes
+
+Crash data from January 2010 to December 2014 was used in this research. Crashes related to red light violation were extracted from the main crash database based on the contributing factor (red light violation). The selected crash data was then spatially overlaid on the street network. A buffer width of 100 feet was generated around each selected signalized intersection to extract the total number of crashes (related to red light violation) that occurred on all the approaches of each selected signalized intersection. The 100-feet buffer width was selected considering typical width of the signalized intersection (60–80 feet) in the study area. Considering 100-feet buffer width also minimizes concerns related to overlapping of buffers. The number of red light violation crashes at each intersection (within 100 feet) was considered as the dependent variable.
+
+# 2.3.2. Extract on-network, traffic, and signal characteristics
+
+The data pertaining to on-network (speed limit and the number of lanes), traffic, and signal characteristics was obtained from the city of Charlotte Department of Transportation (CDoT).
+
+2.3.2.1. Speed limit. Typically, a signalized intersection has one major approach and one minor approach. Therefore, for each signalized intersection, speed limit on the major approach and the minor approach were considered separately in the analysis.
+
+2.3.2.2. Number of lanes. Different approaches have different number of lanes. The number of lanes on the major approach and minor the approach were considered separately in the analysis.
+
+2.3.2.3. Traffic volume. The number of crashes at a signalized intersection are related to the traffic volume (exposure). In this research, similar to crash data (five years), average traffic volume during the five-year period was considered for analysis. Here, the average AADT for five years from all the approaches was considered to represent traffic volume for an intersection.
+
+2.3.2.4. Yellow change time. Yellow change time depends on the width of the signalized intersection and approach speed limit. It was computed for each approach and used to compute the average yellow change time for the signalized intersection. Eq. [\(1\)](#page-3-1) was used to compute the yellow change time ([North Carolina Department of Transportation](#page-12-28) [\(NCDOT, 2004](#page-12-28)). The minimum yellow change time is 3.0 s.
+
+<span id="page-3-1"></span>Yellow Change Time = 
+$$t + \frac{V}{2a + 64.4 \times g}$$
+ (1)
+
+where, t = perception reaction time, typically 1.5 s, V = design speed, in feet per seconds, a = deceleration rate, typically 11.2 feet /seconds<sup>2</sup> and, g = grade.
+
+<span id="page-4-1"></span>![](_page_4_Figure_2.jpeg)
+
+Fig. 2. Spatial overlay of census block and land use data on a 0.5-mile buffer around a signalized intersection.
+
+2.3.2.5. All-red clearance time. Each signal is equipped with an all-red clearance time. It is provided to clear the vehicles in the signalized intersection. As per the traffic signal design manual of the North Carolina Department of Transportation (NCDOT), all-red clearance time was computed for each signalized intersection using Eq. [\(2\)](#page-4-0). The minimum all-red clearance time is 1.0 s.
+
+<span id="page-4-0"></span>All-red Clearance Time = 
+$$\frac{W}{V}$$
+ (2)
+
+where, W = width of the intersection, in feet, and, V = design Speed, in feet per seconds.
+
+## 2.3.3. Extract demographic characteristics
+
+Demographic characteristics within the vicinity of each selected signalized intersection were extracted by generating 0.5-mile buffers and then overlaying census data [\(Fig. 2\(](#page-4-1)a)).
+
+It was assumed that population and the number of households are uniformly distributed in each census block. Therefore, the proportion of census block area that falls in a buffer was considered to estimate the aforementioned predictor variables (Eq. [\(3\)\)](#page-4-2).
+
+<span id="page-4-2"></span>
+$$P_{i} = \sum_{j} \frac{A_{j,i}}{A_{j}} P_{j}$$
+(3)
+
+where, Pi = total population or the number of households in buffer "i", Aj,i = actual area of census block "j" in the buffer "i", Aj = area of census block "j", and, Pj= total population or number of households in census block "j".
+
+## 2.3.4. Extract land use characteristics
+
+Land use characteristics were extracted within 0.5-mile buffer generated around each selected signalized intersection, using 'intersect' feature in GIS. [Fig. 2\(](#page-4-1)b) illustrates overlaying of existing land use layer on a 0.5-mile buffer around a signalized intersection. All the land use areas were represented in square feet.
+
+## 2.4. Check for multicollinearity between predictor variables
+
+The selected predictor variables may be correlated to each other. To identify the influence of predictor variables and accurately estimate the dependent variable (the number of red light violation crashes), only one predictor variable is selected between the two correlated predictor variables, at a time for model development. In addition to a strong correlation, the included and excluded predictor variables pair must have a meaningful relationship (example, traffic volume and speed limit or the number of lanes).
+
+The multicollinearity between predictor variables and collinearity between dependent and predictor variables was checked by computing Pearson correlation coefficients for each dataset. The range of Pearson correlation coefficient is from −1 to +1. The cut-off value of Pearson correlation coefficient was set up as −0.3 and +0.3 (i.e., Pearson correlation coefficient value less than or equal to −0.3 or greater than or equal to 0.3 are assumed to be correlated to each other) to avoid multicollinearity between the predictor variables (at least at a 95% confidence level).
+
+Either of the two correlated predictor variables could be selected for model development. This leads to multiple models with combinations of predictor variables.
+
+## 2.5. Develop crash estimation models by area type
+
+Crash estimation models by area type (CBD, urban, and suburban area) were developed to investigate the factors contributing to the number of red light violation crashes by area type.
+
+The model selection depends on linear or non-linear correlation between the dependent and predictor variables. Generalized linear model with Negative Binomial distribution, log-normal distribution, Gamma distribution, and Poisson distribution are generally used to develop crash estimation models. The best model was selected based on the Quasi Likelihood under Independence Model Criterion (QIC) and Corrected Quasi Likelihood under Independence Model Criterion (QICC). The lower the QIC and QICC, the better is the goodness of fit. Moreover, the difference between QIC and QICC should be generally low for a good model. In addition, the predictor variables with p-value less than 0.05 are considered as significant at a 95% confidence level.
+
+# 2.6. Validate the models
+
+Each crash estimation model was validated using Mean Absolute Deviation (MAD). MAD was computed using Eq. [\(4\)](#page-4-3). The values of MAD close to zero resembles the ability of models to predict the number of crashes close to the actual number of crashes ([Oh et al., 2003](#page-12-29)).
+
+<span id="page-4-3"></span>
+$$MAD = \frac{1}{n} \sum_{i=1}^{n} |\hat{c}_{j} - c_{j}|$$
+(4)
+
+where, n = the number of observations, *ĉj*= the predicted number of red light violation crashes, and, *cj*= the observed number of red light violation crashes.
+
+<span id="page-5-0"></span>Table 3 Number of Red Light Violation Crashes.
+
+| Year  | CBD             |                  | Urban           |                  | Suburban        |                  |
+|-------|-----------------|------------------|-----------------|------------------|-----------------|------------------|
+|       | # of<br>Crashes | # of<br>Injuries | # of<br>Crashes | # of<br>Injuries | # of<br>Crashes | # of<br>Injuries |
+| 2010  | 184             | 215              | 221             | 259              | 226             | 267              |
+| 2011  | 150             | 173              | 200             | 228              | 206             | 235              |
+| 2012  | 153             | 149              | 194             | 206              | 192             | 198              |
+| 2013  | 192             | 213              | 270             | 296              | 271             | 298              |
+| 2014  | 180             | 222              | 231             | 283              | 233             | 282              |
+| Total | 859             | 972              | 1,116           | 1,272            | 1,128           | 1,280            |
+
+#### 3. Results
+
+The results obtained from analysis and model development are presented in this section.
+
+## 3.1. Descriptive analysis
+
+The total number of red light violation crashes and the associated number of injuries over the five-year period (January 2010 to December 2014) are presented in [Table 3](#page-5-0). The maximum number of crashes was observed in the year 2013, in all the area types. Moreover, the total number of crashes from year 2010 to year 2014 is ∼30% more in urban and suburban areas when compared to the CBD area. Descriptive statistics pertaining to all the variables considered in this research are presented in [Table 4.](#page-6-0)
+
+#### 3.2. Examine the correlation between predictor variables
+
+A correlation matrix was developed for each area type and for all the signalized intersections irrespective of area type. As an example, [Table 5](#page-7-0) summarizes Pearson correlation coefficient matrix computed using all the signalized intersections data. It was observed that some of the predictor variables were highly correlated to each other. For example, traffic volume, speed limit on the major approach, and speed limit on the minor approach are highly correlated to each other. As stated previously, to minimize the effect of multicollinearity, only one of these three predictor variables were considered in a model. As any of these three predictor variables could be considered, possible combinations of predictor variables were identified for the development of models. The possible combinations of selected predictor variables for each area type and complete dataset are summarized in [Table 6](#page-9-0). As an example, traffic volume was considered in Model 1, speed limit on the minor approach was considered in Model 2, and speed limit on the major approach was considered in Model 3, when developing models using all the signalized intersections data.
+
+# 3.3. Crash estimation models
+
+The number of red light violation crashes was considered as the dependent variable. A total of twelve crash estimation models were developed in this research. In all the developed models, Negative Binomial distribution-based log-link model was found to be a better fit than log-normal, Gamma or Poisson distribution-based log-link models.
+
+When excluding correlated predictor variables, caution must be exercised to ensure that the estimated coefficients represent the combined effects of both the included and excluded predictor variables ([Tay, 2017\)](#page-12-30). In addition to examining strong and meaningful relationships between included and excluded predictor variables, the computed standard errors, QIC, and QICC for the twelve models were compared with those obtained for models considering all the predictor variables (irrespective of correlations). The computed standard errors, QIC, and QICC are lower for the twelve models than those obtained for models considering all the predictor variables.
+
+Predictor variables with p-value greater than 0.05 were removed, one after another, when developing models. All the predictor variables with p-value less than 0.05 were considered as significant (at a 95% confidence interval) in the final model for each dataset. The general form of the negative binomial log-link distribution based model is represented as follows.
+
+$$Y = EXP^{(\beta + \alpha_1 \times X_1 + \alpha_2 \times X_2 + \dots + \alpha_n \times X_n)}$$
+(5)
+
+where,
+
+Y = predicted number of red light violation crashes,
+
+α1, α2, …, α<sup>n</sup> = coefficients of predictor variables, and,
+
+X1, X2, …, Xn = predictor variables considered for model development.
+
+### 3.3.1. CBD area
+
+Three crash estimation models were developed for the CBD area. The results obtained are summarized in [Table 7](#page-9-1). The results obtained indicate that the number of red light violation crashes increase as the traffic volume increases. However, areas with large lots of residential, retail, utility, and industrial type land use within the vicinity of a signalized intersection have a negative effect on the number of red light violation crashes.
+
+# 3.3.2. Urban area
+
+The results obtained from the three crash estimation models developed for urban area are summarized in [Table 8](#page-10-0). The results obtained from Model 1 indicate that the number of red light violation crashes increase as the traffic volume and all-red clearance time increase. Likewise, both, Model 2 and Model 3 indicate that the number of red light violation crashes increase as the areas with horizontal mixed nonresidential and open space/recreational type land use increase within the vicinity of a signalized intersection. However, areas with large lots of residential, warehouse/distribution, and industrial type land use have a negative effect on the number of red light violation crashes.
+
+## 3.3.3. Suburban area
+
+The models developed for suburban area are summarized in [Table 9](#page-10-1). The results obtained indicate that the number of red light violation crashes increase as the traffic volume, speed limit on the major approach, the number of lanes on the minor approach, and all-red clearance time increase within the vicinity of a signalized intersection. Further, the number of red light violation crashes increase as areas with horizontal mixed non-residential and retail type land use increase within the vicinity of a signalized intersection. However, areas with single-family attached, single-family detached, and transportation type land use have a negative effect on the number of red light violation crashes.
+
+## 3.3.4. All signalized intersections
+
+Crash estimation models were developed by considering all the 180 randomly selected signalized intersections. The results obtained are presented in [Table 10.](#page-11-0) The results obtained indicate that the number of red light violation crashes increase as the traffic volume, speed limit on the minor approach, the number of lanes on the minor approach, and areas with horizontal mixed non-residential and retail type land use increase within the vicinity of a signalized intersection. However, areas with industrial, large lots of residential, single-family attached, and transportation type land use have a negative effect on the number of red light violation crashes.
+
+# 3.4. Model validation
+
+Each area type model was validated by randomly selecting 10 signalized intersections in the respective areas. In addition, 30 signalized intersection were randomly selected for validation of complete dataset
+
+Table 4 Descriptive Statistics.
+
+<span id="page-6-0"></span>
+
+| Variables                                          | N  | CBD        |               |              |              | Urban    |               |              |              | Suburban |               |              |              |
+|----------------------------------------------------|----|------------|---------------|--------------|--------------|----------|---------------|--------------|--------------|----------|---------------|--------------|--------------|
+|                                                    |    | Min.       | Max.          | Mean         | SD           | Min.     | Max.          | Mean         | SD           | Min.     | Max.          | Mean         | SD           |
+| # Crashes                                          | 60 | 2          | 30            | 7.65         | 6.09         | 2        | 26            | 6.98         | 4.98         | 2        | 29            | 6.32         | 5.19         |
+| Traffic Volume                                     | 60 | 11,700     | 64,000        | 28,811.67    | 13,395.27    | 16,500   | 153,000       | 50,995.83    | 24,295.30    | 18,700   | 174,000       | 71,093.33    | 41,939.61    |
+| Speed Limit on the Major Approach                  | 60 | 25         | 55            | 33.50        | 5.39         | 35       | 45            | 39.33        | 4.91         | 30       | 55            | 44.17        | 5.06         |
+| Speed Limit on the Minor Approach                  | 60 | 25         | 35            | 29.08        | 3.74         | 25       | 45            | 33.33        | 4.66         | 25       | 45            | 35.83        | 5.46         |
+| # of Lanes on the Major Approach                   | 60 | 2          | 7             | 4.12         | 1.03         | 2        | 6             | 4.53         | 0.79         | 2        | 7             | 4.58         | 1.27         |
+| # of Lanes on the Minor Approach                   | 60 | 1          | 7             | 2.90         | 1.08         | 1        | 5             | 2.75         | 1.07         | 1        | 6             | 2.67         | 1.17         |
+| Yellow Change Time                                 | 60 | 3.00       | 3.63          | 3.15         | 0.17         | 3.00     | 3.95          | 3.37         | 0.26         | 3.13     | 4.28          | 3.55         | 0.27         |
+| All-Red Clearance Time                             | 60 | 1.00       | 2.66          | 1.56         | 0.35         | 1.19     | 2.27          | 1.65         | 0.27         | 1.19     | 2.69          | 1.87         | 0.32         |
+| Civic / Institutional                              | 60 | 254,398.00 | 5,878,422.83  | 3,342,770.44 | 1,789,053.70 | 0.00     | 6,341,518.07  | 2,929,702.45 | 1,855,887.42 | 0.00     | 6,394,974.05  | 2,120,006.58 | 1,755,776.83 |
+| Horizontal Mixed Use Residential & Non-Residential | 60 | 0.00       | 118,103.06    | 40,852.51    | 51,980.63    | 0.00     | 223,278.53    | 21,057.36    | 48,593.25    | 0.00     | 27,866.67     | 2,956.20     | 8,341.75     |
+| Horizontal Mixed Use Non-Residential               | 60 | 0.00       | 1,565,545.52  | 397,896.75   | 365,021.90   | 0.00     | 2,143,985.21  | 414,594.11   | 502,554.75   | 0.00     | 2,231,985.01  | 363,938.64   | 502,167.02   |
+| Industrial                                         | 60 | 0.00       | 5,664,335.24  | 544,377.90   | 1,063,859.30 | 0.00     | 5,804,397.68  | 616,315.63   | 1,180,094.55 | 0.00     | 4,351,364.30  | 533,015.15   | 994,555.81   |
+| Large Lots of Residential                          | 60 | 0.00       | 960,354.56    | 53,833.89    | 172,563.62   | 0.00     | 3,659,272.62  | 200,428.68   | 581,828.08   | 0.00     | 3,297,486.10  | 367,681.23   | 735,645.05   |
+| Multi-family                                       | 60 | 0.00       | 11,142,309.35 | 1,985,797.82 | 1,608,239.40 | 8,957.11 | 7,268,312.44  | 1,723,486.34 | 1,436,299.20 | 0.00     | 7,486,753.76  | 2,196,695.22 | 2,043,201.23 |
+| Office                                             | 60 | 0.00       | 3,030,190.50  | 1,126,151.82 | 740,476.84   | 0.00     | 3,767,315.20  | 1,230,128.82 | 930,743.24   | 0.00     | 7,757,025.70  | 1,192,769.77 | 1,337,266.67 |
+| Open Space / Recreation                            | 60 | 912.79     | 5,282,417.54  | 829,294.68   | 831,171.13   | 0.00     | 4,518,764.78  | 930,169.29   | 872,017.18   | 0.00     | 4,243,499.02  | 998,271.91   | 889,121.75   |
+| Parking                                            | 60 | 0.00       | 2,841,049.05  | 1,026,864.72 | 958,257.33   | 0.00     | 2,638,204.48  | 510,244.72   | 807,845.54   | 0.00     | 2,003,667.72  | 202,007.54   | 415,918.27   |
+| Retail                                             | 60 | 0.00       | 5,380,914.18  | 933,930.46   | 745,264.06   | 0.00     | 8,771,509.35  | 1,806,618.13 | 1,508,168.42 | 0.00     | 7,408,903.41  | 2,179,961.61 | 1,801,904.00 |
+| Single-family Attached                             | 60 | 0.00       | 1,388,892.85  | 199,693.95   | 281,364.27   | 0.00     | 1,092,918.80  | 244,121.01   | 280,718.71   | 0.00     | 2,115,459.34  | 197,877.16   | 392,879.57   |
+| Single-family Detached                             | 60 | 0.00       | 10,542,842.52 | 2,956,778.84 | 3,395,938.48 | 3,641.96 | 10,444,370.27 | 3,267,718.65 | 2,713,917.96 | 0.00     | 14,069,605.22 | 3,957,467.24 | 3,838,236.01 |
+| Transportation                                     | 60 | 0.00       | 1,730,455.37  | 496,043.77   | 450,324.36   | 0.00     | 2,942,897.52  | 352,494.20   | 538,863.30   | 0.00     | 6,199,481.80  | 250,880.04   | 834,313.51   |
+| Utility                                            | 60 | 0.00       | 940,531.65    | 125,806.28   | 154,769.14   | 0.00     | 567,933.68    | 89,192.57    | 117,216.42   | 0.00     | 624,869.12    | 77,224.29    | 123,883.99   |
+| Vertical Mixed Use                                 | 60 | 0.00       | 2,197,209.47  | 866,151.00   | 819,917.81   | 0.00     | 2,032,848.74  | 441,082.59   | 632,955.28   | 0.00     | 1,731,480.37  | 223,652.94   | 370,779.75   |
+| Warehouse Distribution                             | 60 | 0.00       | 4,592,676.78  | 594,300.38   | 883,998.59   | 0.00     | 5,685,340.07  | 743,654.24   | 1,164,114.17 | 0.00     | 6,129,426.78  | 700,656.13   | 1,257,063.39 |
+| Population                                         | 60 | 472.15     | 19,336.44     | 10,915.47    | 6,114.53     | 547.99   | 63,008.94     | 10,501.72    | 9,803.55     | 954.82   | 65,561.64     | 12,101.61    | 13,863.15    |
+| # Households                                       | 60 | 189.46     | 13,279.96     | 6,473.03     | 4,195.23     | 196.25   | 19,827.32     | 5,265.20     | 4,725.83     | 394.36   | 30,323.35     | 5,444.06     | 6,185.06     |
+
+(continued on next page)
+
+Table 5 Correlation Matrix - All Signalized Intersections.
+
+<span id="page-7-0"></span>
+
+| IND                    | −0.226**<br>−0.257**<br>−0.279**<br>−0.217**<br>−0.238**<br>−0.175*<br>−0.151*<br>−0.003<br>−0.083<br>0.361**<br>0.642**<br>0.183*<br>0.080<br>0.008<br>1                                                                                                                | PL                                                                                                                                                                                            |
+|------------------------|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
+| HMNR                   | −0.211**<br>−.0217**<br>−0.230**<br>−0.072<br>−0.016<br>−0.026<br>−0.106<br>0.226**<br>0.226**<br>0.232**<br>0.184*<br>0.140<br>0.082<br>0.031<br>0.041<br>1                                                                                                             | WH                                                                                                                                                                                            |
+| HMRNR                  | −0.204**<br>−0.222**<br>−0.383**<br>−0.148*<br>−0.180*<br>0.512**<br>0.293**<br>0.534**<br>0.244**<br>0.004<br>0.018<br>0.059<br>0.057<br>0.053<br>0.027<br>0.104<br>1                                                                                                   | VMU                                                                                                                                                                                           |
+| Civic                  | −0.369**<br>−0.274**<br>−0.215**<br>−0.266**<br>−0.196**<br>−0.474**<br>−0.460**<br>−0.004<br>−0.111<br>−0.101<br>0.296**<br>0.240**<br>0.686**<br>0.211**<br>0.675**<br>0.308**<br>0.406**<br>1                                                                         | Utility                                                                                                                                                                                       |
+| AR                     | −0.348**<br>−0.400**<br>−0.404**<br>−0.216**<br>−0.395**<br>−0.074<br>−0.071<br>−0.126<br>−0.094<br>−0.109<br>−0.056<br>0.227**<br>0.288**<br>0.356**<br>0.261**<br>0.187*<br>0.028<br>0.091<br>1                                                                        | Trans                                                                                                                                                                                         |
+| Y.C.T.                 | −0.528**<br>−0.285**<br>−0.616**<br>−0.601**<br>−0.279**<br>−0.165*<br>−0.165*<br>−0.129<br>−0.102<br>−0.079<br>−0.076<br>0.263**<br>0.247**<br>0.398**<br>0.192**<br>0.457**<br>0.225**<br>0.162*<br>0.066<br>1                                                         | SFD<br>1                                                                                                                                                                                      |
+|                        | −0.046<br>−0.009<br>0−.043<br>−0.083<br>−0.056<br>−0.050<br>−0.085<br>−0.032<br>−0.114<br>−0.096<br>−0.111                                                                                                                                                               | 0.439**<br>SFA<br>1                                                                                                                                                                           |
+| NLMI                   | 0.063<br>0.079<br>0.059<br>0.133<br>0.008<br>0.057<br>0.015<br>0.018<br>0.002<br>0.009<br>1                                                                                                                                                                              | −0.157*<br>−0.084<br>Retail<br>1                                                                                                                                                              |
+| NLMA                   | −0.171*<br>−0.086<br>−0.055<br>−0.059<br>−0.035<br>−0.077<br>−0.025<br>−0.042<br>−0.114<br>−0.108<br>−0.007<br>0.284**<br>0.214**<br>0.126<br>0.064<br>0.113<br>0.077<br>0.077<br>0.066<br>0.045<br>0.022<br>0.003<br>1                                                  | −0.327**<br>−0.306**<br>−0.605**<br>Park<br>1                                                                                                                                                 |
+| SLMI                   | −0.202**<br>−0.198**<br>−0.198**<br>−0.199**<br>−0.067<br>−0.045<br>−0.123<br>−0.029<br>−0.049<br>−0.049<br>−0.133<br>0.467**<br>0.152*<br>0.153*<br>0.159*<br>0.036<br>0.090<br>0.018<br>0.122<br>0.043<br>0.022<br>0.007<br>0.091<br>1                                 | −0.234**<br>−0.060<br>0.281**<br>0.014                                                                                                                                                        |
+| SLMA                   | −0.210**<br>−0.235**<br>−0.225**<br>−0.088<br>−0.077<br>−0.143<br>−0.050<br>−0.031<br>−0.041<br>−0.081<br>0.586**<br>0.500**<br>0.201**<br>0.230**<br>0.183*<br>0.023<br>0.136<br>0.082<br>0.012<br>0.020<br>0.008<br>0.006<br>0.034<br>0.026<br>1                       | OS<br>1<br>−0.204**<br>−0.463**<br>−0.134<br>0.266**<br>0.205**<br>OFF                                                                                                                        |
+| TV                     | −0.237**<br>−0.335**<br>−0.199**<br>−0.323**<br>−0.188*<br>−.196**<br>−0.108<br>−0.062<br>−0.087<br>−0.124<br>−0.047<br>0.526**<br>0.665**<br>0.247**<br>0.263**<br>0.621**<br>0.345**<br>0.269**<br>0.352**<br>0.149*<br>0.004<br>0.015<br>0.045<br>0.104<br>0.007<br>1 | 1<br>−0.134<br>−0.058<br>−0.119<br>−0.131<br>0.234**<br>0.042<br>MF<br>1                                                                                                                      |
+| # Crashes              | −0.066<br>−0.080<br>−0.007<br>−0.138<br>−0.105<br>−0.105<br>−0.033<br>−0.054<br>−0.100<br>−0.004<br>−0.121<br>0.253**<br>0.174*<br>0.148*<br>0.165*<br>.195**<br>0.093<br>0.007<br>0.029<br>0.145<br>0.074<br>0.013<br>0.130<br>0.020<br>0.026<br>0.030<br>1             | −0.254**<br>−0.166*<br>−0.152*<br>−0.013<br>0.171*<br>0.020<br>0.061<br>LLoR<br>1                                                                                                             |
+| Correlation<br>Pearson | # Crashes<br>HMRNR<br>HMNR<br>NLMA<br>Utility<br>SLMA<br>NAPR<br>Y.C.T.<br>Retail<br>NLMI<br>Trans<br>SLMI<br>Civic<br>LLoR<br>VMU<br>Park<br>OFF<br>IND<br>SFA<br>SFD<br>WH<br>HU<br>AR<br>MF<br>TV<br>OS<br>PL                                                         | Correlation<br># Crashes<br>HMRNR<br>Pearson<br>HMNR<br>NLMA<br>SLMA<br>NAPR<br>Y.C.T.<br>Retail<br>NLMI<br>SLMI<br>Civic<br>LLoR<br>Park<br>OFF<br>IND<br>SFA<br>SFD<br>AR<br>MF<br>TV<br>OS |
+
+| PL      |             |          |          |          |          | 1        | 0.923**  |  |
+|---------|-------------|----------|----------|----------|----------|----------|----------|--|
+| WH      |             |          |          |          | 1        | −0.245** | −0.261** |  |
+| VMU     |             |          |          | 1        | −0.261** | 0.313**  | 0.574**  |  |
+| Utility |             |          | 1        | 0.120    | −0.048   | −0.043   | −0.004   |  |
+| Trans   |             | 1        | 0.085    | 0.142    | 0.337**  | −0.138   | −0.060   |  |
+| SFD     |             | −0.242** | −0.232** | −.614**  | −0.045   | −0.414** | −0.515** |  |
+| SFA     |             | −0.109   | −0.138   | −0.295** | 0.055    | −0.115   | −0.157*  |  |
+| Retail  |             | −0.117   | −0.184*  | −0.294** | 0.064    | 0.366**  | 0.169*   |  |
+| Park    |             | 0.178*   | 0.138    | 0.960**  | −0.260** | 0.241**  | 0.515**  |  |
+| OS      |             | −0.302** | −0.155*  | −0.217** | −0.262** | −0.220** | −0.272** |  |
+| OFF     |             | −0.093   | 0.040    | 0.316**  | −0.171*  | 0.600**  | 0.625**  |  |
+| MF      |             | −0.151*  | −0.248** | −0.107   | −0.148*  | −0.034   | −0.054   |  |
+| LLoR    |             | −.0073   | 0.068    | −0.272** | 0.076    | −0.227** | −0.275** |  |
+| Pearson | Correlation | Trans    | Utility  | VMU      | WH       | PL       | HU       |  |
+
+<span id="page-8-1"></span><span id="page-8-0"></span>TV: Traffic Volume, SLMA: Speed Limit on Major Approach, SLMI: Speed Limit on Minor Approach, NLMA: # of Lanes on Major Approach, NLMI: # of Lanes on Minor Approach, YCT: Average Yellow Change Time, AR: All-Red Clearance Time, Civic: Civic/Institutional, HMRNR: Horizontal Mixed Use Residential & Non-Residential, HMNR: Horizontal Mixed Use Non-Residential, IND: Industrial, LLoR: Large Lots of Residential, MF: Multi-family, OFF: Office, OS: Open Space/Recreational, Park: Parking, SFA: Single-family – Attached, SFD: Single-family – Detached, Trans: Transportation, VMU: Vertical Mixed Use, WH: Warehouse/Distribution, PL:
+
+Population, HU: Total Household Units. \* Correlation is significant at the 0.05 level (2-tailed). \*\* Correlation is significant at the 0.01 level (2-tailed).
+
+(180 signalized intersection) models. These signalized intersections were not considered for model development. The MAD values closer to zero indicate the best crash estimation model. The results obtained from validation are summarized in [Table 11.](#page-11-1) The computed MAD values of all the models are close to zero. Therefore, all the models are acceptable.
+
+# 3.5. Effect of predictor variables
+
+Positive and negative coe fficients of predictor variables in nonlinear models indicate the role of each predictor variable on red light violation crashes. The results obtained indicate that, in most of the cases, the sign of predictor variable was found to be consistent with respect to each area type. For example, in CBD area, the sign of tra ffi c volume coe fficient was positive in all the developed models ([Table 7](#page-9-1)). Likewise, in urban area, the sign of horizontal mixed non-residential coe fficient was same, in both models 2 and 3 ([Table 8\)](#page-10-0). The e ffect of each signi ficant predictor variable from the developed models are summarized in [Table 12](#page-11-2)
+
+The results obtained indicate that in all the area types, tra ffic volume was found to be significant and has a positive effect on the number of red light violation crashes. In urban and suburban areas, allred clearance time and areas with horizontal mixed non-residential type land use have a positive e ffect on the number of red light violation crashes. Similarly, in suburban area, speed limit on the major approach, the number of lanes on the minor approach, and areas with horizontal mixed non-residential and retail type land uses have a positive e ffect on the number of red light violation crashes. Likewise, areas with horizontal mixed non-residential, open space/recreational, and retail type land uses have a positive e ffect on red light violation crashes when complete dataset was considered for model development.
+
+The positive e ffect indicates that these factors contribute more to the number of red light violation crashes than predictor variables with negative coe fficients (areas with industrial, large lots of residential, retail, and utility type land uses in CBD area; areas with industrial, large lots of residential, and warehouse/distribution type land uses in urban area; areas with single-family attached, single-family detached, and transportation type land uses in suburban area; areas with industrial, large lots of residential, single-family attached, and transportation type land uses in the complete dataset).
+
+From [Table 12,](#page-11-2) various predictor variables were observed to in fluence the number of red light violation crashes. [Datta et al. \(2000\)](#page-12-18) stated that a well-designed all-red clearance time would reduce red light violation and right-angle crashes. Similar results were observed from this research. The results obtained indicate that all-red clearance time in fluences the number of red light violation crashes at signalized intersections in the urban and suburban areas; could be associated to the width of the signalized intersections in these areas. Therefore, optimized all-red clearance time should be provided for signalized intersections in urban and suburban areas. However, its influence at signalized intersections in the CBD area is statistically insignificant.
+
+[Mohamedshah et al. \(2000\)](#page-12-26) stated that tra ffi c volume, speed limit, and the number of lanes on the major and minor approach contribute to the number of red light violation crashes. Similar results were observed from this research. Tra ffic volume (exposure) was observed to have an in fluence on the number of red light violation crashes at signalized intersections in all the three area types. However, tra ffic volume is relatively higher at signalized intersections in the suburban area when compared to CBD and urban areas, indicating possible in fluence of other predictor variables. The findings indicate that speed limit on the major approach and the number of lanes on the minor approach in fluence the number of red light violation crashes at only signalized intersections in the suburban area (in addition to tra ffic volume). This could be due to higher speed limits and permitted turning movements with high wait time from the minor approaches in suburban areas. Typically, the variation between the speed limit on major approach and
+
+<span id="page-9-0"></span>Table 6 Selected Predictor Variables to Develop Models by Area Type.
+
+| Predictor Variables                         | CBD Area |         |         | Urban Area |         |         | Suburban Area |         |         | All     |         |         |
+|---------------------------------------------|----------|---------|---------|------------|---------|---------|---------------|---------|---------|---------|---------|---------|
+|                                             | Model 1  | Model 2 | Model 3 | Model 1    | Model 2 | Model 3 | Model 1       | Model 2 | Model 3 | Model 1 | Model 2 | Model 3 |
+| Traffic Volume (TV)                         | ✓        | ✓       | ✓       | ✓          | ✓       | ✓       | ✓             |         |         | ✓       |         |         |
+| Speed Limit on the Major Approach (SLMA)    | ✓        |         |         |            |         |         |               | ✓       | ✓       |         |         | ✓       |
+| Speed Limit on the Minor Approach (SLMI)    |          |         |         |            |         |         |               |         |         |         | ✓       |         |
+| # of Lanes on the Major Approach (NLMA)     |          | ✓       |         | ✓          | ✓       |         | ✓             |         |         |         | ✓       | ✓       |
+| # of Lanes on the Minor Approach (NLMI)     |          |         |         |            |         |         |               |         | ✓       | ✓       |         | ✓       |
+| Average Yellow Change Time (YCT)            |          |         |         |            |         |         |               |         |         |         |         |         |
+| All-Red Clearance Time (AR)                 |          |         |         | ✓          |         |         |               | ✓       |         |         |         |         |
+| Civic/Institutional (Civic)                 |          |         |         | ✓          |         |         | ✓             |         | ✓       |         |         |         |
+| Horizontal Mixed Use Residential & Non      |          |         | ✓       |            | ✓       | ✓       |               |         |         |         | ✓       | ✓       |
+| Residential (HMRNR)                         |          |         |         |            |         |         |               |         |         |         |         |         |
+| Horizontal Mixed Use Non-Residential (HMNR) |          |         | ✓       | ✓          | ✓       | ✓       | ✓             |         |         | ✓       | ✓       | ✓       |
+| Industrial (IND)                            |          |         | ✓       |            | ✓       | ✓       |               |         |         |         |         | ✓       |
+| Large lots of residential (LLoR)            | ✓        |         |         | ✓          | ✓       |         |               |         |         | ✓       | ✓       | ✓       |
+| Multi-family (MF)                           | ✓        | ✓       |         |            | ✓       |         |               |         | ✓       |         | ✓       | ✓       |
+| Office (OFF)                                |          |         | ✓       |            | ✓       | ✓       |               | ✓       |         |         | ✓       | ✓       |
+| Open Space/Recreational (OS)                |          |         |         |            |         |         | ✓             |         |         | ✓       |         | ✓       |
+| Parking (Park)                              |          |         |         |            |         |         |               |         |         |         |         |         |
+| Retail (Retail)                             |          | ✓       |         |            | ✓       | ✓       |               |         | ✓       |         |         | ✓       |
+| Single-family – Attached (SFA)              | ✓        |         |         | ✓          | ✓       | ✓       | ✓             |         | ✓       |         | ✓       |         |
+| Single-family – Detached (SFD)              |          |         |         |            |         |         |               | ✓       |         | ✓       |         |         |
+| Transportation (Trans)                      |          |         |         |            |         |         | ✓             | ✓       | ✓       |         | ✓       |         |
+| Utility (Utility)                           |          | ✓       |         |            |         |         |               |         | ✓       |         | ✓       |         |
+| Vertical Mixed Use (VMU)                    |          | ✓       |         |            |         |         |               |         |         |         |         |         |
+| Warehouse/Distribution (WH)                 | ✓        |         |         | ✓          |         |         |               |         |         | ✓       |         |         |
+| Population (PL)                             |          |         |         |            |         |         |               |         |         |         |         |         |
+| # Households (HU)                           |          |         |         |            |         |         |               |         |         |         |         |         |
+
+<span id="page-9-1"></span>Table 7 Crash Estimation Models – CBD Area.
+
+| Parameters                | Model 1                |         | Model 2               |         | Model 3                |         |
+|---------------------------|------------------------|---------|-----------------------|---------|------------------------|---------|
+|                           | B<br>(S.E.)            | p-value | B<br>(S.E.)           | p-value | B<br>(S.E.)            | p-value |
+| Intercept                 | 1.315<br>(0.19)        | < 0.05  | 1.581 (0.20)          | < 0.05  | 1.444<br>(0.20)        | < 0.05  |
+| Traffic Volume            | 2.473E-05<br>(< 0.01)  | < 0.05  | 2.282E-05<br>(< 0.01) | < 0.05  | 2.157E-05<br>(< 0.01)  | < 0.05  |
+| Industrial                |                        |         |                       |         | −1.709E-07<br>(< 0.01) | < 0.05  |
+| Large Lots of Residential | −9.450E-07<br>(< 0.01) | < 0.05  |                       |         |                        |         |
+| Retail                    |                        |         | −1.45E-07<br>(< 0.01) | 0.02    |                        |         |
+| Utility                   |                        |         | −1.050E-06 (< 0.01)   | 0.03    |                        |         |
+| QIC                       | 23.578                 |         | 23.386                |         | 23.375                 |         |
+| QICC                      | 27.920                 |         | 29.367                |         | 27.589                 |         |
+
+minor approach at signalized intersections in suburban area is higher than in the CBD and urban areas. Hence, higher priority is given for major approaches than minor approaches. The high traffic volume, high speed limit on the major approach, and more green time for the major approach traffic at signalized intersections in suburban area influence some drivers from minor approach to take risk and violate the signal.
+
+An increase in area with retail type land use within 0.5 miles from a signalized intersection in suburban areas increases the number of red light violation crashes. This could be attributed to increased traffic activity at such signalized intersections. However, in CBD area, an increase in area with retail type land use within 0.5 miles from a signalized intersection decreases the number of red light violation crashes (could be attributed to the presence of mixed use activity and alternate mode users). Likewise, an increase in areas with utility type land use in CBD area, areas with warehouse/distribution land use in urban area, and areas with single-family detached land use in suburban area decrease the number of red light violation crashes.
+
+Overall, the predictor variables and their effect on red light violation crashes differ by area type. The developed crash estimation models or safety performance functions (SPFs) by area type can be used to predict the number of red light violation crashes at a signalized intersection, by area type, within the city of Charlotte boundary. Also, these SPFs can be used to identify and prioritize signalized intersections for safety improvements.
+
+<span id="page-10-0"></span>Table 8 Crash Estimation Models – Urban Area.
+
+| Parameters                           | Model 1                |         | Model 2               |         | Model 3                |         |
+|--------------------------------------|------------------------|---------|-----------------------|---------|------------------------|---------|
+|                                      | B<br>(S.E.)            | p-value | B<br>(S.E.)           | p-value | B<br>(S.E.)            | p-value |
+| Intercept                            | −0.071<br>(0.48)       | 0.88    | 0.900<br>(0.17)       | < 0.05  | 1.297<br>(0.17)        | < 0.05  |
+| Traffic Volume                       | 1.216E-05<br>(< 0.01)  | < 0.05  | 1.173E-05<br>(< 0.01) | < 0.05  | 1.157E-05<br>(< 0.01)  | < 0.05  |
+| All-red Clearance Time               | 0.898<br>(0.28)        | < 0.05  |                       |         |                        |         |
+| Horizontal Mixed Use Non-Residential |                        |         | 3.591E-07<br>(< 0.01) | < 0.05  | 2.200E-07<br>(< 0.01)  | < 0.05  |
+| Industrial                           |                        |         |                       |         | −1.207E-07<br>(< 0.01) | < 0.05  |
+| Large Lots of Residential            | −2.808E-07<br>(< 0.01) | < 0.05  |                       |         |                        |         |
+| Open Space/Recreational              |                        |         | 2.564E-07<br>(< 0.01) | 0.02    |                        |         |
+| Warehouse/Distribution               | −1.349E-07<br>(< 0.01) | < 0.05  |                       |         |                        |         |
+| QIC<br>QICC                          | 14.693<br>22.921       |         | 16.359<br>22.250      |         | 18.015<br>24.196       |         |
+
+## 4. Conclusions
+
+Red light running violation is a major contributing factor of crashes at signalized intersections. This paper investigates red light violation crashes by categorizing the study area into CBD, urban, and suburban areas. The on-network, traffic, signal, demographic, and land use characteristics were considered as predictor variables. Crash estimation models were developed by randomly selecting sixty signalized intersections in each area. Models were also developed using data for all the 180 signalized intersections.
+
+As the physical property of predictor variables vary by area type, different predictors were observed to be significant and influence the number of red light violation crashes in the three different area types. CBD, urban, and suburban areas have different land use patterns that contribute to the number of red light violation crashes. For example, industrial, large lots of residential, retail, and utility type land uses are observed as significant predictor variables in the CBD area. They attract people from all over the study area. These areas seem to have a direct but negative influence on the number of red light violation crashes. On the other hand, in an urban area, areas with horizontal mixed nonresidential land use influence the number of red light violation crashes. Likewise, in suburban area, the number of red light violation crashes are higher at signalized intersections with high traffic volume, speed limit, and areas surrounded with horizontal mixed non-residential and
+
+<span id="page-10-1"></span>Table 9 Crash Estimation Models – Suburban Area.
+
+| Parameters                           | Model 1                |         | Model 2                |         | Model 3                |         |
+|--------------------------------------|------------------------|---------|------------------------|---------|------------------------|---------|
+|                                      | B<br>(S.E.)            | p-value | B<br>(S.E.)            | p-value | B<br>(S.E.)            | p-value |
+| Intercept                            | 1.084<br>(0.14)        | < 0.05  | 0.080<br>(0.62)        | 0.90    | −0.262<br>(0.44)       | 0.55    |
+| Traffic Volume                       | 8.03E-06<br>(< 0.01)   | < 0.05  |                        |         |                        |         |
+| Speed Limit on Major Approach        |                        |         | 0.022<br>(0.01)        | 0.05    | 0.031<br>(0.01)        | < 0.05  |
+| # of Lanes on Minor Approach         |                        |         |                        |         | 0.152<br>(0.61)        | 0.01    |
+| All-red Clearance Time               |                        |         | 0.542<br>(0.21)        | < 0.05  |                        |         |
+| Horizontal Mixed Use Non-Residential | 4.537E-07<br>(< 0.01)  | < 0.05  |                        |         |                        |         |
+| Retail                               |                        |         |                        |         | 1.374E-07<br>(< 0.01)  | < 0.05  |
+| Single-family – Attached             | −3.937E-07<br>(< 0.01) | 0.04    |                        |         | −4.021E-07<br>(< 0.01) | 0.02    |
+| Single-family - Detached             |                        |         | −5.975E-08<br>(< 0.01) | 0.01    |                        |         |
+| Transportation                       |                        |         | −1.609E-07<br>(< 0.01) | 0.01    |                        |         |
+| QIC<br>QICC                          | 16.090<br>21.966       |         | 22.386<br>29.944       |         | 17.098<br>24.708       |         |
+
+<span id="page-11-0"></span>Table 10 Crash Estimation Model – All Signalized Intersections.
+
+| Parameters                           | Model 1                |         | Model 2                |         | Model 3                |         |
+|--------------------------------------|------------------------|---------|------------------------|---------|------------------------|---------|
+|                                      | B<br>(S.E.)            | p-value | B<br>(S.E.)            | p-value | B<br>(S.E.)            | p-value |
+| Intercept                            | 1.047<br>(0.09)        | < 0.05  | 1.351<br>(0.07)        | < 0.05  | 1.566<br>(0.20)        | < 0.05  |
+| Traffic Volume                       | 6.720E-06<br>(< 0.01)  | < 0.05  |                        |         |                        |         |
+| Speed Limit on Minor Approach        |                        |         | 0.02<br>(< 0.01)       | < 0.05  |                        |         |
+| #of Lanes on Minor Approach          |                        |         |                        |         | 0.120<br>(0.06)        | 0.03    |
+| Horizontal Mixed Use Non-Residential | 2.361E-07<br>(< 0.01)  | < 0.05  | 1.673E-07<br>(< 0.01)  | < 0.05  |                        |         |
+| Industrial                           |                        |         |                        |         | −1.026E-07<br>(< 0.01) | < 0.05  |
+| Large Lots of Residential            | −2.409E-07<br>(< 0.01) | < 0.05  | −2.051E-07<br>(< 0.01) | < 0.05  | −1.948E-07<br>(< 0.01) | < 0.05  |
+| Open Space/Recreational              | 1.279E-07<br>(< 0.01)  |         |                        |         |                        |         |
+| Retail                               |                        |         |                        |         | 7.106E-08<br>(< 0.01)  | 0.05    |
+| Single-family - Attached             |                        |         | −3.249E-07<br>(< 0.01) | < 0.05  |                        |         |
+| Transportation                       |                        |         | −1.486E-07<br>(< 0.01) | < 0.05  |                        |         |
+| QIC<br>QICC                          | 63.303<br>71.776       |         | 67.438<br>77.823       |         | 69.289<br>75.294       |         |
+
+<span id="page-11-1"></span>Table 11 Model Validation – Results Summary.
+
+| Area Type | Model 1 | Model 2 | Model 3 |
+|-----------|---------|---------|---------|
+| CBD       | 0.81    | 0.77    | 0.78    |
+| Urban     | 0.26    | 0.54    | 0.24    |
+| Suburban  | 0.29    | 0.28    | 0.33    |
+| All       | 0.61    | 1.28    | 1.29    |
+|           |         |         |         |
+
+retail type land use. In addition, traffic volume is the most significant predictor variable related to the number of red light violation crashes. It was found to be a significant predictor variable in all the developed models.
+
+Based on the findings, manual enforcement or RLCs can be an effective measure at signalized intersections with high traffic volume in the CBD area. Implementing manual enforcement or installing RLCs at signalized intersections with high all-red clearance time, near high density of horizontal mixed non-residential and open space/recreational type land uses in urban area; at signalized intersections with high traffic volume, speed limit on the major approach, the number of lanes on the minor approach, all-red clearance time and areas surrounded with horizontal mixed non-residential and retail type land use in suburban areas would lead to a reduction in the number of red light violation crashes.
+
+The all-red clearance time could be checked and optimized, if found inadequate, at signalized intersections with higher number of red light violation crashes in urban and suburban areas. Likewise, the speed limit on the major approaches, and possible influence of traffic, signal phases and signal timings attributed to minor approaches could be examined at signalized intersections with higher number of red light violation crashes in suburban areas.
+
+Data for the city of Charlotte were used for this research. Similar studies should be conducted using data for other cities to increase the
+
+<span id="page-11-2"></span>Table 12 Effect of Predictor Variables.
+
+| Predictor Variables                  | CBD Area | Urban Area | Suburban Area | All |
+|--------------------------------------|----------|------------|---------------|-----|
+| Traffic Volume                       | P        | P          | P             | P   |
+| Speed Limit on the Major Approach    |          |            | P             |     |
+| Speed Limit on the Minor Approach    |          |            |               | P   |
+| # of Lanes on the Minor Approach     |          |            | P             | P   |
+| All-red Clearance Time               |          | P          | P             |     |
+| Horizontal Mixed Use Non-Residential |          | P          | P             | P   |
+| Industrial                           | N        | N          |               | N   |
+| Large Lots of Residential            | N        | N          |               | N   |
+| Open Space / Recreational            |          | P          |               | P   |
+| Retail                               | N        |            | P             | P   |
+| Single-family - Attached             |          |            | N             | N   |
+| Single-family - Detached             |          |            | N             |     |
+| Transportation                       |          |            | N             | N   |
+| Utility                              | N        |            |               |     |
+| Warehouse/Distribution               |          | N          |               |     |
+
+P: Positive EffectN: Negative Effect.
+
+understanding of the role of predictor variables on the number of red light violation crashes. Further, geospatial characteristics within 0.5 miles were only captured and used in the analysis. Considering other buffer widths and detailed demographic and socio-economic characteristics (age, gender, ethnicity, income groups, etc.) warrant further investigation.
+
+#### Funding
+
+This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
+
+#### Acknowledgement
+
+The authors sincerely thank the staff of the city of Charlotte Department of Transportation (CDoT) for their help with providing data required for the study.
+
+#### References
+
+- <span id="page-12-16"></span>[Alver, Y., Demirel, M.C., Mutlu, M.M., 2014. Interaction between socio-demographic](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0005) characteristics: traffic rule violations and traffi[c crash history for young drivers.](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0005) [Accid. Anal. Prevent. J. 72, 95](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0005)–104.
+- <span id="page-12-24"></span>[Baguley, C.J., 1988. Running the red at signals on high-speed roads. J. Tra](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0010)ffic Eng. [Control 29 \(7](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0010)–8), 415–420.
+- <span id="page-12-27"></span>[Bonneson, J.A., Zimmerman, K., Brewer, M.A., 2002. Engineering Countermeasures to](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0015) [Reduce Red-Light-Running. Publication # FHWA/TX-03/4027-2. FHWA. United](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0015) [States Department of Transportation](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0015).
+- <span id="page-12-14"></span>[Chapman, E.A., Masten, S.V., Browning, K.K., 2014. Crash and tra](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0020)ffic violation rates [before and after licensure for novice California drivers subject to di](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0020)fferent driver [licensing requirements. J. Saf. Res. 50, 125](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0020)–138.
+- <span id="page-12-18"></span>[Datta, T., Schattler, K., Datta, S., 2000. Red light violations and crashes at Urban inter](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0025)[sections. Transportation Research Record: Journal of the Transportation Research](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0025) [Board, No. 1734. Transportation Research Board of the National Academies,](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0025) [Washington, D.C, pp. 52](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0025)–58.
+- <span id="page-12-7"></span>[Garber, N.J., Miller, J.S., Abel, R.E., Eslambolchi, S., Korukonda, S.K., 2007. The Impact](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0030) [of Red Light Cameras \(Photo-Red Enforcement\) on Crashes in Virginia. Publication](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0030) [No. FHWA/VTRC 07-R2. United States Department of Transportation](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0030).
+- <span id="page-12-22"></span>Gates, T.J., Noyce, D.A., 2010. Dilemma zone driver behavior as a function of vehicle type, time of day, and platooning. Transportation Research Record: Journal of the Transportation Research Board, No. 2149. Transportation Research Board of the National Academies, Washington, DC, pp. 84–93. (Accessed 20 June 2016). [https://](https://connect.ncdot.gov/resources/safety/its%20and%20signals%20resources/its%20and%20signals%20unit%20design%20manual.pdf) [connect.ncdot.gov/resources/safety/its%20and%20signals%20resources/its%20and](https://connect.ncdot.gov/resources/safety/its%20and%20signals%20resources/its%20and%20signals%20unit%20design%20manual.pdf) [%20signals%20unit%20design%20manual.pdf.](https://connect.ncdot.gov/resources/safety/its%20and%20signals%20resources/its%20and%20signals%20unit%20design%20manual.pdf)
+- <span id="page-12-0"></span>[Insurance Institute for Highway Safety \(IIHS\), 2007. Red Light Cameras in Philadelphia](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0040) [All but Eliminate Violations. Status Report, vol. 42 No. 1](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0040).
+- <span id="page-12-23"></span>[Kamyab, A., McDonald, T., Stribiak, J.J., Storm, B., 2000. Red Light Running in Iowa: The](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0045) [Scope, Impact, and Possible Implications. Ames IA. Center for Transportation](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0045) [Research and Education, Iowa State University December](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0045).
+- <span id="page-12-6"></span>[Kamyab, A., McDonald, T., Stribiak, J.J., 2002. The scope and impact of Red light running](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0050) [in Iowa. the 81st Annual Meeting of the Transportation Research Board Compendium](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0050) [of Papers. Transportation Research Board of the National Academies,](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0050)
+
+- [Washington, D.C](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0050).
+- <span id="page-12-25"></span>[Kulanthayan, S., Phang, W.K., Hayati, K.S., 2007. Tra](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0055)ffic light violation among motorists [in Malaysia. IATSS Res. 31 \(2\), 67](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0055)–73.
+- <span id="page-12-4"></span>Maisey, G.E., 1981. The Eff[ects of Mechanical Surveillance Device on Urban Signalized](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0060) [Intersection Accidents. Research and Statistics Report No.17. Road Tra](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0060)ffic Authority, [Perth, Australia](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0060).
+- <span id="page-12-13"></span>[McGwin Jr., G., Brown, D.B., 1999. Characteristics of tra](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0065)ffic crashes among young, [middle-aged, and older drivers. Accid. Anal. Prevent. J. 31 \(No. 3\), 181](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0065)–198.
+- <span id="page-12-26"></span>[Mohamedshah, Y.M., Chen, L.W., Council, F.M., 2000. Association of selected intersection](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0070) [factors with Red light running crashes. Highway Safety Information System \(HSIS\)](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0070) [Summary Report. United States Department of Transportation](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0070).
+- <span id="page-12-28"></span>[North Carolina Department of Transportation \(NCDOT\), 2004. Tra](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0075)ffic Management & [Signal Systems Unit Design Manual](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0075).
+- <span id="page-12-29"></span>[Oh, J., Lyon, C., Washington, S., Persaud, B., Bared, J., 2003. Validation of FHWA crash](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0080) [models for rural intersections: lessons learned. Transp. Res. Rec.: J. Transp. Res.](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0080) [Board 1840, 41](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0080)–49.
+- <span id="page-12-3"></span>Porter, [B.E., England, K.J., 2000. Predicting red-light running behavior: a tra](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0085)ffic safety [study in three urban settings. J. Saf. Res. 31 \(1\), 1](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0085)–8.
+- <span id="page-12-10"></span>Pulugurtha, S.S., Otturu, R., 2014. Eff[ectiveness of red light running camera enforcement](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0090) [program in reducing crashes: evaluation using](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0090) "before the installation", "after the installation", and "after the termination" [data. Accid. Anal. Prevent. J. 64, 9](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0090)–17.
+- <span id="page-12-11"></span>[Retting, R.A., Kyrychenko, S.Y., 2002. Reductions in injury crashes associated with Red](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0095) [light camera enforcement in Oxnard, California. The 81st Annual Meeting of the](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0095) [Transportation Research Board Compendium of Papers. Transportation Research](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0095) [Board of the National Academies, Washington, D.C](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0095).
+- <span id="page-12-1"></span>[Retting, R.A., Williams, A.F., 1996. Characteristics of red light violators: results of a](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0100) field [investigation. J. Saf. Res. 27 \(1\), 9](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0100)–15.
+- <span id="page-12-2"></span>[Retting, R.A., Williams, A.F., Greene, M.A., 1998. Red-light running and sensible coun](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0105)termeasures: summary of research fi[ndings. Transportation Research Record: Journal](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0105) [of the Transportation Research Board, No. 1640. Transportation Research Board of](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0105) [the National Academies, Washington, D.C, pp. 23](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0105)–26.
+- <span id="page-12-19"></span>[Schattler, K., Datta, T., 2004. Driver behavior characteristics at urban signalized inter](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0110)[sections. Transp. Res. Rec.: J. Transp. Res. Board 1862, 17](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0110)–23.
+- <span id="page-12-20"></span>[Schattler, K., Datta, T., Hill, C., 2003. Change and clearance interval design on red-light](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0115) [running and late exits. Transp. Res. Rec.: J. Transp. Res. Board 1856, 193](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0115)–201.
+- <span id="page-12-15"></span>[Shell, D.F., Newman, I.M., Córdova-Cazar, A.L., Heese, J.M., 2015. Driver education and](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0120) teen crashes and traffic violations in the fi[rst two years of driving in a graduated](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0120) [licensing system. Accid. Anal. Prevent. J. 82, 45](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0120)–52.
+- <span id="page-12-5"></span>[South, D.R., Harrison, W.A., Portans, I., King, M., 1998. Evaluation of the Red Light](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0125) [Camera Program and the Owner Onus Legislation. Victoria Transport: Road Tra](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0125)ffic [Authority, Hawthorn, Victoria, Australia.](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0125)
+- <span id="page-12-30"></span>Tay, R., 2017. Correlation, variance infl[ation and multicollinearity in regression model. J.](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0130) [Eastern Asia Soc. Transp. Stud. 12, 2006](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0130)–2015.
+- <span id="page-12-8"></span>[Tay, R., De Barros, A., 2009. Minimizing red light violations: how many cameras do we](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0135) [need for a given number of locations? J. Transp. Saf. Secur. 1 \(1\), 18](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0135)–31.
+- <span id="page-12-9"></span>Tay, R., De Barros, A., 2011. Should traffi[c enforcement be unpredictable? The case of red](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0140) [light cameras in Edmonton. Accid. Anal. Prevent. J. 43 \(3\), 955](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0140)–961.
+- <span id="page-12-17"></span>[Van der Horst, A., Richard, A., Wilmink, A., 1986. Drivers](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0145)' decision-making at signalised [intersections: an optimisation of the yellow timing. J. Tra](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0145)ffic Eng. Control 27 (12), 615–[617](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0145).
+- <span id="page-12-12"></span>Weldegiorgis, [Y., Jha, M.K., 2009. Driver behavior, dilemma zone, and capacity at red](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0150) [light camera equipped intersections. Transportation and Tra](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0150)ffic Theory 2009: Golden [Jubilee. pp. 481](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0150)–494.
+- <span id="page-12-21"></span>[Zimmerman, K., 2007. Additional dilemma zone protection for trucks at High-speed](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0155) [signalized intersections. Transportation Research Record: Journal of the](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0155) [Transportation Research Board, No. 2009. Transportation Research Board of the](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0155) [National Academies, Washington, DC, pp. 82](http://refhub.elsevier.com/S0001-4575(18)30414-7/sbref0155)–88.

Zotero/001_artiklid/Road Safety Indicators/martinrelationshipbetweencrashrate2002.md

@@ -0,0 +1,265 @@
+---
+category: literaturenote
+citekey: martinrelationshipbetweencrashrate2002
+title: Relationship between crash rate and hourly traffic flow on interurban motorways
+authors: "Martin, Jean-Louis"
+year: 2002
+date: 2002-00-00 2002
+doi: 10.1016/s0001-4575(01)00061-6
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/s0001-4575(01)00061-6"
+zotero_key: 3MW7H8KU
+zotero_storage: PT8EARZW
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Martin, Jean-Louis"
+status: converted
+---
+![](_page_0_Picture_0.jpeg)
+
+Accident Analysis and Prevention 34 (2002) 619–629
+
+![](_page_0_Picture_2.jpeg)
+
+# Relationship between crash rate and hourly traffic flow on interurban motorways
+
+Jean-Louis Martin \*
+
+*Institut National de Recherche sur les Transports et leur Se´curite´*, <sup>25</sup> *a*. *Franc¸ois Mitterrand*, <sup>69675</sup> *Bron*, *France* Received 14 November 2000; received in revised form 18 April 2001; accepted 24 April 2001
+
+#### **Abstract**
+
+This paper describes the relationship between crash incidence rates and hourly traffic volume and discusses the influence of traffic on crash severity, based on observations made on 2000 km of French interurban motorways over 2 years. Incidence rates involving property damage-only crashes and injury-crashes are highest when traffic is lightest (under 400 vehicles/h). These incidence rates are at their lowest when traffic flows at a rate of 1000–1500 vehicles/h. For heavier traffic flows, crash incidence rates increase steadily as traffic increases on 2- and 3-lane motorways and inflect on 2-lane motorways when traffic increases to a level of 3000 vehicles/h. For an equivalent light traffic level, the number of crashes is higher on three-lane than on 2-lane motorways and higher at weekends (when truck traffic is restricted) than on weekdays. In heavy traffic, the number of crashes is higher on weekdays. We found no significant difference between the number of daytime and night-time crashes, whatever the traffic. No difference was observed in crash severity by number of lanes or period in the week for a given level of traffic. However, severity is greater at night and when hourly traffic is light. Compared to the number of vehicles on the road, light traffic is a safety problem in terms of frequency and severity, and road safety campaigns targeting motorway users to influence their behavior in these driving conditions should be introduced. © 2002 Elsevier Science Ltd. All rights reserved.
+
+*Keywords*: Hourly traffic flow; Incidence rate; Interurban motorway; Modeling; Negative-binomial regression; Relationship; Road accident; Severity
+
+#### **1. Introduction**
+
+When you consider road crashes, it seems normal to try to correlate the number of crashes occurring in a given zone during a given observation period with the number of vehicles in this zone at the same period and exposed to a risk of crash.
+
+Some authors have already examined this problem (Satterthwaite, 1981), although there are not many recent references. There is an abundant literature on traffic modeling from the traffic flow point of view (for regulation or forecasting, for example), but few studies simultaneously address the relationship between road traffic and crash occurrence. Some articles discuss traffic–crash relationships for a geographical area as a
+
+\* Tel.: +33-4-7214-2512; fax: +33-4-7214-2520 *E*-*mail address*: [jean-louis.martin@inrets.fr](mailto:jean-louis.martin@inrets.fr) (J.-L. Martin). whole on all the various types of roads (Jadaan, 1992). They use average traffic calculated on an annual (Turner and Thomas, 1986; McGuigan, 1987) or hourly basis (Gwynn and Baker, 1970; Ceder and Livneh, 1982; Zhou and Sisiopiku, 1997; Hall and Pendleton, 1990), which significantly influences the type of phenomenon demonstrated.
+
+Other authors have developed models to predict crashes and traffic, using the *V*/*C* ratio (volume of traffic divided by the theoretical capacity of the road observed) to characterize the infrastructure used and its theoretical capacity to handle given traffic (Frantzeskakis and Iordanis, 1987). Others have tried to derive a Safety Performance Function to predict the effect of a change of exposure on the number of crashes (Mensah and Hauer, 1998) and to group together very different highway types. Most authors only consider injury-crashes.
+
+This study is based on the observation of all crashes involving damage to vehicles and structures and all crashes causing injuries, occurring on most of the French motorway network. Knowledge about damageonly crashes enables the study of crash frequency and crash severity (as observed inside all vehicles involved), as a function of traffic. More specifically, as traffic flows are considered to be uninterrupted, the objectives of this study were: (1) firstly, to describe the relationship between the observed crash rate and hourly traffic volume and to examine the effects of the few exposure measurements we possess; and (2) to address the influence of traffic on the crash severity.
+
+# **2. Material**
+
+Most French interurban motorways are toll roads. The data used in this study concerns crashes which occurred on the motorways managed by the Socie´te´ des Autoroutes du Sud de la France (ASF). This network comprises approximately 2000 km of motorways located in plains. Two thirds of these roads have two lanes in each direction and one third have three lanes. Average traffic is between 10,000 and 60,000 vehicles per day, 10–20% of which is trucks.
+
+A crash report form is completed whenever a vehicle on or off the motorway pavement cannot resume its journey, without being towed away, after a crash. This is how we were able to obtain data concerning injurycrashes and damage-only crashes. For all these crashes, the crash circumstances, the condition of the vehicle occupants and a detailed description of the damage incurred following between-vehicle collisions and crashes into infrastructures are recorded.
+
+Data relates to 2 years of observation—1997 and 1998. We identified 92 motorway sections considered homogeneous in terms of traffic between two motorway entry points and for which we possess hourly traffic measurements. The counting stations measure traffic flow at a point on each of these sections every 6 min, but only the average hourly figures are archived for the year. When no crash occurs on a section, we use the average hourly traffic volume. However, when a crash occurs, we replace this volume with the traffic volume noted on the 6-min traffic count recorded before the crash occurred This ensures that the traffic value is not modified by the crash, which would not be the case if 'uninterrupted' hourly rates were used. In what follows, in both cases we will call the result the hourly traffic flow. As the two traffic streams are considered separately, and as hourly traffic measurements are insufficiently reliable for only very few hours on some motorway sections, the study relates to over 2,900,000 'section-hour' observations.
+
+## **3. Method**
+
+We addressed traffic flows per motorway rather than by lane. The *V*/*C* ratio frequently used for traffic management to combine different network types, seems less relevant for crash studies. In fact the assumption that the number of crashes is proportional to the number of lanes for a given volume of traffic on each lane does not seem very realistic as the traffic conditions are not always identical and crash typologies can be different. This leads us to distinguish between 2-lane and 3-lane motorways for one part of the analysis. A first stage of the study underlines the existence of a correlation between the average number of crashes and average corresponding traffic. We are then more specifically interested in the relationship between the hourly traffic flow and the incidence rates of all crashes as well as of vehicles damaged in these crashes (Hauer, 1995). Incidence rates are obtained by adding together, for all the hours during which a certain volume of traffic is measured, the number of events as the numerator, and the number of vehicles multiplied by the number of km as the denominator. For each traffic section, the 95% confidence interval (CI) around the average incidence rate (IR) is calculated by the conventional formula: IRVAR(IR), with variance of IR, var(IR) esti-
+
+mated by the number of events (the number of vehicles×km)<sup>2</sup> . This estimate is valid as the number of events is very low according to the number of vehicles exposed, and we can therefore assume that the number of events is Poisson distributed (Rice, 1995).
+
+Some graphs presented were smoothed with cubic spline functions as the objective is to give their form, but not to use them subsequently for predictive purposes. The method utilizes a selected parameter value, making it possible to obtain a solution closest to the mean points while smoothing the curve as much as possible (Reinsch, 1967).
+
+The relationship between traffic flow and the crash incidence rate is examined by the number of lanes per motorway and by the two other 'continuous' dimensions available, relating to traffic volume and traffic conditions: the day–night and weekend–weekday variables.
+
+A regression model was then used to include these different variables simultaneously. The systematic component of the model used to explain the number of crashes (na) on each section and taking account of the number of vehicles exposed (nve) has the form (with and *<sup>k</sup>* as the parameters to estimate): Ln(na)= Ln(nve)++*k k*=1*kxk*.
+
+Conventionally the random component is supposed to have a Poisson distribution (Nicholson and Wong, 1993) and the model used is a Poisson regression. This law of distribution supposes the equality of the mean and the variance of the outcome. For a correctly specified model, the ratio obtained by dividing the deviation by the number of degrees of freedom must be close to 1. The fact that all values for adjusted models exceed 1 means that the data present an overdistribution (Mc-Cullagh and Nelder, 1989). To take account of this, the random component of the model is assumed to have a negative binomial distribution (Miaou, 1994; Maher and Summersgill, 1996; Poch and Mannering, 1996). This model of negative-binomial regression enables a 'best fit' to our data with a theoretical variance varying with the mean although it is necessary to estimate an additional parameter : Var(na)=*E*(na)+*E*(na)<sup>2</sup> .
+
+It should also be noted that the assumption of an underlying Poisson distribution is only valid if the events considered are independent. This is not always the case for two crashes occurring at the same time in the same section, either because one crash causes another or two crashes are both caused by the same specific traffic conditions. However, as this problem concerned under 4% of all the crashes we observed, we can consider our working distribution hypothesis as valid. All the calculations were carried out with SAS (1999) software and STATA (1999) software for the negative binomial regression.
+
+After addressing the influence of traffic on crash occurrence, we go on to describe crash severity and the traffic observed when the crash occurred based on the same factors as above. To explain the probability, *P*, of observing at least one injury-crashes in a crash we use a logistic regression with the form: Ln(*P*/(1−*P*))= + -*K k*=1*kxk*, the random component having a binomial distribution. In the different types of regression used, the contribution of each variable is tested by the ratio of the likelihood *L*1 of the model containing the variable and of the likelihood *L*0 of the model without it. The test is based on the property that the expression −2Ln(*L*1/*L*0) has a <sup>2</sup> distribution with a number of degrees of freedom equal to the difference in the number of parameters between two nested models (McCul-
+
+![](_page_2_Figure_5.jpeg)
+
+Fig. 1. Crash number and traffic flow versus time of day for 1997 and 1998.
+
+lagh and Nelder, 1989). Only significant variables (-5%) are included in the final model.
+
+We then address severity and the number of vehicles involved in the crash.
+
+#### **4. Results**
+
+During 1997 and 1998, our observations covered 2,937,886 motorway sections×hours and 33.8 billion vehicles×km. During this period 11,172 crashes occurred, involving 17,448 vehicles. Approximately 15% of the crashes injured people.
+
+Fig. 1 shows, for each hour in the day, the annual average traffic flow and the annual number of crashes observed on the corresponding sections.
+
+As expected, variations throughout a day between the least busy and most busy hours is high—from under 400 up to 1400 vehicles/h, and the average number of crashes follows this variation extremely closely.
+
+Fig. 2 identifies each motorway section observed (1 point per direction) and shows the number of crashes observed per kilometer and traffic flow averaged over 1 year.
+
+The linear trend is very clear, except for some specific sections where the number of crashes appears abnormally high. These are very specific sections like ring roads around large cities with a high number of access spurs used by vehicles on long and medium length trips as well as people travelling from their homes to work, for example. All are retained to ensure the continuity of the analysis as the impact of these sections is negligible within the complete set of observations.
+
+Fig. 3 shows the distribution of the number of crashes per hour and per number of lanes.
+
+Crashes observed on 2-lane motorways generally occur at traffic flows of under 1000 vehicles/h, with a definitely skewed distribution, whereas crashes on 3 lane motorways occur at different traffic flow rates with a flatter distribution and a mean of 1500 vehicles/h. This corresponds to a specific feature of the observed network, as the major highway from Lyon to the Spanish border constitutes the main part of the 3-lane motorways in our data, with a higher traffic flow than the rest of the network, and a higher percentage of trucks (about 20%).
+
+## <sup>4</sup>.1. *Incidence rate and hourly traffic flow*
+
+Fig. 4 highlights the relationship between the crash rate, calculated as the ratio between the number of crashes involving damage-only and injury crashes on 10<sup>8</sup> times the number of vehicles×km and traffic per hour. The continuous line corresponds to the confidence intervals represented by rectangles and the dotted line corresponds to the intervals symbolized by seg-
+
+![](_page_3_Figure_2.jpeg)
+
+Fig. 2. Number of crashes per kilometer for each motorway section versus average hourly traffic flow over 1 year.
+
+![](_page_3_Figure_4.jpeg)
+
+Fig. 3. Number of crashes per number of lanes versus hourly traffic flow.
+
+ments. This convention applies to all the following graphs.
+
+Several points of interest emerge: (1) the relationship is non-linear and the curve passes through a minimum, approximately 1000 vehicles/h for the 2-lane, and 1500 vehicles/h for the 3-lane, i.e. identical values in terms of volume/capacity; (2) the crash rate is significantly higher on 3-lane motorways when traffic is light. This remains true from a volume/capacity point of view too; and (3) the curve for 2-lane motorways inflects beyond 3000 vehicles/h. This could indicate the beginning of saturation phenomena, although this assertion should be considered with prudence given the confidence intervals (impact of the small number of crashes observed when traffic is heavy, as shown in Fig. 3).
+
+As the number of vehicles involved in a crash tends to increase with traffic flow, it is also interesting to examine the incidence of crashed vehicles (Fig. 5).
+
+As expected, the gradient is steeper with, for example, an average number of damage-only vehicles twice as high at 4000 than at 1500 vehicles an hour on 3-lane sections. Fig. 6 shows that the number of vehicles involved in a crash increases with traffic flow.
+
+To some extent the shape of the graph in Fig. 5 appears to result from the overlay of the two curves in Fig. 6, as previously described by some authors as Ceder and Livneh (1982) or Zhou and Sisiopiku (1997).
+
+The number of vehicles involved is one variable characterizing a crash, but many others vary with traffic flow. Fig. 7 shows the distribution, and traffic flow, of five identified crash types: crashes involving a single vehicle in the right-hand lane; crashes involving a
+
+![](_page_4_Figure_6.jpeg)
+
+Fig. 4. Crash incidence rates (crashes/100 million vehicles×km) and hourly traffic flow.
+
+![](_page_4_Figure_8.jpeg)
+
+Fig. 5. Crashed vehicles incidence rates (crashed vehicles/100 million vehicles×km) per hourly traffic flow.
+
+![](_page_5_Figure_2.jpeg)
+
+Fig. 6. Crash incidence rates for crashes involving one vehicle and crashes involving two or more vehicles per hourly traffic flow.
+
+![](_page_5_Figure_4.jpeg)
+
+Fig. 7. Distribution of five crash types per hourly traffic flow.
+
+single vehicle in the left-hand or central lane if any; crashes involving two vehicles in the same lane; crashes involving two vehicles in different lanes and crashes occurring at motorway entrances and exits (all the other categories being in running sections).
+
+As expected, crashes only involving one vehicle decrease as traffic flow increases and are replaced by crashes involving several vehicles and, more specifically, at the highest traffic peaks, by crashes involving vehicles in different lanes.
+
+Two other exposure factors related to crash incidence rates and traffic flows are also known for all our observations—the sections-hours: day/night and weekday/weekend.
+
+As Fig. 8 shows, when hourly traffic is considered the crash rate is not appreciably different by day and by night below 3400 vehicles/h. The part of the graph describing the trend for night-time crashes cannot be interpreted for traffic flows over 3400 vehicles/h as there are so few observations available.
+
+Fig. 9 shows a slightly higher crash rate in light traffic at weekends, whereas the rate increases during the week as traffic builds up. The distinction between periods in which trucks are forbidden to use motorways and periods in which they are authorized was also addressed (in France from Saturday 22:00 h to Sunday 22:00 h, plus specific days when very heavy traffic is expected). The result is very close to the weekend–weekday distinction, which includes the previous one: weekend traffic with few trucks and, more generally, fewer professional drivers. This combination of dimensions—vehicle type and driver type—makes Fig. 9 difficult to interpret. No precision concerning the proportion of trucks in traffic were available from the data. The influence of trucks in the traffic flow on crash occurrence will be studied more specifically in a later work.
+
+![](_page_6_Figure_4.jpeg)
+
+Fig. 8. Accident incidence rates per night and day versus hourly traffic flow.
+
+![](_page_6_Figure_6.jpeg)
+
+Fig. 9. Accident incidence rates per weekend-weekday periods versus hourly traffic flow.
+
+Table 1 Crash incidence rate ratios fitted by negative-binomial regression (with significant figures in bold type)
+
+| Traffic flow (vehicles/h) | Weekdays and<br>2-lanes | 3-lanes | Weekend |
+|---------------------------|-------------------------|---------|---------|
+| 1–400                     | 1.61                    | 2.43    | 2.05    |
+| 400–800                   | 1.03                    | 1.41    | 1.28    |
+| 800–2000                  | 1.00                    | 1.01    | 0.92    |
+| 2000–3000                 | 1.27                    | 1.16    | 1.23    |
+| 3000+                     | 1.40                    | 1.55    | 1.02    |
+
+Table 2 Accident severity expressed as a percentage of accidents with injuries and relative risks (RR) of severity according to different variables
+
+| Factor              | %    | RR   | 95% CI        |
+|---------------------|------|------|---------------|
+| Night               | 16.9 | 1.17 | 1.06–1.28     |
+| Day                 | 14.5 | 1    |               |
+| Weekend             | 15.1 | 1.0  | NS            |
+| Weekdays            | 15.1 | 1    |               |
+| 2×3 lanes           | 15.6 | 1.07 | NS            |
+| 2×2 lanes           | 14.7 | 1    |               |
+| 1–400 vehicle/h     | 16.3 | 1.14 | 1.01<br>–1.28 |
+| 400–800 vehicle/h   | 15.7 | 1.10 | NS            |
+| 800–2000 vehicle/h  | 14.3 | 1    |               |
+| 2000–3000 vehicle/h | 15.2 | 1.06 | NS            |
+| 3000+ vehicle/h     | 14.2 | 0.99 | NS            |
+
+#### <sup>4</sup>.2. *Multiariate analysis*
+
+Prior to the introduction in a negative-binomial regression, the traffic flow was made discrete and split into five traffic flow classes to make it possible to avoid the linear constraint underlying the use of the continuous variable. This split is made because the relationship studied is roughly U-shaped, and because this will help for interpretation. The reference class selected is the mean traffic flow (800–2000 vehicles/h).
+
+The traffic flow is then introduced in the model and the likelihood ratio test shows its significant contribution. We then introduce each of the three variables already analyzed in a univariate way. The variable night and day period is not significant and is not kept in the model. On the other hand, number of lanes and weekend–weekday period variables are significant both separately and together. As the main effects are significant, we introduce class 1 interactions—number of lanes×traffic flow class—and then weekend–weekday×traffic flow class. The likelihood ratio test shows the significant contribution of both interactions which leads us to present the complete model in Table 1 below. The Risk Ratios are the exponential of the parameter estimates.
+
+This model confirms the two separate analyses of Figs. 4 and 5. As the reference class is a flow of traffic from 800 to 2000 vehicles/h on a 2×2-lane motorway on weekdays, Table 1 shows that, for example, the crash incidence rate is multiplied, on average, by 2.43 in light traffic (1–400 vehicles/h) on 3-lane motorways and by almost 5 (2.43×2.05) at weekends on 3-lane motorways. Similar results are obtained if the model is used to determine the number of damaged vehicles rather than the number of crashes as the outcome, with higher incidence ratios at higher traffic flows.
+
+# <sup>4</sup>.3. *Accident seerity*
+
+The preceding analysis concerns crash incidence rates. The same analysis was also carried out on injurycrash incidence rates alone. It reveals similar trends, but the shapes of the curves obtained are difficult to interpret for the lightest and heaviest traffic densities, because the small number of injury-crashes generate very wide confidence intervals. However, knowledge about damage-only crashes makes it possible to break down the probability of observing an injury-crash when one is exposed to a risk of crash in a two- term product:
+
+injury crashes vehicles×km <sup>=</sup> injury crashes injuries+damage only crashes <sup>×</sup> injuries+damage only crashes vehicles×km .
+
+The first part of the analysis addresses the second term, i.e. the probability of observing a crash and the number of vehicles exposed. The second part of the analysis addresses the other term, i.e. the probability of observing at least one injury-crash or fatality once the crash has occurred, and traffic density at the time of the crash. We examined crash severity and traffic density using this method and the same variables as above the number of lanes, the day–night difference and the weekend–weekday difference. Table 2 shows the proportions of injury-crashes according to these three dimensions and traffic density.
+
+No significant difference in severity is observed according to the number of lanes or the period of the week. On the other hand, crashes are 1.17 times more serious at night and 1.14 times more serious in light traffic, when separated into five classes, as above. A logistic regression is then used to model the probability that a crash involves injury-crashes. The explanatory variables are the day–night difference, the traffic and the interaction between these two factors. This relationship is not significant, which means that the increase in severity observed at night does not seem to depend on traffic density. The same test made with the two other factors used above and their interaction with traffic, does not reveal a significant difference.
+
+In all the analyses, the statistical unit considered was the crash and not the damage-only vehicle. For this reason the number of vehicles involved in a crash is directly related to the severity criterion used. For example, the probability of observing injury-crashes in a crash involving two vehicles is equal to *k* times the probability of observing injury-crashes in each vehicle, with *k*=2 if we assume that severity is appreciably the same whether the crash involves another vehicle or any other obstacle, even if the vehicle overturns on the road. Using the same method as above, the logistic regression shows that severity increases with the number of vehicles involved (one, two and three or more), and differently with traffic density (the term expressing the interaction of the number of vehicles×traffic is significant). This is illustrated in Fig. 10.
+
+For average traffic of 800 to 2000 vehicles/h, the proportion of injury-crashes depends on the number of vehicles involved (10% for crashes involving a single vehicle, 20% if two vehicles are involved, 29% per three vehicles and more). This multiplying factor is smaller for high traffic flows, and much higher for light traffic flows, with a severity 45% greater for crashes involving at least three vehicles and a volume of traffic of under 400 vehicles/h.
+
+# **5. Discussion**
+
+Given the importance of traffic fluctuations inside 1 day, the time scale used appears especially relevant when studying the impact of traffic on crash occurrence. However it does not seem to be precise enough to highlight some extreme phenomena. For example, there are no observations of zero traffic during 1 h (for which the number of crashes could be only null) which means the crash/traffic flow rate should pass through the origin point of the graph. More problematically, the hourly traffic count does not make it possible to demonstrate total saturation (i.e. with total immobilization of all vehicles), although this would probably be possible with the original 6 min traffic count. Traditionally, traffic management studies address three dimensions considered as essential (May, 1990): traffic flow *Q*, speed *U* (average speed of all the vehicles on the road) and traffic density *k*. Fundamental relationships exist between these variables (TRB Highway Capacity Manual, 1994). For our study, only 'uninterrupted' traffic flow was available and the other dimensions were only determined from spot estimates. The fundamental relationship between volume and density shows that density increases in a nonlinear way with traffic up to a saturation threshold. However, on the motorway sections observed, hourly flows showing saturated traffic (during the great summer migrations) are very rare, and it seems likely we are always in the part of the graph in which traffic flow increases when the traffic density increases. In the same way, the threshold beyond which traffic flow starts to decrease at the same time as the average all-vehicle speed decreases is seldom reached in an hourly traffic count. In practice, the lack of data for these two dimensions does not prevent interpretation of the relationship between traffic flow and crash incidence rates, but does eliminate reasoning from a 'level of service' point of view.
+
+Of the three factors examined during the analysis, only the day–night distinction does not appear to influence the crash incidence rate if hourly traffic is taken into account. In other words, crash frequency is not higher at night than during the day at equal traffic flows. Let us note that this study concerns only motorways, which means, compared to the rest of the network, a better visibility at night, and less problems with alcohol during long journeys (as is often the case on motorways). The proportion of injury-crashes is, on the other hand, higher by a little over 15%. A partial
+
+![](_page_8_Figure_8.jpeg)
+
+Fig. 10. Accident severity and hourly traffic flow versus the number of vehicles involved.
+
+explanation of higher severity at night is probably that it corresponds to a specific crash typology, with, in particular, crashes due to driver drowsiness or falling asleep, often leading to crashes at full speed.
+
+The difference between the incidence rate—traffic curves and the period of the week is more difficult to interpret. At weekends, the crash rate is higher than in the remainder of the week for light traffic, but relatively stable beyond 1000 vehicles/h. Conversely, during the week, it rises regularly after 1500 vehicles/h. Perhaps one explanation of this rise is the increase in crash-generating configurations when traffic flow increases with a high proportion of trucks.
+
+Finally, the impact of the number of lanes on crash incidence rates was analyzed carefully. It appears that when traffic is light, crashes are more frequent on 3-lane motorways. This remains true, although less significantly so, if *V*/*C* is considered. This result may appear surprising because the possibilities of crash avoidance are higher than on 2-lane motorways, but may also be due to higher average speeds or more unfavorable crash configurations. This issue will be studied in another work as the hourly traffic speed should soon be available.
+
+We have prioritized the crash dimension rather than the damage-only vehicle dimension by analyzing data from an external viewpoint, like that of a motorway operator. This means that we are modeling the probability of observing a crash in a given set of traffic conditions. To reason as a vehicle occupant, the result shown in Fig. 5 is the most pertinent, i.e. for a given flow, the probability of being the victim of a crash is higher when traffic is light, and more likely on a 3-lane motorway. After moving through the minimum for hourly traffic of 1000 vehicles/h for 2-lane and 1500 for 3-lane motorways, it increases regularly at the same rate as the traffic, as does the probability of being in a crash involving several vehicles.
+
+## **6. Conclusion**
+
+The estimate of the average number of vehicles exposed to the risk of crashes and knowledge about damage-only crashes made it possible to determine the probability of observing a crash involving several vehicles per kilometer and per time period and also the probability of injury in the event of a crash. The results obtained show that crash rates are highest, both for damage-only and injury-crashes, in light traffic, the lowest between 1000 and 1500 vehicles/h, and that both increase proportionally to the average number of vehicles involved per crash. In light traffic, the number of crashes is higher on 3-lane than on 2-lane motorways, and higher at weekends (or when truck traffic is restricted). When traffic is heavy, the number of crashes is higher on weekdays. On the other hand, there is no significant difference between daytime and night-time whatever the traffic.
+
+Concerning crash severity, no difference was observed for the number of lanes or the period of the week for a given set of traffic. Conversely, severity is 17% worse at night, 14% worse in hourly traffic of under 400 vehicles/h and more particularly when traffic is light, for crashes involving several vehicles.
+
+In comparison with the vehicles passing through, light traffic is a safety problem for crash frequency and severity. Although traffic flows of under 400 vehicles/h only account for 15% of all traffic, this problem is nevertheless worrying when, particularly in France, motorways with relatively light forecast traffic are starting to be built. This is why we believe it is important to introduce road safety campaigns targeting motorway users to influence their behavior in these driving conditions, in which average speeds measured are high and distances between vehicles often surprisingly small (Aron et al., 1999). The increased relaxation felt by the road user when traffic volume is light on motorways, probably leads to overcompensation in speed, to the detriment of his safety. This is probably similar on rural roads, although driving distances are generally longer, and driving while intoxicated less frequent.
+
+In a subsequent research program, we will try to include those dimensions unavailable for this study, such as the percentage of trucks in hourly traffic and flow speed estimates.
+
+# **Acknowledgements**
+
+We would like to thank the ASF (Southern France Motorways Operator) with whom we work for crash data base maintenance and more specifically Jean-Marie Be´rier and Nicolas Schwab who made the traffic data available. Our thanks also go to LICIT/INRETS researcher Christine Buisson for his expert advice, the members of UMRETTE, my laboratory, Mireille Chiron, Sylviane Lafont and Robert Quincy for their pertinent comments and Pierre Bloch and Yves Derrien for updating the database.
+
+# **References**
+
+Aron, M., Biecheler, M.B., Peytavin, J.F., 1999. Temps interve´hiculaires et vitesses, quels enjeux de se´curite´ sur l'autoroute? Recherche Transports Se´curite´ 64, 3–20.
+
+Ceder, A., Livneh, M., 1982. Relationship between road accidents and hourly traffic flow (I and II). Accident Anal. Prev. 14 (1), 19–44.
+
+Frantzeskakis, J.M., Iordanis, D.I., 1987. Volume-to-capacity ratio and traffic accidents on interurban four-lane highways in Greece. Transp. Res. Record 1112, 29–38.
+
+- Gwynn, D.W., Baker, W.T., 1970. Relationship of accident rates with hourly traffic volume. Traffic eng. 40 (5), 42–47.
+- Hall, J.W., Pendleton, O.J., 1990. Rural accident rate variations with traffic volume. Transp. Res. Record 1281, 62–70.
+- Hauer, E., 1995. On exposure and accident rate. Traffic Eng. Control 36 (3), 134–138.
+- Jadaan, K.S., Nicholson, A.J., 1992. Relationships between road accidents and traffic flows in an urban network. Traffic Eng. Control 33 (9), 507–511.
+- Maher, M.J., Summersgill, I., 1996. A comprehensive methodology for the fitting of predictive accident models. Accident Anal. Prev. 28 (3), 281–296.
+- May, A.D., 1990. Traffic Flow Fundamentals. Prentice-Hall, Englewood Cliffs, N.J, pp. 284–315.
+- McCullagh, P., Nelder, J.A., 1989. Generalized Linear Models, 2nd. Chapman & Hall, London.
+- McGuigan D.R.D, 1987. An examination of relationships between road accidents and traffic flow, Thesis, University of Newcastle upon Tyne.
+- Mensah, A., Hauer, E., 1998. Two problems of averaging arising in the estimation of the relationship between accidents and traffic flow. Transp. Res. Record 1635, 37–43.
+- Miaou, S.P., 1994. The relationship between truck accidents and
+
+- geometric design of road sections: Poisson versus negative binomial regressions. Accident Anal. Prev. 26 (4), 471–482.
+- Nicholson, A., Wong, Y.D., 1993. Are accidents Poisson distributed, a statistical test. Accident Anal. Prev. 25 (1), 91–97.
+- Poch, M., Mannering, F., 1996. Negative binomial analysis of intersection-accident frequencies. J. Transp. Eng. 122 (2), 105–113.
+- Reinsch, C.H., 1967. Smoothing by Spline Functions. Numerische Mathematik 10, 177–183.
+- Rice, J.A., 1995. Mathematical Statistics and Data Analysis, 2nd. Duxbury Press, Belmond.
+- Satterthwaite S.P., 1981. A survey of research into relationships between traffic accidents and traffic volumes, TRRL Report SR692, Transport and Road Research Laboratory, Crowthorne.
+- SAS version 8, 1999. SAS Institute corporation, Cary NC.
+- STATA release 6, 1999. Stata corporation, College station, TX.
+- TRB Highway Capacity Manual, Special Report 209, 3rd ed., 1994. Transportation Research Board, National Research Council, Washington, D.C.
+- Turner, D.J., Thomas, R., 1986. Motorway accidents: an examination of accident totals, rates and severity and their relationship with traffic flow. Traffic Eng. Control 27 (7–8), 377–387.
+- Zhou, M., Sisiopiku, V.P., 1997. Relationship between volume-to-capacity ratios and crash rates. Transp. Res. Record 1581, 47–58.

Zotero/001_artiklid/Road Safety Indicators/moeinaddinianalyzingrelationshipsbetweennumber2015.md

@@ -0,0 +1,329 @@
+---
+category: literaturenote
+citekey: moeinaddinianalyzingrelationshipsbetweennumber2015
+title: Analyzing the relationships between the number of deaths in road accidents and the work travel mode choice at the city level
+authors: "Moeinaddini, Mehdi; Asadi-Shekari, Zohreh; Sultan, Zahid; Zaly Shah, Muhammad"
+year: 2015
+date: 2015-00-00 2015
+doi: 10.1016/j.ssci.2014.09.015
+publication: Safety Science
+url: "https://doi.org/10.1016/j.ssci.2014.09.015"
+zotero_key: VQINFARA
+zotero_storage: C2FADKD3
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Moeinaddini, Mehdi"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+Contents lists available at [ScienceDirect](http://www.sciencedirect.com/science/journal/09257535)
+
+# Safety Science
+
+journal homepage: [www.elsevier.com/locate/ssci](http://www.elsevier.com/locate/ssci)
+
+![](_page_0_Picture_5.jpeg)
+
+# Analyzing the relationships between the number of deaths in road accidents and the work travel mode choice at the city level
+
+![](_page_0_Picture_7.jpeg)
+
+Mehdi Moeinaddini <sup>a</sup> , Zohreh Asadi-Shekari b,⇑ , Zahid Sultan <sup>a</sup> , Muhammad Zaly Shah <sup>a</sup>
+
+- <sup>a</sup>Department of Urban and Regional Planning, Faculty of Built Environment, Universiti Teknologi Malaysia, Skudai, Johor, Malaysia
+- <sup>b</sup> Centre for Innovative Planning and Development (CiPD), Faculty of Built Environment, Universiti Teknologi Malaysia, Skudai, Johor, Malaysia
+
+## article info
+
+Article history: Received 7 May 2014 Received in revised form 22 September 2014 Accepted 22 September 2014 Available online 10 October 2014
+
+Keywords: Work travel mode choice Urban transport safety Crash fatality City level European cities
+
+### abstract
+
+Currently, several efforts and strategies are used to reduce deaths in road accidents. However, only a few studies have considered the effectiveness of work travel mode choice on the number of accident-related fatalities at the city level. This study introduces a city-level (across cities) model to estimate the relationships between the number of deaths in road accidents (as the dependent variable) and several work travel mode choice indicators, including walking, cycling and public transport (as independent variables). Generalized linear modeling (GLM), which is a common technique for modeling crash data, was used to estimate this relationship. Data sets from various European cities were used to develop this city level model. Overall, the percentages of the journeys to work by public transport, motorcycle, bicycle and foot were effective variables in the model.
+
+> -2014 Elsevier Ltd. All rights reserved.
+
+### 1. Introduction
+
+According to World Health [Organization \(2013\)](#page-5-0) data, road traffic crashes cause nearly 3400 deaths each day and injure or disable more than ten million people every year worldwide. Although approximately 70% of traffic-related deaths occur in developing countries ([Augustus, 2012](#page-4-0)), nearly 28,000 road fatalities were reported in European countries in 2012 ([European Commission,](#page-4-0) [2013\)](#page-4-0). These statistics indicate that concerns about road traffic crashes are very serious in developing countries and still exist in European countries ([Moeinaddini et al., 2014a\)](#page-5-0). Among all of the road fatalities in the EU, approximately 50% are car occupants, 15% are motorcyclists, 10% are cyclists or moped riders, and 20% are pedestrians ([European Commission, 2013](#page-4-0)). In the past two decades, bicycle use by adults has increased in some European cities. In some countries where cycling is common, such as the Netherlands, cyclists account for more fatal accidents than pedestrians do ([Mohan, 2002](#page-5-0)).
+
+In many countries, especially industrial countries, road traffic accidents are a major type of fatal work accident ([Charbotel](#page-4-0) [et al., 2001 and Charbotel et al., 2010; Murray, 2007](#page-4-0)). The proportion of work-related traffic fatalities among all accidental fatalities in Finland has increased by 63% ([Statistics Finland, 1997\)](#page-5-0). In 2012, the number of casualties in Britain was higher during working hours [\(RRCGB,](#page-5-0) 2013). The number of daily work trips is the highest in Mashhad City, Iran ([Naderan and Shahi, 2010](#page-5-0)). [Christoforou](#page-4-0) [et al. \(2010\)](#page-4-0) found that approximately 93% of the individuals that are injured on working days in accidents at the A4-A86 junction in the Paris region are seriously injured. The annual crash number and traffic flow are higher during working hours according to observations that were made of 2000 km of French interurban motorways in 1997 and 1998 ([Martin, 2002](#page-5-0)).
+
+In the past several decades, a large number of empirical mode choice studies have considered a large range of factors for the transportation mode used to reach different destinations (university or school, work, favorite leisure activities and shopping). The majority of these studies considered several socio-economic and demographic factors, including age, gender, income, education level, employment status, family size, marital status, vehicle ownership, the number of workers in the household, the presence of children in the household, and the number of household motor vehicles (e.g., [Whalen et al., 2013; Vega and Reynolds-Feighan,](#page-5-0) [2009; Can, 2013; Carse et al., 2013; Muhs, 2013\)](#page-5-0). In addition, several studies considered the environmental factors that impact travel mode choice, such as the availability of parking, sidewalk density, bike density, street density, availability of park and ride facilities, shelter, weather and daylight (e.g., [Whalen et al., 2013;](#page-5-0) [Klöckner and Friedrichsmeier, 2011; Commins and Nolan, 2011\)](#page-5-0). Furthermore, some trip characteristics, such as travel cost, travel time, travel distance, comfort, safety, car availability, public
+
+<sup>⇑</sup> Corresponding author. Tel.: +60 129410543. E-mail addresses: [asadi41360@gmail.com](mailto:asadi41360@gmail.com), [aszohreh2@live.utm.my](mailto:aszohreh2@live.utm.my) (Z. Asadi-Shekari).
+
+transport availability and waiting time, influence travel behavior (e.g., [Habib and Sasic, 2014; Whalen et al., 2013; Vega and](#page-4-0) [Reynolds-Feighan, 2009; Can, 2013; Bhatta and Larsen, 2011\)](#page-4-0).
+
+Although safety is one of the most important factors to consider when investigating mode choice [\(Lovegrove et al., 2010; Lovegrove](#page-5-0) [and Sayed, 2006a,b, and 2007; Can, 2013; Deka, 2013\)](#page-5-0), few studies have focused on this important factor in Europe ([Asadi-Shekari](#page-4-0) [et al., in press\)](#page-4-0). Several efforts have been made to explain the different factors that affect work travel mode choice (e.g., [Vega and](#page-5-0) [Reynolds-Feighan, 2009; Commins and Nolan, 2011; Liu, 2007;](#page-5-0) [Habib, 2012; Pooley and Turnbull, 2000; Palma and Rochat,](#page-5-0) [2000; Xie et al., 2003](#page-5-0)). However, limited studies have focused on travel safety (e.g., [Amoh-Gyimah and Aidoo, 2013](#page-4-0)). The need to go to school or work is the main reason why many people travel. Nearly all previous efforts have focused on safe routes to schools (e.g., [Nevelsteena et al., 2012; National Center for Safe Routes to](#page-5-0) [School, 2012](#page-5-0)) and the mode choice when traveling to a school or university (e.g., [Whalen et al., 2013; Mitra et al., 2010; Muller](#page-5-0) [et al., 2008; Ewing et al., 2004\)](#page-5-0). In contrast, only a few studies have evaluated the impacts of work travel mode on traffic safety. This study attempts to compensate for this shortcoming by analyzing the effects of work travel mode choice on traffic fatalities.
+
+Work travel mode impacts are important because urban traffic congestion is worse during the periods of travel to and from work ([Quarmby, 1967](#page-5-0)). Traffic congestion appears to significantly impact road safety. However, [Quddus et al. \(2010\)](#page-5-0) found that the effects of traffic congestion on accident severity are limited. Earlier studies (e.g., [Shefer, 1994; Shefer and Rietveld, 1997; Baruya,](#page-5-0) [1998\)](#page-5-0) suggested that because speed would be lower in congested areas, there should be a negative relationship between traffic congestion and road accidents. However, recent studies (e.g., [Noland](#page-5-0) [and Quddus, 2005](#page-5-0)) have revealed that a negative relationship does not exist between traffic congestion and road accidents. [Kononov](#page-5-0) [et al. \(2008\)](#page-5-0) found that the total accident rate and fatality and injury rates increased in congested areas. These authors used data from Texas, California and Colorado in the US. In addition, [Wang](#page-5-0) [et al. \(2013\)](#page-5-0) found a positive relationship between traffic congestion and fatalities and serious injuries from accidents in London. This positive relationship may be due to poor driving behaviors and higher speed variances within and between the lanes.
+
+Nearly all people who work make two trips per day. Thus, the trip to work (one of the most commonly experienced types of travel) is a special scenario ([Quarmby, 1967\)](#page-5-0). To our knowledge, little is known in the transportation literature regarding the impacts of key work travel modes (e.g., walking, cycling, public transport and car) on traffic fatalities. This research represents an attempt to understand this phenomenon better in a quantitative sense. In addition, limited studies are available regarding macro-level collision prediction models at the city or neighborhood level (e.g., [Hadayeghi et al., 2003, 2010; Lovegrove and Sayed, 2006a;](#page-4-0) [Lovegrove, 2007; Hadayeghi, 2009](#page-4-0)). [Lovegrove et al. \(2010\)](#page-5-0) used data from 400 Greater Vancouver neighborhoods in British Columbia, Canada to evaluate a regional transportation plan regarding road safety. These authors utilized previous community-based macro-level models. In addition, [Lovegrove and Sayed \(2006a\)](#page-5-0) proposed macro-level collision models. These authors used data from 577 neighborhoods in the Greater Vancouver Regional District and found that traffic collisions were related to several neighborhood characteristics (e.g., network shape). [Moeinaddini et al. \(2014a\)](#page-5-0) observed a relationship between macro-level indicators, including traffic fatalities and the street network, at the city level. Although these studies proposed important factors that affect travel safety, they cannot represent travel safety and the choice of travel mode at the city level.
+
+The city level can result in better planning decisions and strategies. Thus, as part of continuing efforts in this area, this study examines how work travel modes, including the percentages of journeys to work by public transport, motorcycle, bicycle and foot, contribute to traffic fatalities at the city level. This research uses data from different European cities to estimate the relationships between work travel modes and the numbers of transport fatalities in different European cities (city level). Finally, this research attempts to explain the use of macro-level models in the city-level planning process.
+
+# 2. Data and methodology
+
+This study estimated the strengths of relationships between work travel mode indicators and transport fatalities in various European cities at a macro-level scale (across cities). The numbers of deaths that occurred due to road accidents per one-hundred thousand people was used as the dependent variable, and the work travel mode choice indicators, including walking, cycling and public transport, were used as the independent variables to estimate this relationship (refer to [Table 1](#page-2-0)). Zero traffic was tested by adding the numbers of registered private cars as an exposure variable. The Eurostat transport data from 2011 were used in this research (from the [Eurostat](#page-4-0) [database website](#page-4-0)). The Eurostat database has different mobility indicators for more than 1000 cities from different parts of Europe. The cities were chosen based on data availability for all of the variables after removing the outliers. Overall, four indicators for work travel mode choices were used (i.e., the percentages of journeys to work by public transport, motorcycle, bicycle and foot).
+
+A collision is a random, rare and multi-factor event. Various methods of analysis have been used previously in different approaches for crash data. To model crash data, the Poisson (PO) distribution and the NB models have been used widely (e.g., [Wei](#page-5-0) [and Lovegrove, 2013; Mitra and Washington, 2007; Hadayeghi](#page-5-0) [et al., 2007, 2003; Memon, 2006; Lovegrove and Sayed, 2006b;](#page-5-0) [Noland and Quddus, 2004; Miaou and Lord, 2003; Abdel-Aty and](#page-5-0) [Radwan, 2000; Kim et al., 2002\)](#page-5-0). However, the Poisson model is not suitable for handling over- or underdispersed data. The NB model is more suitable for overdispersion but is not suitable for handling underdispersion ([Lord and Mannering, 2010](#page-5-0)).
+
+Both the NB and Poisson distributions have been used to predict accidents related to counting data ([Kibria, 2006](#page-5-0)). Counting data are distributed as non-negative integers. The NB distribution is a more flexible extension of the Poisson distribution [\(Kibria, 2006](#page-5-0)). The Poisson distribution is suitable for modeling crash data for a single site [\(Hadayeghi, 2009](#page-4-0)). Crash data from a series of sites often exhibit large variances and small means that result in overdispersion when the variance-to-mean ratio is greater than 1 ([Hadayeghi,](#page-4-0) [2009\)](#page-4-0). For modeling crash data at a series of sites, the NB distribution, which is also known as the Poisson–gamma distribution, is the most commonly used method [\(Hadayeghi, 2009\)](#page-4-0).
+
+Several studies have used other analysis methods, such as linear regression (e.g., Clark [and Cushing, 2004](#page-4-0)), geographically weighted regressions (e.g., [Hadayeghi et al., 2003](#page-4-0)), and log-linear models using ordinary least squares (OLS) estimations (e.g., [Wier et al.,](#page-5-0) [2009; Washington et al., 2006](#page-5-0)). Linear regression models that have normal distributed error structures cannot handle road collisions data because the data are non-negative and discrete ([Moeinaddini et al., 2014a](#page-5-0)). The Poisson and Negative Binomial (NB) regressions in the generalized linear model (GLM) framework have been more successfully adopted for these count data ([Hadayeghi, 2009](#page-4-0)). Previously, GLMs were commonly used in collision models because they are useful for interpreting random, rare, sporadic and non-negative data ([Lovegrove et al., 2010; Wei and](#page-5-0) [Lovegrove, 2013\)](#page-5-0).
+
+The analysis methods are selected based on the measurement scale, the number of groups, the natures of the relationships
+
+<span id="page-2-0"></span>Table 1 Research data.
+
+| Cities                | Percentage of journeys to work by<br>public transport (rail, metro, bus,<br>tram) | Percentage of<br>journeys to work by<br>motor cycle | Percentage of<br>journeys to work<br>by bicycle | Percentage of<br>journeys to work<br>by foot | Number of<br>private cars<br>registered | Number of deaths in road<br>accidents per 100,000<br>population |
+|-----------------------|-----------------------------------------------------------------------------------|-----------------------------------------------------|-------------------------------------------------|----------------------------------------------|-----------------------------------------|-----------------------------------------------------------------|
+| Bruxelles/<br>Brussel | 40.3                                                                              | 1.2                                                 | 1.9                                             | 4.1                                          | 513,050                                 | 2.2                                                             |
+| Antwerpen             | 20.7                                                                              | 1.48                                                | 14.81                                           | 2.01                                         | 192,115                                 | 3.41                                                            |
+| Gent                  | 14.27                                                                             | 2                                                   | 14.09                                           | 1.43                                         | 103,265                                 | 4.43                                                            |
+| Liège                 | 8.43                                                                              | 0.92                                                | 1.13                                            | 5.13                                         | 154,495                                 | 5.04                                                            |
+| Brugge                | 9.75                                                                              | 2.21                                                | 24.71                                           | 1.99                                         | 54,307                                  | 2.56                                                            |
+| Namur                 | 15.95                                                                             | 0.85                                                | 1.08                                            | 3.2                                          | 48,704                                  | 11.8                                                            |
+| Leuven                | 17.98                                                                             | 1.13                                                | 15.26                                           | 1.85                                         | 73,667                                  | 3.08                                                            |
+| Mons                  | 11.42                                                                             | 1.19                                                | 1.57                                            | 2.63                                         | 41,605                                  | 15.1                                                            |
+| Kortrijk              | 6.04                                                                              | 1.36                                                | 15.74                                           | 1.23                                         | 38,302                                  | 6.65                                                            |
+| Oostende              | 8.08                                                                              | 2.89                                                | 20.1                                            | 3.46                                         | 26,857                                  | 4.29                                                            |
+| Plovdiv               | 28                                                                                | 10                                                  | 8                                               | 30                                           | 182,602                                 | 4.14                                                            |
+| Varna                 | 32                                                                                | 11                                                  | 9                                               | 21                                           | 185,600                                 | 4.48                                                            |
+| Burgas                | 26                                                                                | 14                                                  | 13                                              | 26                                           | 114,967                                 | 13.98                                                           |
+| Pleven                | 25                                                                                | 11                                                  | 15                                              | 27                                           | 69,510                                  | 10.27                                                           |
+| Ruse                  | 21                                                                                | 9                                                   | 12                                              | 36                                           | 88,820                                  | 8.68                                                            |
+| Vidin                 | 18                                                                                | 9                                                   | 14                                              | 33                                           | 33,526                                  | 6.24                                                            |
+| Stara                 | 25                                                                                | 15                                                  | 17                                              | 17                                           | 84,857                                  | 13.01                                                           |
+| Zagora                |                                                                                   |                                                     |                                                 |                                              |                                         |                                                                 |
+| Sliven                | 25                                                                                | 14                                                  | 15                                              | 20                                           | 65,139                                  | 31.64                                                           |
+|                       |                                                                                   |                                                     |                                                 |                                              |                                         |                                                                 |
+| Dobrich               | 28                                                                                | 13                                                  | 16                                              | 22                                           | 48,245                                  | 2.2                                                             |
+| Shumen                | 19                                                                                | 13                                                  | 16                                              | 30                                           | 48,697                                  | 8.65                                                            |
+| Pernik                | 29                                                                                | 4                                                   | 12                                              | 24                                           | 72,106                                  | 3.74                                                            |
+| Yambol                | 24                                                                                | 13                                                  | 18                                              | 23                                           | 40,031                                  | 4.04                                                            |
+| Haskovo               | 23                                                                                | 15                                                  | 18                                              | 25                                           | 50,844                                  | 7.85                                                            |
+| Pazardzhik            | 27                                                                                | 14                                                  | 18                                              | 18                                           | 62,003                                  | 19.44                                                           |
+| Blagoevgrad           | 22                                                                                | 13                                                  | 17                                              | 22                                           | 41,043                                  | 5.64                                                            |
+| Veliko<br>Tarnovo     | 28                                                                                | 14                                                  | 9                                               | 29                                           | 46,996                                  | 17.44                                                           |
+| Dublin                | 24.4                                                                              | 0.9                                                 | 8.2                                             | 22.7                                         | 217,805                                 | 1.16                                                            |
+| Dublin                | 21                                                                                | 1                                                   | 5.3                                             | 13.5                                         | 411,836                                 | 0.87                                                            |
+| (greater<br>cit)      |                                                                                   |                                                     |                                                 |                                              |                                         |                                                                 |
+| Cork                  | 9.2                                                                               | 0.7                                                 | 2.9                                             | 23.3                                         | 58,357                                  | 2.53                                                            |
+| Limerick              | 5.3                                                                               | 0.3                                                 | 2.8                                             | 22.4                                         | 21,369                                  | 1.76                                                            |
+| Galway                | 6.5                                                                               | 0.3                                                 | 5                                               | 17.3                                         | 30,908                                  | 2.7                                                             |
+| Waterford             | 3                                                                                 | 0.5                                                 | 2                                               | 15.5                                         | 20,057                                  | 2.15                                                            |
+| Helsinki/             | 37.8                                                                              | 0.5                                                 | 9.5                                             | 12.5                                         | 240,175                                 | 1.36                                                            |
+| Helsingf              |                                                                                   |                                                     |                                                 |                                              |                                         |                                                                 |
+| Helsinki/             | 30.8                                                                              | 0.3                                                 | 7.7                                             | 9.8                                          | 468,314                                 | 1.82                                                            |
+| Helsingf              |                                                                                   |                                                     |                                                 |                                              |                                         |                                                                 |
+| Oulu/                 | 7                                                                                 | 0                                                   | 24                                              | 9                                            | 67,644                                  | 4.94                                                            |
+| Uleåborg              |                                                                                   |                                                     |                                                 |                                              |                                         |                                                                 |
+| Espoo/Esbo            | 20                                                                                | 2                                                   | 7                                               | 21                                           | 120,957                                 | 2.42                                                            |
+| Lahti/Lahtis          | 8                                                                                 | 1                                                   | 14                                              | 13                                           | 49,861                                  | 5.91                                                            |
+
+between the groups, the number of variables and the assumptions of the statistical tests [\(Moeinaddini et al., 2014b](#page-5-0)). In this study, the dependent variable is the number of deaths in road accidents per one hundred thousand people. This variable is similar to nonnegative count data, in which the numbers of deaths only take positive and discrete values. In contrast, the dependent variables are scaled factors because they are divided by the population to normalize the variables between cities. Gamma with log link and lognormal models can be used for scale data in the GLM framework. GLMs with a gamma distribution can handle collision data slightly better than the log-transformed model [\(Moeinaddini et al., 2014a\)](#page-5-0). In addition, lognormal is not related to the exponential family (e.g., Gamma, Poisson, Negative Binominal, etc.). The Gamma model assumes a log link, such as the following NB model (refer to Eq. (1)):
+
+$$Y = EXP\left(\beta_0 + \sum_{i=1}^{i} \beta_i \times X_i\right), \tag{1}$$
+
+where Y is the dependent variable, i is a subscript that shows the number of independent variables, X is the independent variable, b<sup>0</sup> is a constant, and b<sup>i</sup> is a coefficient of the independent variable. The constants and coefficients will be calculated during the calibration process.
+
+## 3. Results
+
+In this section, the results of the mode choice fatality GLM analysis are discussed. Descriptive statistics for the variables included in the model are indicated in [Tables 2 and 3.](#page-3-0) No strong correlation occurred between the independent variables that were included in the model (refer to [Table 4\)](#page-3-0). The variables that were included in the model, their parameter estimates and the significance of these parameters (5% level) are indicated in [Table 5.](#page-3-0) The modeled goodness of fit for the Omnibus test, the likelihood ratio, the Chi-Square test statistics, the scaled deviance (SD) and the Pearson Chi-Square statistic are indicated in [Tables 6 and 7](#page-3-0). Therefore, the final model can be defined as follows:
+
+$$D = EXP (-7.789 - 0.102P - 0.031F + 0.238M - 0.044B),$$
+ (2)
+
+where D is the number of road accident deaths per one-hundred thousand people, P is the percentage of journeys to work by public transport, F is the percentage of journeys to work by foot, M is the percentage of journeys to work by motorcycle, and B is the
+
+<span id="page-3-0"></span>Table 2 Continuous variable information.
+
+|                    |      | N  | Minimum | Maximum | Mean    | Std. Deviation |
+|--------------------|------|----|---------|---------|---------|----------------|
+| Dependent Variable | D    | 37 | .87     | 31.64   | 6.6918  | 6.35116        |
+| Covariate          | P    | 37 | 3.00    | 40.30   | 19.6195 | 9.56392        |
+|                    | F    | 37 | 1.23    | 36.00   | 16.4873 | 10.32461       |
+|                    | M    | 37 | .00     | 15.00   | 5.8035  | 5.81870        |
+|                    | B    | 37 | 1.08    | 24.71   | 11.5078 | 6.50596        |
+| Offset             | LNCR | 37 | 9.91    | 13.15   | 11.2568 | .82816         |
+
+Where D is the number of deaths in road accidents per hundred thousand population, P the percentage of journeys to work by public transport, F the percentage of journeys to work by foot, M the percentage of journeys to work by motorcycle, B the percentage of journeys to work by bicycle and CR is the number of private cars registered.
+
+Table 3 Case processing summary.
+
+|                      | N       | Percent (%)  |
+|----------------------|---------|--------------|
+| Included<br>Excluded | 37<br>0 | 100.0<br>0.0 |
+| Total                | 37      | 100.0        |
+
+Table 4 Correlation matrix.
+
+| Coefficientsa |   |                         |       |  |  |  |  |
+|---------------|---|-------------------------|-------|--|--|--|--|
+| Model         |   | Collinearity statistics |       |  |  |  |  |
+|               |   | Tolerance               | VIF   |  |  |  |  |
+| 1             | P | .748                    | 1.337 |  |  |  |  |
+|               | F | .522                    | 1.915 |  |  |  |  |
+|               | M | .354                    | 2.824 |  |  |  |  |
+|               | B | .680                    | 1.471 |  |  |  |  |
+
+Where D is the number of deaths in road accidents per hundred thousand population, P the percentage of journeys to work by public transport, F the percentage of journeys to work by foot, M the percentage of journeys to work by motorcycle and B is the percentage of journeys to work by bicycle.
+
+percentage of journeys to work by bicycle. This model shows that the numbers of deaths in road accidents were significantly affected by the work travel mode indicators in this model. Among these indicators, the percentage of journeys to work by motorcycle had a positive parameter, which indicated a greater effect on the number of road accident deaths in this model. The second most effective indicator of the percentage of journeys to work by public transport had a negative relationship. The percentage of journeys to work by bicycle and the percentage of journeys to work by foot also had negative parameters and were inversely associated with transport fatalities in this model.
+
+Table 5 Parameter estimates.
+
+| Parameter   | B     | Std. error | 95% Wald confidence interval |       | Hypothesis test |    |      |
+|-------------|-------|------------|------------------------------|-------|-----------------|----|------|
+|             |       |            | Lower                        | Upper | Wald Chi-Square | df | Sig. |
+| (Intercept) | 7.789 | .3763      | 8.526                        | 7.051 | 428.454         | 1  | .000 |
+| P           | .102  | .0151      | .132                         | .072  | 45.460          | 1  | .000 |
+| F           | .031  | .0148      | .060                         | .002  | 4.431           | 1  | .035 |
+| M           | .238  | .0320      | .175                         | .300  | 55.091          | 1  | .000 |
+| B           | .044  | .0202      | .084                         | .004  | 4.725           | 1  | .030 |
+| (Scale)     | .513a | .1105      | .336                         | .782  |                 |    |      |
+
+Dependent variable: D.
+
+Model: (Intercept), P, F, M, B, offset = LNCR.
+
+Where D is the number of deaths in road accidents per hundred thousand population, P the percentage of journeys to work by public transport, F the percentage of journeys to work by foot, M the percentage of journeys to work by motorcycle, B the percentage of journeys to work by bicycle and CR is the number of private cars registered.
+
+Table 6 Omnibus test.<sup>a</sup>
+
+| Likelihood ratio Chi-Square | df | Sig. |
+|-----------------------------|----|------|
+| 34.598                      | 4  | .000 |
+
+Dependent variable: D.
+
+Model: (Intercept), P, F, M, B, offset = LNCR.
+
+Where D is the number of deaths in road accidents per hundred thousand population, P the percentage of journeys to work by public transport, F the percentage of journeys to work by foot, M the percentage of journeys to work by motorcycle, B the percentage of journeys to work by bicycle and CR is the number of private cars registered.
+
+Table 7 SD and Pearson Chi-Square goodness of fit.
+
+|                           | Value  | df | Value/df |
+|---------------------------|--------|----|----------|
+| Deviance                  | 20.554 | 32 | .642     |
+| Scaled deviance           | 40.087 | 32 |          |
+| Pearson Chi-Square        | 20.794 | 32 | .650     |
+| Scaled Pearson Chi-Square | 40.555 | 32 |          |
+
+Dependent Variable: D.
+
+Model: (Intercept), P, F, M, B, offset = LNCR.
+
+Where D is the number of deaths in road accidents per hundred thousand population, P the percentage of journeys to work by public transport, F the percentage of journeys to work by foot, M the percentage of journeys to work by motorcycle, B the percentage of journeys to work by bicycle and CR is the number of private cars registered.
+
+Overall, fewer journeys to work by motorcycle and more journeys to work by public transport, bicycle and foot are effective work travel mode indicators that are associated with fewer transport fatalities. This finding suggests that macro-scale urban planning that reduces the number of journeys to work by motorcycle and increases the numbers of journeys to work by public transport, bicycle and foot could contribute to fewer transport fatalities in different cities.
+
+<sup>a</sup> Dependent Variable: D.
+
+<sup>a</sup> Maximum likelihood estimate.
+
+<sup>a</sup> Compares the fitted model against the intercept-only model.
+
+# <span id="page-4-0"></span>4. Discussion and conclusions
+
+Macro-scale work travel mode indicators (as the independent variables) and road accident fatalities (as the dependent variable) were analyzed in this study using GLM analysis. Overall, the percentages of journeys to work by public transport, motorcycle, bicycle and foot were the work travel mode factors that were associated with road accident fatalities.
+
+The model shows that a greater number of journeys to work by motorcycle is correlated with more road accident deaths. Thus, distributing more motorcycles in urban areas could produce more conflicts [\(Nguyen et al., 2014\)](#page-5-0). Motorcycles are not as safe as other travel modes, and their smaller size allows more mobility, especially in congested areas [\(Nguyen et al., 2014](#page-5-0)). Therefore, considering more connected routes for exclusive use by motorcycles may result in fewer conflicts and transport fatalities involving vehicle transport. However, this suggestion is not supported by the proposed model.
+
+In addition, the model shows that the percentage of journeys to work by public transport is negatively associated with road accident fatalities. Consequently, the cities that have more journeys to work by public transport mode have fewer fatalities. Public transportation has an overall low crash risk [\(Lovegrove et al.,](#page-5-0) [2010; Litman, 2014\)](#page-5-0). The total number of traffic fatalities declines significantly as transit ridership increases ([Kenworthy and Laube,](#page-5-0) [2000](#page-5-0)).
+
+This model also indicates that more journeys to work by bicycle and foot are correlated with fewer transport fatalities. Safe walking and cycling can be more attractive options for urban travelers than crowded, dangerous and costly streets that are used by private vehicles (Asadi-Shekari et al., 2013a; Asadi-Shekari et al., 2013b and Asadi-Shekari et al., 2014; Asadi-Shekari and Zaly Shah, 2011). Increasing the number of journeys to work by bicycle and foot may decrease the use of other travel modes. More private motorized facilities, such as freeways, do not contribute to improved street conditions for pedestrians and cyclists because freeways are used exclusively by motorized vehicles. Thus, pedestrians and cyclists do not benefit from freeways. Improving the street conditions for pedestrians and cyclists and improving the distribution of destinations that are within walking distance in mixed-land-use areas may increase walking and cycling [\(Moeinaddini et al., 2013\)](#page-5-0).
+
+This study evaluated the relationships of work travel mode indicators and deaths in road accidents in different cities in Europe at the city level. Therefore, the results of this study can illustrate this relationship regardless of the micro-level effects. In addition, this study attempted to evaluate the relationships between work travel mode indicators and safety, which have not been sufficiently studied in the literature. Moreover, this study addresses work travel mode indicators at the city level instead of the neighborhood level.
+
+A lower number of traffic-related fatalities is an effective indicator of a sustainable and safe city. To promote these types of cities, it is important to conduct more research at the city level. Therefore, this research attempted to indicate the relationships between work travel mode indicators and road accident fatalities at the city level. These relationships can be used in fatality reduction strategies in different cities. In addition, considering the city level can result in better planning decisions for new cities.
+
+Furthermore, this research is limited to European cities in the Eurostat database and work travel mode indicators. Universal scale relationship models may be replicated in future studies. Other urban structure indicators can be included in future studies to identify effective factors for sustainable travel patterns, regardless of the context effects. This research used Eurostat data sources, which are free, due to financial limitations; further studies may enhance the model by using more data. Moreover, travel mode indicators can change rapidly. Future studies may evaluate these changes by updating their data sources and examining the relationships between the travel mode indicators and fatalities in various parts of the world to address changing results.
+
+# Acknowledgments
+
+The authors wish to thank all of those who have supported this research for their useful comments during its completion. In particular, we would like to acknowledge Centre for Innovative Planning and Development (CIPD) and Prof. Dr. Amran Hamzah who helped us during the research process. The research underlying this paper was supported by Universiti Teknologi Malaysia Research Management Centre (Grant Number: Q.J130000.2721.00K96).
+
+### References
+
+- [Abdel-Aty, M.A., Radwan, A.E., 2000. Modeling traffic accident occurrence and](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0005) [involvement. Accid. Anal. Prev. 32 \(5\), 633–642](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0005).
+- [Amoh-Gyimah, R., Aidoo, E.N., 2013. Mode of transport to work by government](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0010) [employees in the Kumasi metropolis, Ghana. J. Transp. Geogr. 31, 35–43.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0010)
+- Asadi-Shekari, Z., Zaly Shah, M., 2011. Practical evaluation method for pedestrian level of service in urban streets. Paper presented at the International Transport Research Conference, Penang, Malaysia.
+- [Asadi-Shekari, Z., Moeinaddini, M., Zaly Shah, M., 2013a. Non-motorized level of](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0020) service methods: addressing [challenges in research and practice. Transport Rev.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0020) [33 \(2\), 166–194](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0020).
+- [Asadi-Shekari, Z., Moeinaddini, M., Zaly Shah, M., 2013b. Disabled pedestrian level](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0025) [of service method for evaluating and promoting inclusive walking facilities on](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0025) [urban streets. J. Transport. Eng. 139 \(2\), 181–192.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0025)
+- [Asadi-Shekari, Z., Moeinaddini, M., Zaly Shah, M., 2014. A pedestrian level of service](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0030) [method for evaluating and promoting walking facilities on campus streets. Land](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0030) [Use Policy 38, 175–193.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0030)
+- Asadi-Shekari, Z., Moeinaddini, M., Zaly Shah, M., in press. A Bicycle Safety Index for Evaluating Urban Street Facilities. Traffic Injury Prevention. [http://dx.doi.org/](http://dx.doi.org/10.1080/15389588.2014.936010) [10.1080/15389588.2014.936010](http://dx.doi.org/10.1080/15389588.2014.936010)
+- [Augustus, A., 2012. Determinants of road traffic accident occurrences in lagos state:](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0040) some lessons for [Nigeria. Int. J. Humanities Social Sci. 2 \(6\), 252–259.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0040)
+- Baruya, A., 1998. MASTER: Speed-Accident Relationships on Different Kinds of European Roads.: Deliverable D7. UK: Transport Research Laboratory.
+- [Bhatta, B.P., Larsen, O.I., 2011. Errors in variables in multinomial choice modeling: a](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0050) simulation study applied [to a multinomial logit model of travel mode choice.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0050) [Transport Policy 18, 326–335](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0050).
+- [Can, V.V., 2013. Estimation of travel mode choice for domestic tourists to Nha Trang](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0055) [using the multinomial probit model. Transp. Res. Part A 49, 149–159.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0055)
+- [Carse, A., Goodman, A., Mackett, R.L., Panter, J., Ogilvie, D., 2013. The factors](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0060) [influencing car use in a cycle-friendly city: the case of Cambridge. J. Transp.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0060) [Geogr. 28, 67–74](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0060).
+- [Charbotel, B., Chiron, M., Martin, J.L., Bergeret, A., 2001. Work related road accidents](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0065) [in France. Eur. J. Epidemiol. 17 \(8\), 773–778.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0065)
+- [Charbotel, B., Martin, J.L., Chiron, M., 2010. Work-related versus non-work-related](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0070) [road accidents, developments in the last decade in France. Accid. Anal. Prev. 42,](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0070) [604–611](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0070).
+- [Christoforou, Z., Cohen, S., Karlaftis, M.G., 2010. Vehicle occupant injury severity on](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0075)
+- [highways: an empirical investigation. Accid. Anal. Prev. 42, 1606–1620.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0075) [Clark, D.E., Cushing, B.M., 2004. Rural and urban traffic fatalities, vehicle-miles and](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0080) [population density. Accid. Anal. Prev. 36 \(6\), 967–972.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0080)
+- [Commins, N., Nolan, A., 2011. The determinants of mode of transport to work in the](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0085) [Greater Dublin Area. Transp. Policy 18, 259–268.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0085)
+- [Deka, D., 2013. An explanation of the relationship between adults' work trip mode](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0090) [and children's school trip mode through the Heckman approach. J. Transp.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0090) [Geogr. 31, 54–63](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0090).
+- European Commission., 2013. Road Safety Vademecum Road Safety Trends, Statistics and Challenges in the EU 2011–2012.
+- Eurostat, 2014. Retrieved 6/4/2014, from <[http://epp.eurostat.ec.europa.eu/portal/](http://epp.eurostat.ec.europa.eu/portal/page/portal/statistics/search_database) [page/portal/statistics/search\\_database>](http://epp.eurostat.ec.europa.eu/portal/page/portal/statistics/search_database).
+- [Ewing, R., Schroeer, W., Greene, W., 2004. School location and student travel](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0105) analysis of factors [affecting mode choice. Transport. Res. Rec.: J. Transport. Res.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0105) [Board 1895, 55–63](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0105).
+- [Habib, K.M.N., 2012. Modeling commuting mode choice jointly with work start time](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0110) [and work duration. Transp. Res. Part A 46, 33–47.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0110)
+- [Habib, K.M.N., Sasic, A., 2014. A GEV model with scale heterogeneity for](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0115) [investigating the role of mobility tool ownership in peak period non-work](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0115) [travel mode choices. J. Choice Modell. 10, 46–59](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0115).
+- Hadayeghi, A., 2009. Use of Advanced Techniques to Estimate Zonal Level Safety Planning Models and Examine Their Temporal Transferability. (Doctor of Philosophy), University of Toronto, Ontario, Canada.
+- [Hadayeghi, A., Shalaby, A., Persaud, B., 2003. Macro-level accident prediction](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0125) models for evaluating [the safety of urban transportation systems. Transp. Res.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0125) [Rec. 1840, 87–95.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0125)
+
+- <span id="page-5-0"></span>[Hadayeghi, A., Shalaby, A., Pesuad, B., 2007. Safety prediction models: a proactive](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0130) [tool for safety evaluation in urban transportation planning applications. Transp.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0130) [Res. Rec. 2019, 225–236.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0130)
+- [Hadayeghi, A., Shalaby, A., Persaud, B., 2010. Development of planning-level](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0135) [transportation safety models using full bayesian semiparametric additive](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0135) [techniques. J. Transport. Safety Security 2 \(1\), 45–68.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0135)
+- Kenworthy, J., Laube, F., 2000. Millennium Cities Database For Sustainable Transport: Institute for Sustainability and Technology Policy, Distributed by the International Union of Public Transport.
+- [Kibria, B.M.G., 2006. Applications of some discrete regression models for count data.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0145) Pakistan J. Stat. [Oper. Res. 2 \(1\), 1–16](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0145).
+- [Kim, H., Sun, D., Tsutakawa, R.K., 2002. Lognormal vs. Gamma: extra variations.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0150)
+- [Biometrical J. 44 \(3\), 305–323.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0150) [Klöckner, C.A., Friedrichsmeier, T., 2011. A multi-level approach to travel mode](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0155) [choice – how person characteristics and situation specific aspects determine car](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0155)
+- [use in a student sample. Transp. Res. Part F 14, 261–277](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0155). [Kononov, J., Bailey, B., Allery, B., 2008. Relationships between safety and both](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0160) [congestion and number of lanes on urban freeways. Transport. Res. Rec.: J.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0160) [Transport. Res. Board 2083, 26–39](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0160).
+- [Litman, T., 2014. Safer Than You Think! Revising the Transit Safety Narrative.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0165) [Victoria Transport Policy Institute.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0165)
+- [Liu, G., 2007. A behavioral model of work-trip mode choice in Shanghai. China Econ.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0170) [Rev. 18, 456–476](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0170).
+- [Lord, D., Mannering, F., 2010. The statistical analysis of crash-frequency data: a](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0175) [review and assessment of methodological alternatives. Transp. Res. Part A 44,](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0175) [291–305](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0175).
+- [Lovegrove, G., 2007. Road Safety planning, new tools for sustainable road safety and](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0180) [community development. Verlag Dr. Müller, Germany](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0180).
+- [Lovegrove, G., Sayed, T., 2006a. Macro-level collision prediction models for](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0185)
+- [evaluating neighbourhood traffic safety. Can. J. Civ. Eng. 33 \(5\), 609–621.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0185) [Lovegrove, G., Sayed, T., 2006b. Using macro-level collision prediction models in](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0190) [road safety planning applications. Transport. Res. Rec.: J. Transport. Res. Board](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0190)
+- [1950, 73–82.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0190) [Lovegrove, G., Sayed, T., 2007. Macro-level collision prediction models to enhance](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0195) [traditional reactive road safety improvement programs. Transport. Res. Rec.: J.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0195) [Transport. Res. Board 2019, 65–73](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0195).
+- [Lovegrove, G., Lim, C., Sayed, T., 2010. Community-based, macrolevel collision](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0200) [prediction model use with a regional transportation plan. J. Transport. Eng. 136](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0200) [\(2\), 120–128.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0200)
+- [Martin, J.-L., 2002. Relationship between crash rate and hourly traffic flow on](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0205) [interurban motorways. Accid. Anal. Prev. 34, 619–629](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0205).
+- Memon, A.Q., 2006. Road Accident Prediction Models Developed from a National Database: Poisson and Negative Binomial Regressions. Paper presented at the Transportation Research Board 85th Annual Meeting CD-ROM Washington DC, United States.
+- [Miaou, S.P., Lord, D., 2003. Modeling traffic crash-flow relationships for](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0215) intersections: dispersion parameter, [functional form, and Bayes versus](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0215)
+- [empirical Bayes. Transp. Res. Rec. 1840, 31–40.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0215) [Mitra, S., Washington, S., 2007. On the nature of over-dispersion in motor vehicle](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0220)
+- [crash prediction models. Accid. Anal. Prev. 39 \(3\), 459–468.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0220) Mitra, R., Buliung, R.N., Roorda, M.J., 2010. The built environment and school travel mode choice in Toronto, Canada. Paper presented at the 89th Annual Meeting of the Transportation Research Board, Washington, DC (January).
+- [Moeinaddini, M., Asadi-Shekari, Z., Ismail, C.R., Zaly Shah, M., 2013. A](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0230) practical method for [evaluating parking area level of service. Land Use Policy](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0230) [33, 1–10](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0230).
+- [Moeinaddini, M., Asadi-Shekari, Z., Zaly Shah, M., 2014a. The relationship between](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0235) [urban street networks and the number of transport fatalities at the city level.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0235) [Saf. Sci. 62, 114–120](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0235).
+- [Moeinaddini, M., Asadi-Shekari, Z., Zaly Shah, M., 2014b. Analysing the relationship](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0240) [between park-and-ride facilities and private motorised trips indicators. Arabian](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0240) [J. Sci. Eng. 39 \(5\), 3481–3488.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0240)
+- [Mohan, D., 2002. Work trips and safety of bicyclists. Indian J. Transport Manage.,](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0245)
+
+- Muhs, C.D., 2013. Understanding Travel Modes to Non-work Destinations: Analysis of an Establishment Survey in Portland, Oregon. (Master of Science), Portland State University.
+- Muller, S., [Tscharaktschiew, S., Haase, K., 2008. Travel-to-school mode choice](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0255) modelling and patterns [of school choice in urban areas. J. Transp. Geogr. 16,](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0255) [342–357](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0255).
+- Murray, W., 2007. Worldwide Occupational Road Safety (WORS) Review Project. Retrieved 1/1/2011, from [<http://www.cdc.gov/niosh/contract-reports/WORS/](http://www.cdc.gov/niosh/contract-reports/WORS/WORS-04-10-%202007.pdf) [WORS-04-10- 2007.pdf](http://www.cdc.gov/niosh/contract-reports/WORS/WORS-04-10-%202007.pdf)>.
+- [Naderan, A., Shahi, J., 2010. Aggregate crash prediction models: introducing crash](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0265) generation concept. Accid. [Anal. Prev. 42, 339–346.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0265)
+- National Center for Safe Routes to School., 2012. Shifting Modes: A Comparative Analysis of Safe Routes to School Program Elements and Travel Mode Outcomes: National Center for Safe Routes to School.
+- [Nevelsteena, K., Steenberghena, T., Rompaeyb, A.V., Uyttersprota, L., 2012.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0275) Controlling factors of [the parental safety perception on children's travel mode](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0275) [choice. Accid. Anal. Prev. 45, 39–49](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0275).
+- [Nguyen, L.X., Hanaoka, S., Kawasaki, T., 2014. Traffic conflict assessment for non](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0280)[lane-based movements of motorcycles under congested conditions. IATSS Res.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0280) [37, 137–147.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0280)
+- [Noland, R.B., Quddus, M.A., 2004. Improvements in medical care and technology](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0285) [and reductions in traffic-related fatalities in Great Britain. Accid. Anal. Prev. 36](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0285) [\(1\), 103–113.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0285)
+- [Noland, R.B., Quddus, M.A., 2005. Congestion and safety: a spatial analysis of](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0290) [London. Transp. Res. Part A 39, 737–754.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0290)
+- [Palma, A.d., Rochat, D., 2000. Mode choices for trips to work in Geneva: an empirical](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0295) [analysis. J. Transp. Geogr. 8, 43–51.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0295)
+- [Pooley, C.G., Turnbull, J., 2000. Modal choice and modal change: the journey to work](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0300) [in Britain since 1890. J. Transp. Geogr. 8, 11–24.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0300)
+- [Quarmby, D.A., 1967. Choice of travel mode for the journey to work. J. Transport](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0305) [Econ. Policy, 273–314 \(September\).](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0305)
+- [Quddus, M.A., Wang, C., Ison, S.G., 2010. Road traffic congestion and crash severity:](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0310) [econometric analysis using ordered response models. J. Transport. Eng. 136,](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0310) [424–435](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0310).
+- RRCGB., 2013. Reported Road Casualties in Great Britain: 2012 Annual Report: Department for Transport.
+- [Shefer, D., 1994. Congestion, air pollution, and road fatalities in urban areas. Accid.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0320) Anal. Prev. 26 [\(4\), 501–509.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0320)
+- [Shefer, D., Rietveld, P., 1997. Congestion and safety on highways: towards an](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0325) [analytical model. Urban Stud. 34 \(4\), 679–692.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0325)
+- Statistics Finland, 1997. Accidents at Work 1994: SVT Labour Market 1997. Helsinki.
+- [Vega, A., Reynolds-Feighan, A., 2009. A methodological framework for the study of](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0335) residential location and [travel-to-work mode choice under central and](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0335) [suburban employment destination patterns. Transp. Res. Part A 43, 401–419.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0335)
+- [Wang, C., Quddus, M.A., Ison, S.G., 2013. The effect of traffic and road characteristics](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0340) [on road safety: a review and future research direction. Saf. Sci. 57, 264–275](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0340).
+- Washington, S., Schalwyk, I.V., Meyer, M., Dumbaugh, E., Zoll, M., 2006. Incorporating Safety into Long-Range Transportation Planning. Washington, DC, NCHRP Report 546, TRB.
+- [Wei, F., Lovegrove, G., 2013. An empirical tool to evaluate the safety of cyclists:](http://refhub.elsevier.com/S0925-7535(14)00225-2/h9000) Community based, macro-level [collision prediction models using negative](http://refhub.elsevier.com/S0925-7535(14)00225-2/h9000) [binomial regression. Accid. Anal. Prev. 61, 129–137.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h9000)
+- [Whalen, K.E., Páez, A., Carrasco, J.A., 2013. Mode choice of university students](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0355)
+- [commuting to school and the role of active travel. J. Transp. Geogr. 31, 132–142](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0355). [Wier, M., Weintraub, J., Humphreys, E.H., Seto, E., Bhatia, R., 2009. An area-level](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0360) [model of vehicle-pedestrian injury collisions with implications for land use and](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0360) [transportation planning. Accid. Anal. Prev. 41 \(1\), 137–145.](http://refhub.elsevier.com/S0925-7535(14)00225-2/h0360)
+- World Health Organization (WHO), 2013. Global Status Report on Road Safety. Time for Action. Geneva: World Health Organization. Department of Violence & Injury Prevention & Disability.
+- Xie, C., Lu, J., Parkany, E., 2003. Work travel mode choice modeling using data mining: decision trees and neural networks. Paper presented at the 82nd Transportation Research Board Annual Meeting, Washington, DC (January 12– 16).

Zotero/001_artiklid/Road Safety Indicators/mussoneanalysisurbancollisionsusing1999.md

@@ -0,0 +1,292 @@
+---
+category: literaturenote
+citekey: mussoneanalysisurbancollisionsusing1999
+title: An analysis of urban collisions using an artificial intelligence model
+authors: "Mussone, Lorenzo; Ferrari, Andrea; Oneta, Marcello"
+year: 1999
+date: 1999-00-00 1999
+doi: 10.1016/s0001-4575(99)00031-7
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/s0001-4575(99)00031-7"
+zotero_key: 6CEGZP49
+zotero_storage: MEN8362I
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Mussone, Lorenzo"
+status: converted
+---
+model
+
+Lorenzo Mussone \*, Andrea Ferrari, Marcello Oneta *Department of Transport Systems and Mobility*, *Polytechnic of Milan*, *Piazza Leonardo da Vinci* 32, *I*-20133 *Milan*, *Italy* Received 20 July 1998; received in revised form 23 February 1999; accepted 2 March 1999
+
+Traditional studies on road accidents estimate the effect of variables (such as vehicular flows, road geometry, vehicular characteristics), and the calculation of the number of accidents. A descriptive statistical analysis of the accidents (those used in the model) over the period 1992–1995 is proposed. The paper describes an alternative method based on the use of artificial neural networks (ANN) in order to work out a model that relates to the analysis of vehicular accidents in Milan. The degree of danger of urban intersections using different scenarios is quantified by the ANN model. Methodology is the first result, which allows us to tackle the modelling of urban vehicular accidents by the innovative use of ANN. Other results deal with model outputs: intersection complexity may determine a higher accident index depending on the regulation of intersection. The highest index for running over of pedestrian occurs at non-signalised intersections at night-time. © 1999 Elsevier Science Ltd. All rights reserved.
+
+*Keywords*: Artificial neural networks; Accidents; Intersections; Safety; Urban environment; Models
+
+A widely used approach in describing accident phenomena involves the creation and calibration of a functional relationship capable of treating the above-mentioned variables. The model is calibrated by samples of accidents which determine both its structure and its forecasting capabilities. The latter one is a fundamental property which a model must possess since intervention policies, aimed at increasing road safety,
+
+The artificial neural network (ANN) model in this paper is the follow up on an approach which we have used before (Mussone and Rinelli, 1996; Ferrari et al., 1997). Though similar to traditional analyses of accidents the ANN has different capabilities and characteristics. The making of accurate predictions depends on the collection and manipulation of accident data and on the choice of variables used in the model structure. Instead of working out an analytical functional relationship, usually a complex and laborious task, the
+
+\* Corresponding author. Tel.: +39-2-2399-6612; fax: +39-2-2399-
+
+0001-4575/99/\$ - see front matter © 1999 Elsevier Science Ltd. All rights reserved.
+
+*E*-*mail address*: mussone@mail.polimi.it (L. Mussone)
+
+PII: S0001-4575(99)00031-7
+
+depend on the forecasting capabilities.
+
+![](_page_0_Picture_1.jpeg)
+
+![](_page_0_Picture_2.jpeg)
+
+neural network model reconstructs, simply by learning from the real data of accidents, the weights by which
+
+The study focuses only on accidents that occurred at intersections and not on links which would require a different approach. The data, therefore, have been desegregated in order to have a clear view of the kind and number of accidents at different types of intersections. We examine only superficially accidents on links. In fact, the aim of this paper is to identify the most significant parameters that determine the possibility of an accident occurring at an intersection. Attention has been paid to the identification of possible causes of accidents: roadway conditions; visibility; weather; and the characteristics of vehicles and drivers. On the other hand, a descriptive analysis enables us to clarify the distribution of the values of variables noted by the authorities, but this analysis does not enable us to say anything about the presumed causes of accidents, or the influence of the various factors involved and in general it is not possible to single out any relationships between the causes and effects of accidents. For this purpose it is necessary to use mathematical models in order to quantify the degree of danger linked to particular flow
+
+each variable must be weighed.
+
+and environmental conditions.
+
+An analysis of urban collisions using an artificial intelligence
+
+**Abstract**
+
+**1. Introduction**
+
+6606.
+
+![](_page_1_Picture_0.jpeg)
+
+![](_page_2_Picture_0.jpeg)
+
+![](_page_3_Picture_0.jpeg)
+
+![](_page_4_Picture_0.jpeg)
+
+![](_page_5_Picture_0.jpeg)
+
+There are several assumptions regarding the number of the lines of vehicles turning at intersections and in circulation on the main roads. It is assumed that a single line of vehicles turns 90° at an intersection. Other criteria were used to improve the database which is the basis on which the neural net model is elaborated. The purpose was to integrate the information contained in TABTOT, with the field survey of flow circulation and complexity of intersections.
+
+#### 6.2. *The NN methodology*
+
+The methodology used to build the model is based on artificial neural networks. We used feed-forward neural networks with a back-propagation learning paradigm. This type of neural network is well known and has been described in Cybenko (1989) paper and in other papers (Hornik et al., 1990; Hornik, 1991; Girosi and Poggio, 1991; Leshno et al., 1993). These papers describe the ANNs capability of approximating with minimum error any function belonging to L<sup>2</sup> space (the Lebesgue two space). Applications regarding transport, planning and control fields are numerous (Dougherty, 1995; Mussone, 1995). This approximation capability is a fascinating tool which does not need any a priori assumption on relationships between variables, linear or non linear. In problems where the phenomena are not well known and an analytical approach could be time consuming, NN offers the opportunity to investigate and create the first discriminant analysis.
+
+Essentially the methodology in building an ANN model consists of four phases. The first is a random extraction from the entire data set of a subset. The number may vary from a few 100 to 1000 cases. The subset must satisfy the law of large numbers. A very large subset may slow down the learning procedure. When data are grouped into classes, the model should recognise a particular classification pattern. Data extraction moreover should be as homogenous as possible for each class both for train and test set and an equal number of cases should be randomly extracted for each class. If not, because of the too low number of cases used which don't cover all possible combinations, ANN performance (evaluated on the test set) can worsen; this may happen when many variables are used and data are not homogeneously distributed. Therefore a lesser number of input variables can improve data extraction and case distribution between test and train set.
+
+In the second phase the data subset is randomly subdivided into two sets, the training and test sets, which will be used for the learning and validation of the ANN. The third and fourth phase consist in the learning and testing of the ANN, iteratively until the optimal network is determined. Optimal means that ANN minimises model error. The number of inputs and outputs of models does not have any theoretical limit. However the exact relationships between input/output vectors and the optimal network are not known. Therefore, it is not known how many cases are needed for the learning phase. When the number of inputs and outputs increases the complexity of the process, the more complicated networks require more data.
+
+The heuristic approach in all four phases increases the amount of computational time needed in order to work out the optimal network. If the heuristic procedure fails to recognise certain network configurations that lead to better performance, results obtained by other means may be considered valid. To build our ANN we first used the neural network toolbox in Matlab to single out the optimal network. The best ten network configurations were analysed using Neural-Works (a neural network shell) to single out the best model based on an evaluation of their performance.
+
+The structure of the ANN used to work out the model is drawn in Fig. 3. The input layer has ten neurones for eight variables: in fact the Crash variable is represented by three binary neurones (0 or 1) to consider the six types of accident (Table 3). The hidden layer has four neurones and the output has one neurone. The neurone transfer functions of the input layer are linear, hyperbolic tangent for the hidden layer and sigmoid for the output one. It is not easy to explain why the transfer functions differ from the hidden to the output layer but it is the result of the learning phase aimed at the reduction of RMSE (root mean square error). Other transfer functions such as sine were tried but without success.
+
+Each neurone belonging to the 'outer' layer represents a variable while those on the hidden layers are directly connected to the values of non physical variables, so that they haven't a physical sense. The input variables of the neural network have either a numerical value (generally normalised) or a binary code such as for the Crash variable. In the latter case, if the whole range of the possible values cannot be coded by a unique binary variable (0/1), it is necessary to use more
+
+![](_page_6_Figure_11.jpeg)
+
+Fig. 3. Structure of the model.
+
+Table 3 Characteristics of the input neurones of the model
+
+| Number | Variable name                                                 | Binary/numerical code                                                  |  |  |  |  |  |
+|--------|---------------------------------------------------------------|------------------------------------------------------------------------|--|--|--|--|--|
+| 1      | Day or night                                                  | 0, night; 1, day                                                       |  |  |  |  |  |
+| 2      | Flows circulating in the intersection                         | Numerical value (normalisation flow 10 000 [v/h])                      |  |  |  |  |  |
+| 3      | Number of virtual conflict points                             | Numerical value (normalisation value 40)                               |  |  |  |  |  |
+| 4      | Number of real conflict points                                | Numerical value (normalisation value 60)                               |  |  |  |  |  |
+| 5      | Type of intersection                                          | 0, not signalised; 1, signalised                                       |  |  |  |  |  |
+| 6      | Crash 1                                                       | 0/1                                                                    |  |  |  |  |  |
+| 7      | Crash 2                                                       | 0/1                                                                    |  |  |  |  |  |
+| 8      | Crash 3                                                       | 0/1                                                                    |  |  |  |  |  |
+|        | Type of accident (combination of the three variables crash 1, | 1-0-0, frontal impact; 1-1-0, frontal/side impact; 0-1-0, side impact; |  |  |  |  |  |
+|        | 2 and 3)                                                      | 0-0-1, bump; 0-0-0, pedestrian running over; 1-1-1, other types        |  |  |  |  |  |
+| 9      | Road bed                                                      | 0, dry; 1, wet                                                         |  |  |  |  |  |
+| 10     | Weather conditions                                            | 0, clear; 1, cloudy or rainy                                           |  |  |  |  |  |
+
+than one neurone. In general with *n* binary neurones, 2*<sup>n</sup>* values of a variable can be represented. The variables related to the ten neurones of the input layer, are described in detail in Table 3. The output layer of the model has one neurone representing the Accident Index (AI) as described by Eq. (1).
+
+The AI, that the model furnishes as an answer, represents the degree of danger associated with a particular set of variables described by the input layer. The value *N*max in Eq. (1) is the number of accidents that occurred at the most dangerous intersection of the zone under examination (69 accidents rounded up to 70). This means that, if the AI for a certain set of input values is, for instance 0.5, the degree of danger is about 35 accidents in four years. The safety of intersections with the same conditions can be estimated with a certain margin of error.
+
+It can be noted that, along with the descriptive variables in the input layer, there is also the accident index in the output layer. This is a characteristic of the type of ANN used. Back-propagation provides a solution to the problem of learning with a feed-forward ANN. This type of ANN has the capacity to reconstruct any continuous function regardless of whether the input–output sequence is known.
+
+The process of iterative learning used many sets of data that represent the real associations between the causes and effects of intersection collisions. The whole set of data is divided in two equal subsets: one for training and another for testing. The reduction in the number of variables in comparison to those contained in the TABTOT was essential for performance. By increasing the number of variables, local minima points increase. Therefore error, which is minimised by searching along an *n*-dimensional surface stopping where reliability of the results obtained by the simulations increases, can be higher. The smaller the number of variables used, the better the generalisation of the results. The loss of information of those dropped variables simplify the overall execution of the ANN.
+
+The number of cases used for the process of learning of the net is 2073 of which 1037 cases were used for the phase of training and 1036 for the testing phase. The best solution which minimises the RMSE calculated on the testing set was found after 50 000 cycles of learning; RMSE is equal to 18.24% equal to an error on AI of about 3.2 accidents per year. The course of the RMSE in relation to the number of learning cycles is shown in Fig. 4; it is also possible to notice the phenomenon of 'overfitting' that occurs when the number of learning cycles exceeds the zone of least error. An example of the test results is reported in Table 4. Each case is characterised by an input data vector, the real AI value and that obtained by the model; the last column is the error. The table shows also how the NN model can be used to obtain answers for a specific input data vector.
+
+The particular choice of the variable involves a good fit of the model to reality. It remains difficult to simulate an AI closer to the unit. This is a consequence of the 'distance' of such values from the middle value of the index in relation to all accidents. Since the ANN minimises the RMSE for all accidents simultaneously, certain isolated cases cannot be accurately estimated.
+
+## 6.3. *Application of techniques other than NN*
+
+Other techniques are applied to the same data, used in the above-mentioned model, to compare achievable performance. In particular multiple linear and exponential regressions are tested by using tools in Excel7 software. Exponential regression is in the form: *y*= *bmi xi* . where *b* is a coefficient, *xi* are the dependent variables and *mi* the coefficient calculated by the regression.
+
+The main difficulties to include into a regression model some variables arise with class variables such as day/night, intersection, road-bed, weather and, above all, with type of accident. The latter is resolved dividing data according to each class present in type of accident; this can lead to a greater computing effort but, above
+
+Table 4Examples of model outputs for some cases of the test file
+
+| Da<br>ig<br>ht<br>y o<br>r n | F<br>low<br>in<br>he<br>in-<br>t<br>ion<br>ter<br>t<br>sec<br>no<br>r-<br>l<br>ise<br>d t<br>ma<br>o<br>1<br>0<br>0<br>0<br>0<br>[ve<br>/<br>h]<br>t | V<br>irt<br>l c<br>f<br>l<br>ict<br>ua<br>on<br>int<br>po<br>s n<br>or-<br>l<br>ise<br>d t<br>4<br>0<br>ma<br>o | Re<br>l c<br>f<br>l<br>ict<br>a<br>on<br>int<br>po<br>s n<br>or-<br>l<br>ise<br>d t<br>6<br>0<br>ma<br>o | Ty<br>f<br>int<br>pe<br>o<br>er-<br>ion<br>t<br>sec | Ty<br>f a<br>i-<br>pe<br>o<br>cc<br>de<br>nt               | Ro<br>d s<br>fac<br>a<br>ur<br>e<br>d<br>it<br>ion<br>co<br>n<br>s | M<br>log<br>ica<br>l<br>ete<br>oro<br>d<br>it<br>ion<br>co<br>n<br>s | Ac<br>l n<br>be<br>tua<br>um<br>r<br>f a<br>i<br>de<br>nt<br>o<br>cc<br>no<br>r-<br>l<br>ise<br>d t<br>7<br>0<br>ma<br>o | Ca<br>lcu<br>lat<br>d n<br>e<br>um<br>-<br>be<br>f a<br>i<br>de<br>nts<br>r o<br>cc<br>l<br>ise<br>d t<br>no<br>rm<br>a<br>o<br>7<br>0 | Er<br>ror              |
+|------------------------------|------------------------------------------------------------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------------------|-----------------------------------------------------|------------------------------------------------------------|--------------------------------------------------------------------|----------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------------------------------------------------|------------------------|
+| Da<br>y                      | 0.<br>4<br>6<br>1                                                                                                                                    | 0.<br>2<br>25                                                                                                   | 0.<br>0<br>5<br>0                                                                                        | ign<br>l<br>ise<br>d<br>S<br>a                      | l<br>/s<br>i<br>de<br>Fr<br>ta<br>on                       | Dr<br>y                                                            | lea<br>C<br>r                                                        | 0.<br>2<br>2<br>9                                                                                                        | 0.<br>2<br>3<br>7<br>8                                                                                                                 | 0.<br>0<br>0<br>8<br>8 |
+| Da<br>y                      | 0.<br>6<br>7<br>4                                                                                                                                    | 0.<br>17<br>5                                                                                                   | 0.<br>0<br>5<br>0                                                                                        | ign<br>l<br>ise<br>d<br>S<br>a                      | l<br>/s<br>i<br>de<br>Fr<br>ta<br>on                       | Dr<br>y                                                            | lea<br>C<br>r                                                        | 0.<br>1<br>2<br>9                                                                                                        | 0.<br>1<br>2<br>6<br>9                                                                                                                 | −0<br>0<br>0<br>2<br>1 |
+| ig<br>ht<br>N                | 1<br>8<br>1<br>0.                                                                                                                                    | 0.<br>2<br>25                                                                                                   | 0.<br>0<br>5<br>0                                                                                        | S<br>ign<br>l<br>ise<br>d<br>a                      | Ot<br>he<br>r                                              | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>25<br>7                                                                                                            | 6<br>0.<br>2<br>3<br>0                                                                                                                 | 6<br>0.<br>0<br>0<br>0 |
+| ig<br>ht<br>N                | 0.<br>1<br>8<br>6                                                                                                                                    | 0.<br>2<br>25                                                                                                   | 0.<br>0<br>0<br>5                                                                                        | S<br>ign<br>l<br>ise<br>d<br>a                      | Ot<br>he<br>r                                              | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>27<br>1                                                                                                            | 0.<br>2<br>6<br>2<br>1                                                                                                                 | −0<br>0<br>0<br>8<br>9 |
+| Da<br>y                      | 0.<br>6<br>17                                                                                                                                        | 0.<br>3<br>7<br>5                                                                                               | 0.<br>0<br>6<br>7                                                                                        | S<br>ign<br>l<br>ise<br>d<br>a                      | l<br>/s<br>i<br>de<br>Fr<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>45<br>7                                                                                                            | 0.<br>4<br>6<br>6<br>1                                                                                                                 | 0.<br>0<br>0<br>9<br>1 |
+| Da<br>y                      | 0.<br>3<br>4<br>3                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>0<br>8<br>3                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Pe<br>de<br>ian<br>str<br>ru<br>n-<br>ing<br>n<br>ov<br>er | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>1<br>1<br>4                                                                                                        | 0.<br>1<br>1<br>8<br>4                                                                                                                 | 0<br>0<br>0<br>4<br>4  |
+| Da<br>y                      | 0.<br>1<br>8<br>8                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>0<br>8<br>3                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Pe<br>de<br>ian<br>str<br>ru<br>n-<br>ing<br>n<br>ov<br>er | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>1<br>2<br>9                                                                                                        | 0.<br>1<br>2<br>8<br>9                                                                                                                 | −0<br>0<br>0<br>0<br>1 |
+| Da<br>y                      | 0.<br>9<br>6<br>4                                                                                                                                    | 0.<br>25<br>0                                                                                                   | 0.<br>0<br>8<br>3                                                                                        | S<br>ign<br>l<br>ise<br>d<br>a                      | Bu<br>mp                                                   | W<br>et                                                            | C<br>lea<br>r                                                        | 0.<br>1<br>0<br>0                                                                                                        | 0<br>0<br>9<br>6<br>4                                                                                                                  | −0<br>0<br>0<br>3<br>6 |
+| Da<br>y                      | 0.<br>2<br>2<br>9                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>0<br>8<br>3                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Bu<br>mp                                                   | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>1<br>1<br>4                                                                                                        | 0.<br>1<br>2<br>0<br>6                                                                                                                 | 0.<br>0<br>0<br>6<br>6 |
+| Da<br>y                      | 0.<br>0<br>1<br>3                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>0<br>8<br>3                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Bu<br>mp                                                   | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>1<br>1<br>4                                                                                                        | 0.<br>1<br>2<br>3<br>0                                                                                                                 | 0.<br>0<br>0<br>9<br>0 |
+| Da<br>y                      | 0.<br>2<br>8<br>8                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>0<br>8<br>3                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | S<br>i<br>de                                               | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>1<br>1<br>4                                                                                                        | 0.<br>1<br>1<br>3<br>7                                                                                                                 | −0<br>0<br>0<br>0<br>3 |
+| Da<br>y                      | 0.<br>15<br>7                                                                                                                                        | 0.<br>0<br>0<br>0                                                                                               | 0.<br>0<br>8<br>3                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Fr<br>l<br>ta<br>on                                        | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>15<br>7                                                                                                            | 0.<br>1<br>6<br>25                                                                                                                     | 0.<br>0<br>0<br>5<br>5 |
+| ig<br>N<br>ht                | 0.<br>15<br>6                                                                                                                                        | 0.<br>0<br>0<br>0                                                                                               | 0.<br>0<br>8<br>3                                                                                        | ign<br>ise<br>No<br>l<br>d<br>t s<br>a              | i<br>Fr<br>l<br>/s<br>de<br>ta<br>on                       | W<br>et                                                            | C<br>lea<br>r                                                        | 0.<br>1<br>4<br>3                                                                                                        | 0.<br>1<br>3<br>7<br>7                                                                                                                 | 0.<br>0<br>0<br>5<br>3 |
+| Da<br>y                      | 0.<br>2<br>3<br>8                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>0<br>8<br>3                                                                                        | ign<br>l<br>ise<br>d<br>No<br>t s<br>a              | l<br>i<br>de<br>Fr<br>/s<br>ta<br>on                       | Dr<br>y                                                            | lea<br>C<br>r                                                        | 0.<br>1<br>4<br>3                                                                                                        | 0.<br>1<br>4<br>17                                                                                                                     | −0<br>0<br>0<br>1<br>3 |
+| Da<br>y                      | 0.<br>1<br>2<br>2                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>0<br>8<br>3                                                                                        | ign<br>l<br>ise<br>d<br>No<br>t s<br>a              | l<br>/s<br>i<br>de<br>Fr<br>ta<br>on                       | Dr<br>y                                                            | lea<br>C<br>r                                                        | 0.<br>15<br>7                                                                                                            | 0.<br>15<br>6<br>4                                                                                                                     | −0<br>0<br>0<br>0<br>6 |
+| Da<br>y                      | 0.<br>2<br>9<br>9                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>0<br>8<br>3                                                                                        | ign<br>l<br>ise<br>d<br>No<br>t s<br>a              | l<br>/s<br>i<br>de<br>Fr<br>ta<br>on                       | Dr<br>y                                                            | lea<br>C<br>r                                                        | 0.<br>1<br>4<br>3                                                                                                        | 0.<br>1<br>3<br>4<br>9                                                                                                                 | −0<br>0<br>0<br>8<br>1 |
+| Da<br>y                      | 0.<br>6<br>9<br>5                                                                                                                                    | 0.<br>4<br>25                                                                                                   | 0.<br>1<br>0<br>0                                                                                        | S<br>ign<br>l<br>ise<br>d<br>a                      | l<br>/s<br>i<br>de<br>Fr<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>47<br>1                                                                                                            | 0.<br>47<br>4<br>2                                                                                                                     | 0.<br>0<br>0<br>3<br>2 |
+| Da<br>y                      | 0.<br>6<br>9<br>5                                                                                                                                    | 0.<br>4<br>25                                                                                                   | 0.<br>1<br>0<br>0                                                                                        | S<br>ign<br>l<br>ise<br>d<br>a                      | de<br>ian<br>Pe<br>str<br>ru<br>n-<br>ing<br>n<br>ov<br>er | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>47<br>1                                                                                                            | 0.<br>4<br>6<br>47                                                                                                                     | −0<br>0<br>0<br>6<br>3 |
+| Da<br>y                      | 0.<br>7<br>5<br>0                                                                                                                                    | 0.<br>27<br>5                                                                                                   | 0.<br>1<br>0<br>0                                                                                        | S<br>ign<br>l<br>ise<br>d<br>a                      | Fr<br>l<br>ta<br>on                                        | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>2<br>8<br>6                                                                                                        | 0.<br>2<br>9<br>3<br>9                                                                                                                 | 0.<br>0<br>0<br>7<br>9 |
+| Da<br>y                      | 0.<br>2<br>6<br>9                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>1<br>0<br>0                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Fr<br>l<br>ta<br>on                                        | W<br>et                                                            | C<br>lou<br>dy<br>/ra<br>iny                                         | 0.<br>1<br>4<br>3                                                                                                        | 0.<br>1<br>4<br>8<br>4                                                                                                                 | 0.<br>0<br>0<br>5<br>4 |
+| Da<br>y                      | 0.<br>6<br>8<br>2                                                                                                                                    | 0.<br>4<br>25                                                                                                   | 0.<br>1<br>0<br>0                                                                                        | S<br>ign<br>l<br>ise<br>d<br>a                      | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>47<br>1                                                                                                            | 0.<br>47<br>3<br>8                                                                                                                     | 0.<br>0<br>0<br>2<br>8 |
+| N<br>ig<br>ht                | 0.<br>0<br>1<br>0                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>1<br>0<br>0                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>1<br>8<br>6                                                                                                        | 0.<br>1<br>8<br>9<br>3                                                                                                                 | 0.<br>0<br>0<br>3<br>3 |
+| N<br>ig<br>ht                | 0.<br>0<br>0<br>7                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>1<br>0<br>0                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>1<br>8<br>6                                                                                                        | 0.<br>1<br>8<br>9<br>8                                                                                                                 | 0.<br>0<br>0<br>3<br>8 |
+| N<br>ig<br>ht                | 0.<br>1<br>0<br>4                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>1<br>0<br>0                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | W<br>et                                                            | C<br>lou<br>dy<br>/ra<br>iny                                         | 0.<br>17<br>1                                                                                                            | 0.<br>17<br>27                                                                                                                         | 0.<br>0<br>0<br>17     |
+| Da<br>y                      | 0.<br>2<br>9<br>9                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>1<br>0<br>0                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>1<br>4<br>3                                                                                                        | 0.<br>1<br>3<br>5<br>7                                                                                                                 | −0<br>0<br>0<br>7<br>3 |
+| Da<br>y                      | 0.<br>47<br>1                                                                                                                                        | 0.<br>0<br>0<br>0                                                                                               | 0.<br>1<br>0<br>0                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lou<br>dy<br>/ra<br>iny                                         | 0.<br>1<br>2<br>9                                                                                                        | 0.<br>1<br>3<br>8<br>5                                                                                                                 | 0.<br>0<br>0<br>9<br>5 |
+| N<br>ig<br>ht                | 0.<br>0<br>0<br>4                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>1<br>0<br>0                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>2<br>0<br>0                                                                                                        | 0.<br>1<br>9<br>0<br>3                                                                                                                 | −0<br>0<br>0<br>9<br>7 |
+| Da<br>y                      | 0.<br>6<br>8<br>2                                                                                                                                    | 0.<br>4<br>25                                                                                                   | 0.<br>1<br>0<br>0                                                                                        | ign<br>l<br>ise<br>d<br>S<br>a                      | l<br>i<br>de<br>Fr<br>/s<br>ta<br>on                       | Dr<br>y                                                            | lea<br>C<br>r                                                        | 0.<br>47<br>1                                                                                                            | 0.<br>47<br>3<br>8                                                                                                                     | 0.<br>0<br>0<br>2<br>8 |
+| Da<br>y                      | 0.<br>2<br>3<br>4                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>1<br>0<br>0                                                                                        | ign<br>l<br>ise<br>d<br>No<br>t s<br>a              | l<br>/s<br>i<br>de<br>Fr<br>ta<br>on                       | Dr<br>y                                                            | lea<br>C<br>r                                                        | 0.<br>1<br>4<br>3                                                                                                        | 0.<br>1<br>4<br>3<br>1                                                                                                                 | 0.<br>0<br>0<br>0<br>1 |
+| ig<br>ht<br>N                | 0.<br>1<br>4<br>2                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>1<br>0<br>0                                                                                        | ign<br>l<br>ise<br>d<br>No<br>t s<br>a              | l<br>/s<br>i<br>de<br>Fr<br>ta<br>on                       | Dr<br>y                                                            | lea<br>C<br>r                                                        | 0.<br>17<br>1                                                                                                            | 0.<br>1<br>6<br>8<br>3                                                                                                                 | −0<br>0<br>0<br>27     |
+| ig<br>ht<br>N                | 1<br>0.<br>0<br>3                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 1<br>0.<br>0<br>0                                                                                        | ign<br>l<br>ise<br>d<br>No<br>t s<br>a              | l<br>/s<br>i<br>de<br>Fr<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 1<br>8<br>6<br>0.                                                                                                        | 1<br>8<br>8<br>9<br>0.                                                                                                                 | 9<br>0.<br>0<br>0<br>2 |
+
+| b<br>le<br>4<br>(<br>Co<br>inu<br>d<br>)<br>Ta<br>nt<br>e |                                                                                                                                                      |                                                                                                                 |                                                                                                          |                                                     |                                                            |                                                                    |                                                                      |                                                                                                                          |                                                                                                                                        |                        |
+|-----------------------------------------------------------|------------------------------------------------------------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------------------|-----------------------------------------------------|------------------------------------------------------------|--------------------------------------------------------------------|----------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------------------------------------------------|------------------------|
+| Da<br>ig<br>ht<br>y o<br>r n                              | F<br>low<br>in<br>he<br>in-<br>t<br>ion<br>ter<br>t<br>sec<br>no<br>r-<br>l<br>ise<br>d t<br>ma<br>o<br>1<br>0<br>0<br>0<br>0<br>[ve<br>/<br>h]<br>t | V<br>irt<br>l c<br>f<br>l<br>ict<br>ua<br>on<br>int<br>po<br>s n<br>or-<br>l<br>ise<br>d t<br>4<br>0<br>ma<br>o | Re<br>l c<br>f<br>l<br>ict<br>a<br>on<br>int<br>po<br>s n<br>or-<br>l<br>ise<br>d t<br>6<br>0<br>ma<br>o | Ty<br>f<br>int<br>pe<br>o<br>er-<br>ion<br>t<br>sec | Ty<br>f a<br>i-<br>pe<br>o<br>cc<br>de<br>nt               | Ro<br>d s<br>fac<br>a<br>ur<br>e<br>d<br>it<br>ion<br>co<br>n<br>s | M<br>log<br>ica<br>l<br>ete<br>oro<br>d<br>it<br>ion<br>co<br>n<br>s | Ac<br>l n<br>be<br>tua<br>um<br>r<br>f a<br>i<br>de<br>nt<br>o<br>cc<br>no<br>r-<br>l<br>ise<br>d t<br>7<br>0<br>ma<br>o | Ca<br>lcu<br>lat<br>d n<br>e<br>um<br>-<br>be<br>f a<br>i<br>de<br>nts<br>r o<br>cc<br>l<br>ise<br>d t<br>no<br>rm<br>a<br>o<br>7<br>0 | Er<br>ror              |
+| Da<br>y                                                   | 0.<br>2<br>25                                                                                                                                        | 0.<br>0<br>0<br>0                                                                                               | 0.<br>1<br>0<br>0                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>1<br>4<br>3                                                                                                        | 0.<br>1<br>4<br>4<br>1                                                                                                                 | 0.<br>0<br>0<br>1<br>1 |
+| Da<br>y                                                   | 0.<br>5<br>3<br>1                                                                                                                                    | 0.<br>4<br>25                                                                                                   | 0.<br>1<br>0<br>0                                                                                        | S<br>ign<br>l<br>ise<br>d<br>a                      | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>47<br>1                                                                                                            | 0.<br>4<br>6<br>9<br>9                                                                                                                 | −0<br>0<br>0<br>1<br>1 |
+| Da<br>y                                                   | 0.<br>4<br>0<br>2                                                                                                                                    | 0.<br>1<br>0<br>0                                                                                               | 0.<br>1<br>3<br>3                                                                                        | S<br>ign<br>l<br>ise<br>d<br>a                      | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>15<br>7                                                                                                            | 0.<br>1<br>6<br>5<br>7                                                                                                                 | 0.<br>0<br>0<br>8<br>6 |
+| ig<br>N<br>ht                                             | 0.<br>3<br>4<br>0                                                                                                                                    | 0.<br>1<br>0<br>0                                                                                               | 0.<br>1<br>3<br>3                                                                                        | ign<br>ise<br>No<br>l<br>d<br>t s<br>a              | i<br>Fr<br>l<br>/s<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>15<br>7                                                                                                            | 0.<br>15<br>2<br>2                                                                                                                     | −0<br>0<br>0<br>4<br>8 |
+| Da<br>y                                                   | 0.<br>3<br>8<br>1                                                                                                                                    | 0.<br>1<br>0<br>0                                                                                               | 0.<br>1<br>3<br>3                                                                                        | ign<br>l<br>ise<br>d<br>S<br>a                      | l<br>i<br>de<br>Fr<br>/s<br>ta<br>on                       | W<br>et                                                            | lou<br>dy<br>iny<br>C<br>/ra                                         | 0.<br>15<br>7                                                                                                            | 0.<br>1<br>6<br>0<br>0                                                                                                                 | 0.<br>0<br>0<br>3<br>0 |
+| Da<br>y                                                   | 0.<br>3<br>8<br>1                                                                                                                                    | 0.<br>1<br>0<br>0                                                                                               | 0.<br>1<br>3<br>3                                                                                        | ign<br>l<br>ise<br>d<br>S<br>a                      | l<br>/s<br>i<br>de<br>Fr<br>ta<br>on                       | W<br>et                                                            | lou<br>dy<br>/ra<br>iny<br>C                                         | 0.<br>15<br>7                                                                                                            | 0.<br>1<br>6<br>0<br>0                                                                                                                 | 0.<br>0<br>0<br>3<br>0 |
+| Da<br>y                                                   | 0.<br>1<br>1<br>9                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>15<br>0                                                                                            | ign<br>l<br>ise<br>d<br>No<br>t s<br>a              | de<br>ian<br>Pe<br>str<br>ru<br>n-<br>ing<br>n<br>ov<br>er | Dr<br>y                                                            | lea<br>C<br>r                                                        | 0.<br>1<br>2<br>9                                                                                                        | 0.<br>1<br>3<br>8<br>3                                                                                                                 | 0.<br>0<br>0<br>9<br>3 |
+| Da<br>y                                                   | 0.<br>1<br>2<br>0                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>15<br>0                                                                                            | ign<br>l<br>ise<br>d<br>No<br>t s<br>a              | de<br>ian<br>Pe<br>str<br>ru<br>n-<br>ing<br>n<br>ov<br>er | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>1<br>4<br>3                                                                                                        | 0.<br>1<br>3<br>8<br>2                                                                                                                 | 0.<br>0<br>0<br>4<br>8 |
+| N<br>ig<br>ht                                             | 0.<br>27<br>3                                                                                                                                        | 0.<br>5<br>25                                                                                                   | 0.<br>15<br>0                                                                                            | S<br>ign<br>l<br>ise<br>d<br>a                      | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>3<br>5<br>7                                                                                                        | 0.<br>3<br>4<br>8<br>7                                                                                                                 | −0<br>0<br>0<br>8<br>3 |
+| Da<br>y                                                   | 0.<br>17<br>5                                                                                                                                        | 0.<br>0<br>0<br>0                                                                                               | 0.<br>1<br>6<br>7                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>15<br>7                                                                                                            | 0.<br>15<br>7<br>1                                                                                                                     | 0.<br>0<br>0<br>0<br>1 |
+| Da<br>y                                                   | 0.<br>1<br>6<br>6                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>1<br>6<br>7                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>15<br>7                                                                                                            | 0.<br>15<br>8<br>4                                                                                                                     | 0.<br>0<br>0<br>1<br>4 |
+| Da<br>y                                                   | 0.<br>17<br>1                                                                                                                                        | 0.<br>0<br>0<br>0                                                                                               | 0.<br>1<br>8<br>3                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Pe<br>de<br>ian<br>str<br>ru<br>n-<br>ing<br>n<br>ov<br>er | W<br>et                                                            | C<br>lou<br>dy<br>/ra<br>iny                                         | 0.<br>1<br>2<br>9                                                                                                        | 0.<br>1<br>3<br>0<br>9                                                                                                                 | 0.<br>0<br>0<br>1<br>9 |
+| Da<br>y                                                   | 0.<br>7<br>0<br>1                                                                                                                                    | 0.<br>5<br>7<br>5                                                                                               | 0.<br>1<br>8<br>3                                                                                        | S<br>ign<br>l<br>ise<br>d<br>a                      | Bu<br>mp                                                   | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>3<br>1<br>4                                                                                                        | 0.<br>3<br>1<br>3<br>8                                                                                                                 | −0<br>0<br>0<br>0<br>2 |
+| N<br>ig<br>ht                                             | 0.<br>7<br>8<br>2                                                                                                                                    | 0.<br>5<br>7<br>5                                                                                               | 0.<br>1<br>8<br>3                                                                                        | S<br>ign<br>l<br>ise<br>d<br>a                      | Bu<br>mp                                                   | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>3<br>1<br>4                                                                                                        | 0.<br>3<br>1<br>3<br>7                                                                                                                 | 0.<br>0<br>0<br>0<br>3 |
+| Da<br>y                                                   | 0.<br>2<br>3<br>4                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>1<br>8<br>3                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | S<br>i<br>de                                               | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>1<br>1<br>4                                                                                                        | 0.<br>1<br>2<br>0<br>8                                                                                                                 | 0.<br>0<br>0<br>8<br>8 |
+| Da<br>y                                                   | 0.<br>3<br>4<br>4                                                                                                                                    | 0.<br>5<br>0<br>0                                                                                               | 0.<br>2<br>0<br>0                                                                                        | S<br>ign<br>l<br>ise<br>d<br>a                      | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>4<br>2<br>9                                                                                                        | 0.<br>4<br>2<br>15                                                                                                                     | −0<br>0<br>0<br>7<br>5 |
+| Da<br>y                                                   | 0.<br>3<br>6<br>4                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>2<br>0<br>0                                                                                        | ign<br>l<br>ise<br>d<br>No<br>t s<br>a              | l<br>i<br>de<br>Fr<br>/s<br>ta<br>on                       | Dr<br>y                                                            | lea<br>C<br>r                                                        | 0.<br>1<br>4<br>3                                                                                                        | 0.<br>1<br>3<br>6<br>7                                                                                                                 | −0<br>0<br>0<br>6<br>3 |
+| Da<br>y                                                   | 0.<br>3<br>5<br>7                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>2<br>0<br>0                                                                                        | ign<br>l<br>ise<br>d<br>No<br>t s<br>a              | l<br>i<br>de<br>Fr<br>/s<br>ta<br>on                       | Dr<br>y                                                            | lea<br>C<br>r                                                        | 0.<br>1<br>4<br>3                                                                                                        | 0.<br>1<br>3<br>7<br>5                                                                                                                 | −0<br>0<br>0<br>5<br>5 |
+| Da<br>y                                                   | 0.<br>3<br>3<br>2                                                                                                                                    | 0.<br>5<br>0<br>0                                                                                               | 0.<br>2<br>0<br>0                                                                                        | ign<br>l<br>ise<br>d<br>S<br>a                      | l<br>/s<br>i<br>de<br>Fr<br>ta<br>on                       | Dr<br>y                                                            | lea<br>C<br>r                                                        | 0.<br>4<br>2<br>9                                                                                                        | 0.<br>4<br>1<br>9<br>8                                                                                                                 | 0.<br>0<br>0<br>9<br>2 |
+| Da<br>y                                                   | 9<br>0.<br>0<br>0                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 17<br>0.<br>2                                                                                            | ign<br>l<br>ise<br>d<br>No<br>t s<br>a              | l<br>/s<br>i<br>de<br>Fr<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 1<br>8<br>6<br>0.                                                                                                        | 1<br>8<br>0.<br>7<br>4                                                                                                                 | 1<br>0.<br>0<br>0<br>4 |
+| Da<br>y                                                   | 0.<br>1<br>0<br>2                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>2<br>17                                                                                            | ign<br>l<br>ise<br>d<br>No<br>t s<br>a              | l<br>/s<br>i<br>de<br>Fr<br>ta<br>on                       | W<br>et                                                            | C<br>lou<br>dy<br>/ra<br>iny                                         | 0.<br>1<br>8<br>6                                                                                                        | 0.<br>1<br>8<br>2<br>9                                                                                                                 | −0<br>0<br>0<br>3<br>1 |
+| Da<br>y                                                   | 0.<br>1<br>1<br>3                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>2<br>17                                                                                            | ign<br>l<br>ise<br>d<br>No<br>t s<br>a              | l<br>/s<br>i<br>de<br>Fr<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>1<br>8<br>6                                                                                                        | 0.<br>1<br>8<br>25                                                                                                                     | 0.<br>0<br>0<br>3<br>5 |
+| Da<br>y                                                   | 0.<br>0<br>9<br>5                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>2<br>17                                                                                            | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>1<br>8<br>6                                                                                                        | 0.<br>1<br>8<br>6<br>3                                                                                                                 | 0.<br>0<br>0<br>0<br>3 |
+| Da<br>y                                                   | 0.<br>1<br>9<br>6                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>2<br>17                                                                                            | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>17<br>1                                                                                                            | 0.<br>1<br>6<br>6<br>0                                                                                                                 | 0.<br>0<br>0<br>5<br>0 |
+| Da<br>y                                                   | 0.<br>3<br>3<br>0                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>2<br>8<br>3                                                                                        | No<br>ign<br>l<br>ise<br>d<br>t s<br>a              | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | Dr<br>y                                                            | C<br>lea<br>r                                                        | 0.<br>17<br>1                                                                                                            | 0.<br>17<br>6<br>6                                                                                                                     | 0.<br>0<br>0<br>5<br>6 |
+| Da<br>y                                                   | 0.<br>4<br>0<br>6                                                                                                                                    | 0.<br>0<br>0<br>0                                                                                               | 0.<br>8<br>6<br>7                                                                                        | S<br>ign<br>l<br>ise<br>d<br>a                      | Fr<br>l<br>/s<br>i<br>de<br>ta<br>on                       | W<br>et                                                            | C<br>lou<br>dy<br>/ra<br>iny                                         | 0.<br>5<br>5<br>7                                                                                                        | 0.<br>5<br>5<br>3<br>1                                                                                                                 | −0<br>0<br>0<br>3<br>9 |
+
+all, in some cases to a reduction of statistical significance. The remaining variables are transformed into a binary variable. For example, the day/night variable is transformed into two variables: day becomes a variable which is 1 when the accident occurred in the daytime, and night becomes another variable which is 1 when the accident occurred at night time. Output is the accident index as previously calculated in Eq. (1). Only some results are reported but they are sufficient to evaluate model performance.
+
+Multiple linear regression, applied to the sets of data, gives an *r* <sup>2</sup> (*r* is the coefficient of correlation) ranging from 0.20 to 0.40. RMSE (root mean square error) calculated on real data and data calculated by the regression model ranges from 0.5 to 0.6. In some cases when output must be close to zero, the regression model gives negative values, too.
+
+Multiple exponential regression gives an *r* <sup>2</sup> ranging from 0.17 to 0.41. RMSE calculated on real data and data calculated by of the regression model ranges from 0.5 to 0.7.
+
+Therefore, considering the low level of these results no further analysis was carried out using these methods.
+
+#### **7. Results and discussion**
+
+The ANN model can include all possible permutations of variables, discrete or continuous. For continuous variables a classification is carried out only in order to reduce the number of possible values and to lessen computational time. As previously mentioned, Table 4 represents a set of possible enquires and outputs. If these enquires are organised by type, certain particular graphs can be obtained. Some of these examples are drawn in Figs. 5 and 6.
+
+The following results are most significant for the discussion:
+
+- 1. the degree of danger is the highest in night-time collisions, for any type of accident, at signalised intersections;
+- 2. the degree of danger is the highest, with respect to all types of accident, for the running over of pedestrians at night-time at non-signalised intersections;
+- 3. meteorological conditions and the state of the road surface are not determinant in the number of the accidents of each intersection; that is, at the level of analysis of a single intersection there is no statistically relevant correlation between the accident index and meteorological conditions or the state of the road surface but this doesn't mean that grouping together the accident data of some intersections a significant relationship cannot be singled out;
+- 4. the accident index is greater for an intersection of average complexity (number of real and virtual points) if not signalised, compared to a signalised one with the same complexity; on the other hand, for signalised intersections of average complexity at night-time it is smaller than at non-signalised ones or ones with flashing yellow intersections;
+
+![](_page_10_Figure_14.jpeg)
+
+Fig. 4. Learning cycles necessary to minimise the root mean square error.
+
+![](_page_11_Figure_2.jpeg)
+
+Fig. 5. Accident Index for different types of intersection and regulation.
+
+- 5. the accident index is greater for an intersection of average dimensions compared to more or less complex ones;
+- 6. the accident index is greater for a small intersection if signalised compared to a small intersection that is not signalised;
+- 7. real conflict points (that do not depend on trafficlights) are more important than virtual conflict points;
+- 8. the accident index of a single intersection does not depend on accident type;
+- 9. the rate of increase of average flow at a single intersection has no influence on the accident index; this means that in average in the same intersection accidents occurred both with high and low flow so that no rule can be singled out. It can usually be observed that by increasing the average flow (hourly flow for example) on a road network the number of accidents increases; but it is rather difficult to understand whether the accidents occurred with precisely those flow conditions or not.
+
+In general the ability of the ANN to simulate road accidents can be attributed, like any other deterministic or statistical models, to the following factors:
+
+1. the choice and number of the variables used. The ANN requires a classification of the variables to be used. The choice is both theoretical (supported by analogous studies already conducted), and is linked to the availability of data already gathered. For instance, it may be easy to survey an intersection or obtain data on climatic conditions, but the same cannot be said for many human factors. The reduction of the number of variables used to describe the problem leads to an increase in the potential error of the model. Simple ANNs cannot fully explain multi-dimensional phenomena.
+
+- 2. data collection factors. Once the variables for the ANN are chosen, the collection of these can frequently introduce other biases and errors. The data collection errors impinge in an unpredictable manner on the ANN model results;
+- 3. stochastic aspect of the phenomenon. Though all the variables necessary to describe the process are employed and collected with a minimal degree of bias, the deterministic nature of the ANN does not assure irrelevance of model error when variables have a great stochastic component.
+
+This last factor needs further clarification. When an ANN model is used it learns from realty expressed by the use of real data. The accident index associated with the particular set of values of variables that describe the accident is heavily influenced by the distribution of values in the actual set of accidents the ANN receives for the learning process. If a particular value of one variable in the learning set has a high accident index value the ANN will assign a higher link to this factor than to one with a lower value. Whether this link is randomly high or low the ANN will output an average value.
+
+In order to solve a statistical problem of large values, it is necessary to make an appropriate extraction from real data to guarantee an equal distribution to all factors. For this purpose a database that represents every learning process should be built which contains a uniform distribution for the different values of the most meaningful variables.
+
+For example, accidents that occurred in daytime are around 70% of the total. The consequence of this imbalance may be that a greater index is associated by the model to daytime cases. Regardless of whether a uniform database was constituted, the ANN classifies the most dangerous conditions only at those places where crashes most frequently occur. Such a selection of data would lead to a reduction of the available sets of real data for learning and consequently a reduced capacity to understand the nature of collisions. For the purposes of this paper it was not possible to satisfy a similar requirement because it would have been necessary to increase the number of intersections.
+
+The fact that the model does not work perfectly (that is the variance the model actually explains better than stochastic character of the output as shown in Section 6.2) is due to the possibility that other variables should have been used. Intuitively this effect may be more serious in those cases exhibiting accident indexes with low values. This explains the apparently random nature of certain situations. For instance, during the nighttimes of festive occasions elevated speeds are adopted by young drivers. Unfortunately speed and flow data were not collected for all collisions at all times. The absence of these variables has contributed to making the daytime cases less 'clear'. They are more linked to a factor that varies in ways our ANN cannot understand. All these aspects help to explain some of the discrepancies between real values and those estimated by our ANN.
+
+Another way of testing the validity of results is to apply the findings to other situations. This is to establish whether a certain model, trained with a certain set of data, is capable of carrying out reliable solutions for contexts different from those where data are collected. Generally the greater the quantity of data used for learning, the better the possibility of generalising the results and extending them to other urban contexts. It is necessary to emphasise, however, that if we want to extend the results of the simulations on a model 'educated' with data from a certain urban context, the similarity of the contexts must be evaluated since:
+
+- 1. the association of the same variables do not necessarily give rise to the same results when a different context is examined; it is probable that in another city or environment the same variable may have a different weight as regards the actual number of accidents at intersections;
+- 2. not using variables considered not fundamental for the designing of the model in some contexts, could lead to errors in model of other places. If, for instance driving in rainy weather represents a normal condition in one city, the same ANN cannot be used for a city where rain is rare.
+
+If we want to extend our generalisation to other urban realities the results obtained by the ANN used in a specific context, attention should be paid to the choice of a 'similar' context. This applies not only to the values of variables used in the ANN model, but also in terms of those variables that were discarded during the learning phase. Another approach is to create a large database containing descriptions of all possible cases. Such a database is more likely to have a similar distribution of values across variables.
+
+#### **8. Conclusions**
+
+To understand which of the three subset factors, human, vehicular or road, is the most important and
+
+![](_page_12_Figure_11.jpeg)
+
+Fig. 6. Accident Index for intersections of average dimensions and not signalised at night-time according to the type of accident.
+
+which one it is necessary to intervene on, is not a simple task. We need to create tools of investigation capable of answering this question with a reasonably low margin of error.
+
+Our paper shows one method to follow in the use of the ANN. Thanks to the clarity of the results, in terms of which factors explain accidents and which factors contribute to a higher degree of danger, the capability of extracting information from data, exhibited by neural networks, makes the proposed methodology of analysis still more interesting, above all for the possibility of using databases that already exist or have been built for other purposes.
+
+It is necessary to continue work on these types of models, to tackle and solve problems connected to the distribution of data with a low average value. In order to make this factor less preponderant, we suggest a procedure that envisages an extraction of preliminary data (for example by a cluster analysis) so as to improve the training phase of ANNs.
+
+Another problem not yet resolved in technical papers concerns the different recurrence of phenomena which may cause an accident; this can lead to the collection data that are not homogenous in time and affect results in an unpredictable way. To avoid these difficulties studies on driver behaviour and peculiarities have already been carried out in order to understand what is the best (minimum) sample interval for the various accident scenarios.
+
+The use of the accident index to compare the degree of danger of the intersections may be criticised. It is a macroscopic value which could lead to a misunderstanding of microscopic effects occurring at the intersection.
+
+# **Acknowledgements**
+
+Thanks are due to the City Hall for providing data and other help. This paper is partially funded by CNR (Murst 40%).
+
+### **References**
+
+- Cybenko, G., 1989. Approximation by superpositions of sigmoidal functions. Math. Control Signals Syst. 2, 303–314.
+- Dionne, G., Desjardins, D., Laverge-Nadeau, C., Maag, U., 1993. Medical conditions, risk exposure and truck drivers' accidents: an analysis with regression models. The 37th Annual Meeting of The Association for the Advancement of Automotive Medicine, San Antonio, Texas.
+
+- Dougherty, M., 1995. A review of neural networks applied to trasnport. Transpn. Res. 3 (4), 247–260.
+- Ferrari A., Mussone L., Oneta M., 1997. Evaluation of accident gravity in urban crossroads (in Italian). The 6th SIDT (Italian Society of Transport Professors) Annual Meeting Proceedings, Bologna, Italy.
+- Girosi, F., Poggio, T., 1991. Networks for learning. In: Antognetti, P., Milutinovic, V.V. (Eds.), Neural Networks: Concepts, Applications and Implementation. Prentice Hall, Inc, Englewood Cliffs, NJ, pp. 110–154.
+- Graham, A., 1996. An application of generalized linear modelling to the analysis of traffic accidents. Traffic Engng. Control 37 (12), 691–696.
+- Hornik, K., Stinchombe, M., White, H., 1990. Universal approximation of an unknown mapping and its derivative using multilayer feedforward networks. Neural Netw. 3, 551–560.
+- Hornik, K., 1991. Approximation capabilities of multilayer feedforward networks. Neural Netw. 4, 251–257.
+- Joshua, S.C., Garber, N.J., 1990. Estimating truck accident installments and involvments using linear and Poisson regression models. Transp. Plan. Technol. Record 15, 41–58.
+- Jovanis, P.P., Chang, H.L., 1986. Modelling the relationship of accidents to miles travelled. Transp. Res. 106, 42–51.
+- Kim, K., Nitz, L., Richardson, J., Li, L., 1995a. Personal and behavioural predictors of automobile crash and injury severity. Accid. Anal. Prev. 27 (4), 1–13.
+- Kim, K., Nitz, L., Richardson, J., Li, L., 1995b. Analyzing the relationship between crash types and injuries in motor vehicle collisions in Hawaii. Transp. Res. Record 1467, 9–13.
+- Leshno, M., Lin, V.Ya., Pinkus, A., Schocken, S., 1993. Multilayer feedforward networks with a non polynomial activation function can approximate any function. Neural Netw. 6, 861–867.
+- Maher, M.J., Summersgill, I., 1996. To comprehensive methodology for the fitting of predictive accident models. Accid. Anal. Prev. 28 (3), 281–296.
+- Maycock G., Hall, R.D., 1984. Accidents at 4-arm roundabout. Laboratory Report LR1120, Transportation Research Laboratory, Crowthorne, Berks, UK.
+- Miaou, S.P., Hu, P.S., 1992. Relationship between truck accidents and highway geometric design: to Poisson regression approach. Transp. Res. Lab. Record 1376, 10–18.
+- Mussone L., 1995. Capabilities and first applications of neural networks in transports (in Italian). In: Cascetta E., Salerno G. (Eds.) (Italian Society of Transport Professors), Developments of Research in Transport Systems, Collana Trasporti, Franco Angeli, Milano, Italy, pp. 536–565.
+- Mussone L., Rinelli S., 1996. An accident analysis for urban vehicular flow, Urban Transport 96, CMP, pp. 583–592.
+- Okamoto, H., Koshi, M., 1989. The method to cope with the random errors of observed accident rates in regression analysis. Accid. Anal. Prev. 21, 317–332.
+- Salvatore, F., 1992. Trattamento statistico dei dati di incidente (in Italian). Autostrade 1, 27–31.
+- Shankar, V., Mannering, F., Barfield, W., 1995. Effect of roadway geometrics and environmental factors on rural freeway accident frequencies. Accid. Anal. Prev. 27 (3), 371–389.
+- Zeeger C.V., Stewart, R., Reinfurt, D., et al., 1990. Effective geometric improvements for safety upgrading horizontal curves. Prepared for the Federal Highway Administration by the University of North Carolina.

Zotero/001_artiklid/Road Safety Indicators/naderanaggregatecrashpredictionmodels2010.md

@@ -0,0 +1,337 @@
+---
+category: literaturenote
+citekey: naderanaggregatecrashpredictionmodels2010
+title: "Aggregate crash prediction models: Introducing crash generation concept"
+authors: "Naderan, Ali; Shahi, Jalil"
+year: 2010
+date: 2010-00-00 2010
+doi: 10.1016/j.aap.2009.08.020
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2009.08.020"
+zotero_key: TPA6ZJQJ
+zotero_storage: KUZ9R2AK
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Naderan, Ali"
+status: converted
+---
+Contents lists available at [ScienceDirect](http://www.sciencedirect.com/science/journal/00014575)
+
+## Accident Analysis and Prevention
+
+journal homepage: [www.elsevier.com/locate/aap](http://www.elsevier.com/locate/aap)
+
+![](_page_0_Picture_5.jpeg)
+
+# Aggregate crash prediction models: Introducing crash generation concept
+
+### Ali Naderan∗, Jalil Shahi
+
+Iran University of Science and Technology, Narmak, Tehran, Iran
+
+#### article info
+
+Article history: Received 15 March 2009 Received in revised form 31 July 2009 Accepted 17 August 2009
+
+Keywords: Safety planning Trip generation Crash generation Negative binomial
+
+#### abstract
+
+Safety conscious planning is a new proactive approach towards understanding crashes. It requires a planning-level decision-support tool to facilitate proactive approach to assessing safety effects of alternative urban planning scenarios. The objective of this research study is to develop a series of aggregate crash prediction models (ACPM) that are consistent with the trip generation step of the conventional four-step demand models. The concept of crash generation models (CGMs) is introduced utilizing trip generation data in a generalized linear regression with the assumption of a negative binomial error structure. The relationship of crash frequencies in traffic analysis zones (TAZ) and number of trips generated by purpose is investigated. This translates into immediate checking of the impact of future trip generations on crash frequencies in comprehensive transportation-planning studies (i.e. ability to forecast crashes at each time-step trips are being forecasted). A good relation was seen between crash frequency and number of trips produced/attracted by purpose per TAZ.
+
+© 2009 Elsevier Ltd. All rights reserved.
+
+#### **1. Introduction**
+
+Aggregate crash prediction models (ACPMs) may still be called state of the art in urban safety planning. Although there are many references on the use of aggregate models for predicting transportation demand in urban areas, little attention has been paid to aggregate safety planning issues. It has been common practice to consider traffic safety as a short-term 3–5 years engineering problem, rather than a long-term 15–20 years planning issue. In other words, safety considerations are traditionally omitted from the long-range transportation studies and relatively little thought is given to how safety could be considered in the planning process [\(Washington et al., 2006; MTTO, 2007\).](#page-7-0) The reason was that authorities blamed human factors and poor road conditions as major causes of crashes. The former needing publicity and education, the latter lots of funds for what was called black spot remedial measures. Nevertheless, proactive safety conscious planning is now considered a priority in many transportation-planning efforts. That is number of crashes need to be predicted/forecasted in the next 20 years or so.
+
+ACPM may be used to develop four-step safety planning models similar to the process followed in four-step demand modeling. [Fig. 1](#page-1-0) depicts the idea of inter-relating transportation and safety planning. The concept behind this figure is that each trip generated
+
+E-mail addresses: [Ali@Naderan.Com](mailto:Ali@Naderan.Com) (A. Naderan), [Jalil@iust.ac.ir](mailto:Jalil@iust.ac.ir) (J. Shahi).
+
+from a TAZ will be subject to a probable risk of crash. In fact, the fundamental cause of a crash is the trip making itself: no trips results no crashes! The concept of four-step safety planning is still underdeveloped and needs to be defined properly, but as starting point, the following conceptualization may apply.
+
+Crash generation models (CGMs) are developed based on results of trip generation models that provide number of trips made from/to a TAZ by different purposes. This includes an inherent measure of exposure: number of trips. It also provides a set of policy-related variables: number of trips by purpose. As will be demonstrated later herein, this part has proved appropriate and practical. The CGMs predict number of crashes in each TAZ based on number of trips by different purposes produced or attracted to that TAZ. Definition of other steps is in progress and needs further investigation.
+
+This research paper proceeds with examining how to relate crash frequencies per urban TAZ with trip generation (i.e. production/attraction) data. This would provide a framework for predicting/forecasting future number of crashes based on changes in the number of trips produced/attracted to a TAZ.
+
+#### **2. Methodology for developing ACPM**
+
+The process for developing ACPM consists of four basic steps [\(Cameron and Trivedi, 1998\)](#page-6-0) as described in the following subsections. Each step is performed considering its interaction with other steps. That is for example, there may be a feedback loop from step 2 to step 1, if a certain variable suggested in step 1 could not be made available.
+
+<sup>∗</sup> Corresponding author at: Iran University of Science and Technology, Civil Engineering, Narmak, Tehran, Iran. Tel.: +98 9123803508.
+
+<span id="page-1-0"></span>![](_page_1_Figure_2.jpeg)
+
+**Fig. 1.** Four-step safety modeling.
+
+#### 2.1. Variable selection
+
+Identifying preliminary variables based on the purpose of model development and theoretical issues is the first step. To date, ACPM have used different explanatory variables to associate with crashes, including land use ([Hadayeghi et al., 2007; Kim et al., 2006\),](#page-6-0) network characteristics [\(Noland, 2002; de Guevara et al., 2004\),](#page-7-0) environmental conditions [\(Aguero-Valverde and Jovanis, 2006\),](#page-6-0) day of year [\(Qadeer Memon, 2006\),](#page-7-0) demographic ([Washington et](#page-7-0) [al., 2006\)](#page-7-0) and traffic related variables [\(Hadayeghi et al., 2003\).](#page-6-0) These variables have been aggregated at different levels such as: state ([Noland, 2002\),](#page-7-0) county ([Aguero-Valverde and Jovanis, 2006\),](#page-6-0) TAZ ([de Guevara et al., 2004\)](#page-6-0) and [\(Hadayeghi et al., 2007\),](#page-6-0) ward [\(Quddus, 2008\),](#page-7-0) census block [\(Levine et al., 1995\) a](#page-6-0)nd even smaller levels [\(Kim et al., 2006\).](#page-6-0) Nevertheless, the logic behind selection of variables is more important than the goodness-of-fit results. For example, if the idea is to use ACPM to develop safety countermeasures, they should consist of variables that may be affected by engineering (e.g. improvement of geometric characteristics) or policymaking (road classification/construction). On the other hand, if they are meant purely for forecasting number of crashes in future, demographic and socio-economic variables may be best to use.
+
+The primary purpose of this research is to investigate how travel demand in urban areas contributes to crash frequencies at TAZlevel, in order to forecast number of crashes in the future. This implies the following characteristics for model variables:
+
+- (1) Indicate a metric for number of trips.
+- (2) Capture the daily pattern of trips.
+- (3) Capture the nature of trips (i.e. why it is made).
+- (4) Credible for future growth in number of trips.
+- (5) Be easy to forecast.
+- (6) Account for exposure.
+- (7) Comply with the process of travel demand management.
+- (8) Do not burden additional costs of data collection.
+- (9) Results easily incorporated into the planning process.
+
+The variables selected to consider here are results of trip generation models. The number of trips produced or attracted to each TAZ is available by purpose categories such as that presented in Table 1. The selected variables pass all the above-mentioned characteristics. Most importantly, results of any TDM scenario directly affects number of trips performed by different purposes to and from a TAZ, and employing them in an ACPM provides a means for the evaluation of these changes. This is in fact where the concept of crash generation starts, parallel to trip generation. Nevertheless, the pro-
+
+**Table 1** Preliminary variables for ACPM.
+
+| Category             | Variable | Description                                    | N   | Minimum | Maximum  | Mean     | Std. deviation |
+|----------------------|----------|------------------------------------------------|-----|---------|----------|----------|----------------|
+| General              | TAZ      | Traffic analysis zone                          | 380 | 1       | 380      | –        | –              |
+|                      | pop      | Population per TAZ                             | 380 | 0       | 26074    | 5942.861 | 4283.858       |
+|                      | area m2  | TAZ area in km2                                | 380 | .072234 | 18.71179 | .7236114 | 1.242645       |
+| Crashes              | pdo      | No. of property damage only crashes            | 380 | 0       | 405      | 56.58421 | 61.63267       |
+|                      | injury   | No. of injury crashes                          | 380 | 0       | 61       | 8.052632 | 7.538984       |
+|                      | fatal    | No. of fatal crashes                           | 380 | 0       | 2        | .1236842 | .3675019       |
+|                      | severe   | No. of fatal + injury crashes                  | 380 | 0       | 61       | 8.176316 | 7.64468        |
+|                      | total    | Total no. of crashes                           | 380 | 0       | 436      | 64.76053 | 66.64325       |
+| Trip productions     | pw       | No. of work trip productions                   | 380 | 0       | 22374    | 1870.637 | 1640.819       |
+|                      | pedu     | No. of educational trip productions            | 380 | 0       | 9941     | 833.7237 | 1013.701       |
+|                      | peduu    | No. of educational university trip productions | 380 | 0       | 493      | 82.75263 | 85.23744       |
+|                      | psh      | No. of shopping trip productions               | 380 | 0       | 2937     | 432.8026 | 387.138        |
+|                      | pre      | No. of recreational trip productions           | 380 | 0       | 2805     | 527.7684 | 411.0565       |
+|                      | pper     | No. of personal trip productions               | 380 | 0       | 3246     | 367.5632 | 360.4802       |
+|                      | ppilg    | No. of pilgrimage trip productions             | 380 | 0       | 319      | 84.93684 | 60.59434       |
+|                      | pnhb     | No. of non-home-base trip productions          | 380 | 0       | 5856     | 529.7026 | 641.9504       |
+|                      | ptotal   | Total no. of trip productions                  | 380 | 108     | 22834    | 4729.824 | 3005.952       |
+| Trip attractions     | aw       | No. of work trip attractions                   | 380 | 0       | 8813     | 1797.261 | 1551.9         |
+|                      | aedu     | No. of educational trip attractions            | 380 | 0       | 8884     | 869.8658 | 927.8295       |
+|                      | ash      | No. of shopping trip attractions               | 380 | 0       | 23609    | 655.7421 | 2188.884       |
+|                      | are      | No. of recreational trip attractions           | 380 | 0       | 7211     | 631.0132 | 948.3777       |
+|                      | aper     | No. of personal trip attractions               | 380 | 0       | 2763     | 272.5    | 412.0852       |
+|                      | anhb     | No. of non-home-base trip attractions          | 380 | 0       | 7633     | 557.4447 | 785.1203       |
+|                      | atotal   | Total no. of trip attractions                  | 380 | 30      | 45430    | 4783.718 | 5170.675       |
+| Trip attractions per | awa      | No. of work trip attractions per km2           | 380 | 0       | 80972.95 | 4794.547 | 6469.753       |
+| unit of area         | aedua    | No. of educational trip attractions per km2    | 380 | 0       | 28670.71 | 2066.45  | 2702.631       |
+|                      | asha     | No. of shopping trip attractions per km2       | 380 | 0       | 24956.26 | 1532.204 | 3673.977       |
+|                      | area     | No. of recreational trip attractions per km2   | 380 | 0       | 14021.96 | 1341.469 | 1851.679       |
+|                      | apera    | No. of personal trip attractions per km2       | 380 | 0       | 1098.409 | 662.5069 | 1098.409       |
+|                      | anhba    | No. of non-home-base trip attractions per km2  | 380 | 0       | 24624.75 | 1405.424 | 2185.383       |
+
+![](_page_2_Figure_2.jpeg)
+
+**Fig. 2.** Spatial distribution of total trips per TAZ.
+
+posed variables are later selected analytically based on significance and goodness-of-fit tests in the regression model.
+
+It has also been suggested that factors affecting crashesmay vary by crash severity ([Hadayeghi et al., 2007; Quddus, 2008\).](#page-6-0) Therefore, TAZ-level crash data were stratified into Property Damage Only, injury and fatality crashes. Total crashes, defined as sum of all crashes reported to the police and severe crashes defined as sum of injury and fatal crashes were also considered in the modeling effort. The latter is the most reliable because all severe crashes are mandated by law to be reported to the police [\(MTTO, 2007\).](#page-7-0) Besides, it compensates for the many zero observations of fatality crashes ([Quddus, 2008\).](#page-7-0)
+
+#### 2.2. Data collection/process
+
+Development of a database that contains variables used for modeling is the second practical step for the modeling effort. This study is based on the transportation database of Mashhad city, Iran, because comprehensive transportation studies are completed and updated for this city [\(MTTO, 2007\).](#page-7-0) The city is divided into 380 TAZs and trip generation model predictions are available for the year 2006. Crash data of the same year are also geo-coded into the GIS environment. The initial databases were in GIS format, Microsoft Access/Excel files. All these data were first entered into a single GIS file, and then were segregated between TAZs. As a result, a random error detested while creating the databases. The error relates to crash data assigned to points. Given the varying accuracy of their measurements, these data may produce seemingly random errors when GIS tools attempted to assign them to TAZs. This error is thought to be small ([Hadayeghi et al., 2003; de](#page-6-0) [Guevara et al., 2004\).](#page-6-0) Descriptive analysis of the variables is presented in [Table 1.](#page-1-0) Figs. 2 and 3 show the spatial distribution of total trips produced/attracted and total crash frequency per TAZ in the city of Mashhad.
+
+#### 2.3. Model postulation
+
+Hypothesizing the mathematical form of the relationship between variables is the foundation of the statistical modeling. The purpose of study, nature of available data, statistical software packages and computational skills required to calibrate statistical models greatly influence the mathematical formulations in previous researches. Linear regression ([Clark and Cushing, 2004\);](#page-6-0) linear regression with a spatial lag parameter [\(Levine et al., 1995\);](#page-6-0) Poisson models ([Chatterjee et al., 2004\);](#page-6-0) negative binomial (NB) models ([Hadayeghi et al., 2003, 2007; Maher and Summersgill,](#page-6-0) [1996; Abdel-Aty and Radwan, 2000; Miaou and Lord, 2003; Qadeer](#page-6-0) [Memon, 2006; Mitra and Washington, 2007; Noland and Quddus,](#page-6-0) [2004\);](#page-6-0) zero-inflated Poisson and NB models (ZIP/ZINB) ([Qin et al.,](#page-7-0) [2004; Shankar et al., 1997\);](#page-7-0) simultaneous estimation of NB models [\(de Guevara et al., 2004\);](#page-6-0) full Bayes hierarchical models with spatial effects [\(Aguero-Valverde and Jovanis, 2006; Quddus, 2008\),](#page-6-0) geographically weighted regression (GWR) ([Hadayeghi et al., 2003\);](#page-6-0) log-linear models using ordinary least squares (OLS) estimation technique [\(Washington et al., 2006; Wier et al., 2009\)](#page-7-0) were used successfully by previous researchers in different approaches to crash studies. [Hadayeghi et al. \(2006\)](#page-6-0) tested temporal transferability of the macro-models and concluded that model forecasts are within acceptable range in application context.
+
+Most of the above-mentioned models were calibrated to explain the underlying process of crashes and the effect of each independent variable on crash frequency. But, some researchers have argued against validity of linear [\(Miaou and Lum, 1993\)](#page-7-0) and ZIP/ZINB ([Lord et al., 2005, 2007\)](#page-6-0) models for explaining crashes. Despite improvements provided by accounting for spatial correlation in ACPMs, the results from the spatial Bayesian hierarchical models were found quite similar to the standard non-spatial NB models in many cases ([Quddus, 2008\).](#page-7-0) That is, the values of
+
+![](_page_2_Figure_12.jpeg)
+
+**Fig. 3.** Spatial distribution of total crashes per TAZ.
+
+coefficients in both spatial and non-spatial models were significantly similar. It was also concluded that spatial dependence among spatial units would decrease at higher levels of spatial aggregation.
+
+On the other hand, similar to [Washington et al. \(2006\), t](#page-7-0)he focus of this paper is prediction/forecasting and not explanation of the cause-effect issues related to crashes. Therefore, this research paper assumes standard NB formulation for the following reasons:
+
+- (1) Increased simplicity and applicability, less computational efforts in the state of practice.
+- (2) Although spatial NB models produce better results for the explanation purposes, the model coefficients are reported to be close to standard NB models ([Quddus, 2008\),](#page-7-0) justifying use NB models in concept development for prediction purposes.
+- (3) Crash data are clustered into TAZs, accounting for some of the heterogeneity ([Karlaftis and Tarko, 1998\).](#page-6-0)
+- (4) Trip generation models are calibrated in the linear form to predict number of trips that themselves are non-negative integer with spatial correlation, but have been accepted worldwide. So the NB formulation with log-link function was thought reasonable for the starting step in the concept development for CGMs.
+- (5) Focus is prediction, not explanation. Models for prediction have fewer restrictions than models for explanation: inference, or effects of isolated variables (estimated coefficients) not too important, multi-colinearity tolerated; goodness of fit and predictive-ability most important [\(Washington et al., 2006\).](#page-7-0)
+
+One requirement of the Poisson distribution is that the mean of the count data equals its variance. When the variance is significantly larger than the mean, the data are said to be over-dispersed, and are successfully modeled using a negative binomial (NB) model ([Washington et al., 2003\).](#page-7-0) In the NB model, the difference between variance and mean is presented by V[u] = u +˛u2, where V[u] is the estimated variance of the crash frequency; u is the estimated mean crash frequency and ˛ is the NB over-dispersion parameter. The Poisson regression model is regarded as a limiting model of the NB regression model as ˛ approaches zero, which means that the selection between these two models is dependent on the value of ˛. The NB distribution has the general form of Eq. (1):
+
+$$P(y_i) = \frac{\Gamma((1/\alpha) + y_i)}{\Gamma(1/\alpha) + y_i!} \left[ \frac{1/\alpha}{(1/\alpha) + \lambda_i} \right]^{1/\alpha} \left[ \frac{\lambda_i}{(1/\alpha)\lambda_i} \right]^{y_i} \tag{1}$$
+
+where - (.) is a gamma function. The NB model was developed mathematically by assuming that unobserved crash heterogeneity (variation) across sites (intersections, road segments, etc.) is gamma distributed, while crashes within sites are Poisson distributed [\(Washington et al., 2006\).](#page-7-0) Following a review of various model forms used in the literature and applying selected model forms to the data, the model form in Eq. (2) was used where y = predicted crash frequency; Xi = explanatory variables and bi = model parameters:
+
+$$y = \exp \sum_{i=1}^{n} b_i X_i \tag{2}$$
+
+#### 2.4. Model calibration and validation
+
+Model calibration, also known as model estimation, is estimating the parameters of the postulated model based on observed data. This is done by various common software packages for statistical analysis such as SAS [\(Hadayeghi et al., 2007\),](#page-6-0) GenStat [\(Qadeer](#page-7-0) [Memon, 2006\),](#page-7-0) Stata [\(Noland, 2002; Wier et al., 2009\),](#page-7-0) LIMDEP ([Washington et al., 2006\),](#page-7-0) aML ([de Guevara et al., 2004\),](#page-6-0) R Statistical Software [\(Aguero-Valverde and Jovanis, 2006\),](#page-6-0) WinBUGS [\(Quddus, 2008\),](#page-7-0) SPSS, SPlus, etc. All these softwares include modules for regression analysis and provide most of the statistical tests required for model calibration. The regression coefficients as well as the over-dispersion parameter were estimated by the maximum likelihoodmethod of the statistical software ([GenStat, 2008; STATA,](#page-6-0) [2007\)](#page-6-0) following a forward procedure. In this procedure, a simple model with only an intercept term is used as a starting point and the independent variables are added to the model one by one. Keeping these variables in the model was based on whether they would improve goodness-of-fit results.
+
+Goodness-of-fit tests for model calibration and the decision on whether or not to keep a variable in the model is based on the following criteria [\(Washington et al., 2003; Cameron and](#page-7-0) [Trivedi, 1998\):](#page-7-0) (1) t-statistic significant at the 5% level; (2) ratio of deviance or Pearson chi-square over degrees of freedom in the range of 0.8–1.2; (3) Akaike's information criteria (AIC) defined as AIC = [−2 Ln(ˇ)+2b]/N, where Ln(ˇ) is the log-likelihood value of model at convergence, b is the number of variables in the model, and N is number of TAZs. The model with the minimum value of AIC is the best model.
+
+#### **3. Modeling results**
+
+In order to investigate how each trip purpose affects crash occurrence, different models were calibrated. These models are regarded as CGMs in correspondence with the conventional trip generation models in the four-step travel demand modeling technique. The modeling procedure complies with the methodology presented in the preceding section.
+
+CGMs developed here are twofolds: crash production and crash attraction. The former employs trip production data and reports the relationship between number of trips produced in each TAZ and number of crashes observed in that TAZ, while the latter relies on trip attraction data for model calibration and reports the relationship between number of trips attracted to each TAZ and number of crashes observed in that TAZ. In order to capture the combined effect of total number of trips generated in each TAZ, a third model is calibrated consisting of only two variables: total trip production (all purposes) and total trip attraction (all purposes) per TAZ. For convenience, this category is designated crash generation model.
+
+#### 3.1. Models for total crashes
+
+The regression results of the crash production and crash attraction models for total number of crashes per year are presented in [Table 2. A](#page-4-0)s shown, the relationship between crashes and selected variables is highly significant at the 5% level. Personal trips produced (pper) were highly correlated with other variables, hence omitted from the crash production models. The dispersion parameter is 0.87 for the crash production model and 0.98 for the crash attraction model, proving the significant justification of NB over the Poisson formulation. The ratios of Pearson chi-square and deviance over degrees of freedom are within the acceptable range of 0.8–1.2, another indication that the choice of the NB error structure is appropriate and the calibrated model is valid. It is also seen from [Table 2](#page-4-0) that crash production model is a better fit based on the larger loglikelihood value and lower AIC and significance (p-) values. The mean absolute deviance for this model is about 0.938.
+
+The crash generation model consisting of total trips produced and attracted to each TAZ was developed too, which was a good fit based on the log-likelihood and deviance/df ratio results. This model is represented in Eq. (3):
+
+$$y = \exp(3.38057) + \exp(.000104 \times \text{ptotal}) + \exp(.000044 \times \text{atotal})$$
+
+<span id="page-4-0"></span>**Table 2** Results of total crash generation models.
+
+|                                                             | Crash production model                                                            |                                                                             |                                                                 |                                                              | Crash attraction model                                 |                                                                         |                                                                    |                                                        |                                                      | Crash generation model    |                              |                            |                       |                      |  |
+|-------------------------------------------------------------|-----------------------------------------------------------------------------------|-----------------------------------------------------------------------------|-----------------------------------------------------------------|--------------------------------------------------------------|--------------------------------------------------------|-------------------------------------------------------------------------|--------------------------------------------------------------------|--------------------------------------------------------|------------------------------------------------------|---------------------------|------------------------------|----------------------------|-----------------------|----------------------|--|
+| Var.                                                        | Coeff.                                                                            | Std err                                                                     | t-stat                                                          | Sig.                                                         | Var.                                                   | Coeff.                                                                  | Std err                                                            | t-stat                                                 | Sig.                                                 | Var.                      | Coeff.                       | Std err                    | t-stat                | Sig                  |  |
+| cons.<br>pw<br>pedu<br>peduu<br>psh<br>pre<br>ppilg<br>pnhb | 3.54888<br>.00010<br>−.00145<br>.00367<br>.00609<br>−.00222<br>−.00479<br>.000276 | .12031<br>.00004<br>.00053<br>.00133<br>.00233<br>.00103<br>.00159<br>.0001 | 29.5<br>2.46<br>−2.72<br>2.76<br>2.60<br>−2.15<br>−3.00<br>2.76 | .000<br>.014<br>.007<br>.006<br>.009<br>.031<br>.003<br>.006 | cons.<br>awa<br>aedua<br>ash<br>area<br>anhba<br>apera | 4.04834<br>−.77e−4<br>.81e−4<br>.64e−4<br>.000103<br>−.00011<br>.000339 | .07451<br>.17e−4<br>.35e−4<br>.34e−4<br>.53e−4<br>.48e−4<br>.15e−3 | 54.3<br>−4.43<br>2.30<br>1.87<br>1.93<br>−2.30<br>2.29 | .000<br>.000<br>.022<br>.062<br>.053<br>.021<br>.022 | cons.<br>ptotal<br>atotal | 3.38057<br>.000104<br>.44e−4 | .11785<br>.21e−4<br>.12e−4 | 28.69<br>4.93<br>3.46 | .000<br>.000<br>.001 |  |
+| Goodness of fit                                             |                                                                                   |                                                                             |                                                                 |                                                              |                                                        |                                                                         |                                                                    |                                                        |                                                      |                           |                              |                            |                       |                      |  |
+|                                                             | No. of observations                                                               |                                                                             | 380                                                             |                                                              | No. of observations                                    |                                                                         |                                                                    | 380                                                    |                                                      | No. of observations       |                              |                            | 380                   |                      |  |
+|                                                             | Degree of freedom (df)                                                            |                                                                             | 372                                                             |                                                              | Degree of freedom (df)                                 |                                                                         |                                                                    | 373                                                    |                                                      | Degree of freedom (df)    |                              |                            | 377                   |                      |  |
+| Deviance/df                                                 |                                                                                   |                                                                             | 1.2102                                                          |                                                              | Deviance/df                                            |                                                                         |                                                                    | 1.3569                                                 |                                                      | Deviance/df               |                              |                            | 1.1954                |                      |  |
+|                                                             | Pearson/df<br>0.8857                                                              |                                                                             | Pearson/df                                                      |                                                              |                                                        | 1.0608                                                                  |                                                                    | Pearson/df                                             |                                                      |                           | 0.8928                       |                            |                       |                      |  |
+| ˛                                                           | 1.02                                                                              |                                                                             | ˛                                                               |                                                              |                                                        | 1.14                                                                    |                                                                    | ˛                                                      |                                                      |                           | 1.12                         |                            |                       |                      |  |
+| Log-likelihood                                              |                                                                                   |                                                                             | −1915.3                                                         |                                                              | Log-likelihood                                         |                                                                         |                                                                    | −1937.1                                                |                                                      | Log-likelihood            |                              |                            | −1934.2               |                      |  |
+| AIC                                                         |                                                                                   |                                                                             | 10.12                                                           |                                                              | AIC                                                    |                                                                         |                                                                    | 10.24                                                  |                                                      | AIC                       |                              |                            |                       | 10.19                |  |
+
+**Table 3** Results of PDO crash generation models.
+
+| Crash production model                              |                                                                           |                                                                    |                                                         |                                                      |                                                        | Crash attraction model                                                 | Crash generation model                                             |                                                        |                                                      |                           |                            |                           |                       |                      |
+|-----------------------------------------------------|---------------------------------------------------------------------------|--------------------------------------------------------------------|---------------------------------------------------------|------------------------------------------------------|--------------------------------------------------------|------------------------------------------------------------------------|--------------------------------------------------------------------|--------------------------------------------------------|------------------------------------------------------|---------------------------|----------------------------|---------------------------|-----------------------|----------------------|
+| Var.                                                | Coeff.                                                                    | Std err                                                            | t-stat                                                  | Sig.                                                 | Var.                                                   | Coeff.                                                                 | Std err                                                            | t-stat                                                 | Sig.                                                 | Var.                      | Coeff.                     | Std err                   | t-stat                | Sig                  |
+| cons.<br>pw<br>pedu<br>peduu<br>psh<br>pre<br>ppilg | 3.38287<br>.000105<br>−.00167<br>.004085<br>.007012<br>−.00250<br>−.00622 | .12482<br>.43e−4<br>.00056<br>.00138<br>.00245<br>.00108<br>.00166 | 27.1<br>2.43<br>−2.98<br>2.96<br>2.86<br>−2.31<br>−3.74 | .000<br>.015<br>.003<br>.003<br>.004<br>.021<br>.000 | cons.<br>awa<br>aedua<br>ash<br>area<br>anhba<br>apera | 3.88208<br>−.87e−4<br>.99e−4<br>.66e−4<br>.00011<br>−.00012<br>.000377 | .07776<br>.18e−4<br>.37e−4<br>.36e−4<br>.56e−4<br>.51e−4<br>.16e−3 | 49.9<br>−4.72<br>2.66<br>1.86<br>1.97<br>−2.37<br>2.37 | .000<br>.000<br>.008<br>.065<br>.049<br>.018<br>.018 | cons.<br>ptotal<br>atotal | 3.1828<br>.00011<br>.48e−4 | .1245<br>.22e−4<br>.13e−4 | 25.56<br>5.01<br>3.55 | .000<br>.000<br>.000 |
+| pnhb                                                | .000293                                                                   | .00010                                                             | 2.83                                                    | .005                                                 |                                                        |                                                                        |                                                                    |                                                        |                                                      |                           |                            |                           |                       |                      |
+| Goodness of fit                                     |                                                                           |                                                                    |                                                         |                                                      |                                                        |                                                                        |                                                                    |                                                        |                                                      |                           |                            |                           |                       |                      |
+|                                                     | No. of observations                                                       |                                                                    | 380                                                     |                                                      | No. of observations                                    |                                                                        |                                                                    | 380                                                    |                                                      | No. of observations       |                            |                           | 380                   |                      |
+|                                                     | Degree of freedom (df)                                                    |                                                                    | 372                                                     |                                                      | Degree of freedom (df)                                 |                                                                        |                                                                    | 373                                                    |                                                      | Degree of freedom (df)    |                            |                           | 377                   |                      |
+| Deviance/df                                         |                                                                           |                                                                    | 1.2078                                                  |                                                      | Deviance/df                                            |                                                                        |                                                                    | 1.3515                                                 |                                                      | Deviance/df               |                            |                           | 1.2009                |                      |
+|                                                     | Pearson/df<br>0.9102                                                      |                                                                    | Pearson/df                                              |                                                      |                                                        | 1.0877                                                                 |                                                                    | Pearson/df                                             |                                                      |                           | 0.9388                     |                           |                       |                      |
+| ˛                                                   | 1.10                                                                      |                                                                    | ˛                                                       |                                                      |                                                        | 1.24                                                                   |                                                                    | ˛                                                      |                                                      |                           | 1.23                       |                           |                       |                      |
+|                                                     | Log-likelihood<br>−1851.9                                                 |                                                                    | Log-likelihood                                          |                                                      |                                                        | −1877.7                                                                |                                                                    | Log-likelihood                                         |                                                      |                           | −1875.8                    |                           |                       |                      |
+| AIC                                                 |                                                                           |                                                                    | 9.79                                                    |                                                      | AIC                                                    |                                                                        |                                                                    | 9.92                                                   |                                                      | AIC                       |                            |                           |                       | 9.89                 |
+
+#### 3.2. Models for PDO crashes
+
+Property Damage Only (PDO) crashes include non-injury crashes that involve monetary loss and are reported by the police. The regression results of the crash production and crash attractionmodels for PDO crashes per year are presented in Table 3. As can be seen both by log-likelihood and AIC, the results of the PDO crash models indicate goodness-of-fit improvements over the total crash models. It is interesting to see the same set of variables as those of the total CGMs became significant here. Coefficient estimates are also close and sign of variables are the same in both models. This is due to the large influence of number of PDO crashes on total number of crashes.
+
+#### 3.3. Models for severe crashes
+
+Severe crashes include fatal and injury crashes but exclude PDO crashes. The regression results of the crash production and crash attraction models for severe crashes per year are presented in [Table 4.](#page-5-0) As can be seen, the results of the severe CGMs indicate major goodness-of-fit improvements over the total crash models and PDO crash models. This is partly because data on severe crashes are more accurately collected and maintained by the police. On the other hand, number of variables proved significant to be kept in the model has decreased. Still, the model includes major trip purposes (i.e. work, shopping, educational and non-home-based trips) that are mostly influenced by travel demand-management scenarios. The dispersion parameter, ˛, is also half that of the total and PDO CGMs, which also indicates less dispersed and more accurate data.
+
+#### 3.4. Models for injury crashes
+
+The regression results of the crash production and crash attraction models for injury crashes per year are presented in [Table 5.](#page-5-0) As can be seen, the results of the injury crash models indicate goodness-of-fit improvements over the previous models. The injury models show mostly the same characteristics as that of the severe crash models. As would have been expected, work, educational, shopping and non-home-based trip purposes were the significant variables in thesemodels. This indicates themost important trip purposes that may be affected by future policies and travel demand-management plans. Coefficient estimates here are almost the same as the severe CGMs, indicating the significant influence of number of injury crashes on number of severe crashes. In fact, the small number of fatality crashes seems to have little effect on parameter estimations for severe and injury crash models.
+
+<span id="page-5-0"></span>**Table 4** Results of severe crash generation models.
+
+| Crash production model |                           |         |        |                |                        | Crash attraction model |         |            | Crash generation model |                        |         |         |        |      |
+|------------------------|---------------------------|---------|--------|----------------|------------------------|------------------------|---------|------------|------------------------|------------------------|---------|---------|--------|------|
+| Var.                   | Coeff.                    | Std err | t-stat | Sig.           | Var.                   | Coeff.                 | Std Err | t-stat     | Sig.                   | Var.                   | Coeff.  | Std Err | t-stat | Sig  |
+| cons.                  | 1.63069                   | .09769  | 16.7   | .000           | cons.                  | 1.9756                 | .07698  | 25.7       | .000                   | cons.                  | 1.686   | .09588  | 17.6   | .000 |
+| pw                     | .68e−4                    | .32e−4  | 2.18   | .029           | aw                     | .000137                | .41e−4  | 3.35       | .001                   | ptotal                 | .607e−4 | .18e−4  | 3.44   | .001 |
+| pedu                   | −.00022                   | .00012  | −1.88  | .060           | aedua                  | −.46e−4                | .21e−4  | −2.16      | .031                   | atotal                 | .208e−4 | .98e−5  | 2.12   | .034 |
+| psh                    | .000889                   | .00314  | 2.83   | .005           | ash                    | .42e−4                 | .25e−4  | 1.67       | .095                   |                        |         |         |        |      |
+| pnhb                   | .000209                   | .85e−4  | 2.47   | .014           | anhba                  | −.56e−4                | .26e−4  | −2.16      | .031                   |                        |         |         |        |      |
+| Goodness of fit        |                           |         |        |                |                        |                        |         |            |                        |                        |         |         |        |      |
+|                        | No. of observations       |         | 380    |                | No. of observations    |                        |         | 380        |                        | No. of observations    |         |         | 380    |      |
+|                        | Degree of freedom (df)    |         | 375    |                | Degree of freedom (df) |                        |         | 375        |                        | Degree of freedom (df) |         |         | 377    |      |
+| Deviance/df            |                           |         | 1.1917 |                | Deviance/df            |                        |         | 1.3493     |                        | Deviance/df            |         |         | 1.1740 |      |
+|                        | Pearson/df<br>1.0065      |         |        | Pearson/df     |                        |                        | 1.0592  | Pearson/df |                        |                        | 1.0009  |         |        |      |
+| ˛                      | 0.71                      |         | ˛      |                |                        | 0.73                   |         | ˛          |                        |                        | 0.72    |         |        |      |
+|                        | Log-likelihood<br>−1182.3 |         |        | Log-likelihood |                        |                        | −1186.3 |            | Log-likelihood         |                        |         | −1184.5 |        |      |
+| AIC<br>6.25            |                           | AIC     |        |                | 6.28                   |                        | AIC     |            |                        | 6.25                   |         |         |        |      |
+
+**Table 5** Results of injury crash generation models.
+
+| Crash production model |                        |         |         |      |                     | Crash attraction model |         |        |            |                     | Crash generation model |         |        |         |  |  |
+|------------------------|------------------------|---------|---------|------|---------------------|------------------------|---------|--------|------------|---------------------|------------------------|---------|--------|---------|--|--|
+| Var.                   | Coeff.                 | Std err | t-stat  | Sig. | Var.                | Coeff.                 | Std err | t-stat | Sig.       | Var.                | Coeff.                 | Std err | t-stat | Sig     |  |  |
+| cons.                  | 1.61185                | .09781  | 16.5    | .000 | cons.               | 1.9576                 | .07719  | 25.4   | .000       | cons.               | 1.6676                 | .09588  | 17.4   | .000    |  |  |
+| pw                     | .68e−4                 | .31e−4  | 2.17    | .030 | aw                  | .000139                | .41e−4  | 3.38   | .001       | ptotal              | .61e−4                 | .18e−4  | 3.45   | .001    |  |  |
+| pedu                   | −.00022                | .00012  | −1.90   | .057 | aedua               | −.46e−4                | .21e−4  | −2.14  | .032       | atotal              | .21e−4                 | .98e−5  | 2.13   | .033    |  |  |
+| psh                    | .000899                | .00314  | 2.86    | .004 | ash                 | .42e−4                 | .25e−4  | 1.66   | .097       |                     |                        |         |        |         |  |  |
+| pnhb                   | .000212                | .85e−4  | 2.50    | .012 | anhba               | −.57e−4                | .26e−4  | −2.15  | .031       |                     |                        |         |        |         |  |  |
+| Goodness of fit        |                        |         |         |      |                     |                        |         |        |            |                     |                        |         |        |         |  |  |
+|                        | No. of observations    |         | 380     |      | No. of observations |                        |         | 380    |            | No. of observations |                        |         | 380    |         |  |  |
+|                        | Degree of freedom (df) |         | 375     |      |                     | Degree of freedom (df) |         |        | 375        |                     | Degree of freedom (df) |         |        | 377     |  |  |
+| Deviance/df            |                        |         | 1.1904  |      |                     | Deviance/df            |         |        | 1.3487     |                     | Deviance/df            |         |        | 1.1735  |  |  |
+|                        | Pearson/df<br>1.0003   |         |         |      | Pearson/df          |                        |         | 1.0554 | Pearson/df |                     |                        | 0.9957  |        |         |  |  |
+| ˛                      | 0.71                   |         | ˛       |      |                     |                        | 0.73    |        | ˛          |                     | 0.72                   |         |        |         |  |  |
+| Log-likelihood         |                        |         | −1176.7 |      |                     | Log-likelihood         |         |        | −1180.9    |                     | Log-likelihood         |         |        | −1179.0 |  |  |
+| AIC                    | 6.22                   |         | AIC     |      |                     |                        | 6.26    |        | AIC        |                     |                        | 6.22    |        |         |  |  |
+
+#### 3.5. Models for fatal crashes
+
+Efforts to develop crash production/attraction/generation models for fatal crashes failed. This is in part because only 42 out of 380 TAZs had non-zero fatal crash observations. Inability to calibrate fatal crash prediction models may not be a big issue, because the climate of crashes is described well enough by crash prediction models developed for the severe/injury crashes.
+
+## **4. Discussion**
+
+As was mentioned before, previous researches suggested very similar coefficients for the spatial and non-spatial models [\(Quddus,](#page-7-0) [2008\).](#page-7-0) So the choice of NB formulation to develop CGM was based on the stated interest in developing a model for practical application that can readily be applied and interpreted. CGMs developed in previous sections provide distinct characteristics that would be discussed here:
+
+(1) Trip purposes have directly entered the crash production models, while they have been converted to trips per square kilometer of TAZ in crash attraction model. This was because trip attraction models were calibrated based on areas of different attractions (e.g. shopping area) and thus became correlated in the process of estimation of crash attraction models. To compensate for this, trip attractions by purpose were normalized as trips per unit area. [Levine et al. \(1995\)](#page-6-0) also suggested the same approach to account for the differences between areas of census blocks.
+
+- (2) Compared to total/PDO crash models, injury/severe crash models were less dispersed based on lower ˛ values and better fit based on smaller AIC and larger log-likelihood values. This is in part because data on injury/severe crashes are more accurate. Not all PDO crashes are captured/reported by the police, which may result biased data for some TAZs and consequently increased over-dispersion in total/PDO crash models.
+- (3) In order to cover for the over-dispersion, more independent variables became significant in the total/PDO crash models. As was mentioned, multi-colinearity can be tolerated in prediction models, but use of the models for explanation purposes must be with cautious.
+- (4) Trip productions and trip attractions by the purposes of work, shopping, educational (university) and none home-based trips show a positive effect on number of crashes. This was expected because: (a) they are mostly made by private vehicles, (b) in the rush hours, and (c) on a routine basis, and thus bear a more exposure to risk.
+- (5) Trip productions and attractions by the purposes of education and recreation show a negative effect, because: (a) recreational trips are made with less hurriedness and in off-peak hours and (b) because of proximity of schools to residential areas, educational (school) trips are mostly made on foot/public transit and so they are less involved in crashes.
+- (6) Although in the crash attraction models, trip attractions per unit of area by the purposes of work, shopping and none homebased trips show a negative sign, itmust be noted that the sign is in fact induced by the denominator (i.e. TAZ area). The negative effect of TAZ area was checked by calibrating a separate model
+
+<span id="page-6-0"></span>including only TAZ area and the intercept and is according to previous researches [\(Washington et al., 2006;Wier et al., 2009\).](#page-7-0) The model is estimated as <sup>y</sup> = [exp(4.702 <sup>−</sup> .2302)/area m2]. As can be seen, the sign of inverse TAZ area is negative in this model. This also explains the positive sign of trip attractions per unit of area by the purposes of education, recreation and personal. The possible meaning for the negative effect of area on crashes is that increase in area normally results decreasing density in population and other trip producing/attracting variables.
+
+(7) As a usage example, if work trips produced in a TAZ are forecasted to increase by 1000 in the future, given all other trip purposes are constant, number of total crashes in that TAZ may be calculated using crash production models. This results approximately 10% increase in total crashes of that TAZ (exp[.0000997 × 1000] = 1.10) and 6% increase in severe crashes (exp[.0000614 × 1000] = 1.06). Knowledge of this growth in crashes in essential in order to develop long-term safety plans and establish counter measures of a specific type.
+
+#### **5. Conclusions and recommendations**
+
+The primary purpose of this paper was to investigate how travel demand in urban areas contributes to crash frequencies at TAZlevel. To this end, a series of macro-level ACPMs designated CGMs, were developed hiring number of trips by purpose generated per TAZ as the primary variables affecting crash occurrence. Thesemodels provide the basic tool for evaluation of TDM scenarios in urban transportation planning. They would predict number of crashes based on number of trips produced or attracted to a TAZ. A certain TDM scenario may be developed to reduce trip productions of a specific purpose and the related number of crashes could be predicted. These models are also appropriate for forecasting future number of crashes because no statistical model could capture changes in future trends better than the number of trips itself. Though proven a good measure of exposure (Hadayeghi et al., 2007; Quddus, 2008), VKT and traffic flow are both second-hand results of trip generation models, so it seems more appropriate to use the first-hand results, namely trip productions and attractions by purpose.
+
+Other note-worthy issues include:
+
+- (1) The NB formulation for CGMs provides good consistency with travel demand models, which are normally calibrated by the least square method in the linear form.
+- (2) Aggregate models deal with macroscopic data and their results may not to be considered to evaluate a single corridor/project. On the other hand, disaggregate models are at microscopic level and are developed for the purpose of corridor/project evaluation. In other words, aggregate (macro) models are for transportation (safety) planning and disaggregate (micro) models are for traffic (safety) engineering.
+- (3) CGMs are suited for predicting/forecasting future trends in crashes of specific types. This helps targeted focus of stakeholders towards the problematic areas of transportation safety.
+- (4) CGMs may be used cautiously on identifying TAZs that suffer more crashes of a specific type in the future, and developing countermeasures based on approved methods.
+- (5) CGMs may not be used for developing safety countermeasures based on the coefficients of model variables [\(Washington et al.,](#page-7-0) [2006\).](#page-7-0) This needs careful considerations and further investigations utilizing micro-level or disaggregated crash prediction models.
+- (6) Although the underlying process of CGMs is somewhat the same as trip generation models, their ability to forecast into
+
+- the future yet needs to be verified. Hadayeghi et al. (2006) put an step in that direction.
+- (7) Definition of other steps in four-step safety planning needs further investigation. In essence, the role of modal share should be investigated. This would help evaluation of TDM scenarios involving introduction/increase in specific modes of travel.
+- (8) In order to cover for the over-dispersion, redundant and/or multi-collinear variables may become significant in the crash production/attraction models. If the models are to be used for explanatory purposes, a multi-colinearity analysis must be performed for models at TAZ-level to account for this.
+- (9) Spatial correlation and heterogeneity among adjacent TAZs should be tested to justify the use of standard NB models and/or determine the spatial effects on model calibration results. Again, that is if the models are to be considered for explanatory purposes.
+
+#### **Acknowledgments**
+
+This paper is part of an ongoing research on Ph.D. dissertation at Faculty of Civil Engineering, Iran University of Science and Technology, Tehran, Iran. We would like to thank Prof. Hamid Behbahani, Dr. Mahmud Ameri and Dr. M. Ahmadinejad, Faculty of Civil Engineering, Iran University of Science and Technology and Dr. Amir-Abbas Rasafi, Imam Khomeini International University, Faculty of Engineering, as members of the dissertation steering committee for their helpful comments. We also acknowledge invaluable comments from Dr. Kamal Shoar, Tarhe Haftom consulting engineers. Part of data was provided by Mashhad Transportation and Traffic Organization. The writers wish to extend their thanks and appreciation to the two reviewers for their constructive comments to enhance this paper.
+
+#### **References**
+
+- Abdel-Aty, M.A., Radwan, A.E., 2000. Modeling traffic accident occurrence and involvement. Accid. Anal. Prev. 32 (5), 633–642.
+- Aguero-Valverde, J., Jovanis, P.P., 2006. Spatial analysis of fatal and injury crashes in Pennsylvania. Accid. Anal. Prev. 38 (3), 618–625.
+- Cameron, A.C., Trivedi, P.K., 1998. Regression Analysis of Count Data. Cambridge University Press.
+- Chatterjee, A., Everett, J.D., Reiff, B., Schwetz, T.C., Seaver, W.L., Wegmann, F.J., 2004. Tools for assesing safety impact of long-range transportation plans in urban areas. In: Presented at the 80th Annual Meeting of the Transportation Research Board, TRB, Washington, DC.
+- Clark, D.E., Cushing, B.M., 2004. Rural and urban traffic fatalities, vehicle-miles and population density. Accid. Anal. Prev. 36 (6), 967–972.
+- de Guevara, F.L., Washington, S.P., Oh, J., 2004. Forecasting crashes at planning level: a simultaneous negative binomial crash model applied in Tucson, Arizona. Transport. Res. Rec. 1897, 191–199.
+- GenStat, 2008. GenStat Discovery Edition 3, [http://www.vsni.co.uk.](http://www.vsni.co.uk/)
+- Hadayeghi, A., Shalaby, A.S., Persaud, B., 2003. Macro-level accident prediction models for evaluating safety of urban transportation systems. Transport. Res. Rec. 1840, 87–95.
+- Hadayeghi, A., Shalaby, A.S., Pesuad, B., 2007. Safety prediction models: a proactive tool for safety evaluation in urban transportation planning applications. Transport. Res. Rec. 2019, 225–236.
+- Hadayeghi, A., Shalaby, A.S., Persaud, B.N., Cheung, C., 2006. Temporal transferability and updating of zonal-level accident prediction models. Accid. Anal. Prev. 38 (3), 579–589.
+- Karlaftis, M.G., Tarko, A.P., 1998. Heterogenity considerations in accident modeling. Accid. Anal. Prev. 30 (4), 425–433.
+- Kim, K., Brunner, I.M., Yamashita, E.Y., 2006. The influence of land use, population, employment and economic activity on accidents. Transport. Res. Rec. 1953, 56–64.
+- Levine, N., Kim, K.E., Nitz, L.H., 1995. Spatial analysis of Honolulu motor vehicle crashes. II. Zonal generators. Accid. Anal. Prev. 27 (5), 675–685.
+- Lord, D., Washington, S.P., Ivan, J.N., 2007. Further notes on the application of zeroinflated models in highway safety. Accid. Anal. Prev. 39 (1), 53–57.
+- Lord, D.,Washington, S.P., Ivan, J.N., 2005. Poisson, Poisson-gamma and zero-inflated regression models of motor vehicle crashes: balancing statistical fit and theory. Accid. Anal. Prev. 37 (1), 35–46.
+- Maher, M.J., Summersgill, I., 1996. A comprehensive methodology for the fitting of predictive accident models. Accid. Anal. Prev. 28 (3), 281–296.
+
+- <span id="page-7-0"></span>Miaou, S.P., Lum, H., 1993.Modeling vehicle accidents and highway geometric design relationships. Accid. Anal. Prev. 25 (6), 689–709.
+- Miaou, S.P., Lord, D., 2003. Modeling traffic crash-flow relationships for intersections: dispersion parameter, functional form, and Bayes versus empirical Bayes. Transport. Res. Rec. 1840, 31–40.
+- Mitra, S., Washington, S., 2007. On the nature of over-dispersion in motor vehicle crash prediction models. Accid. Anal. Prev. 39 (3), 459–468.
+- MTTO, 2007. Comprehensive Transportation Studies of Mashhad City. Mashhad Transportation and Traffic Organization (MTTO) (update).
+- Noland, R.B., Quddus, M.A., 2004. Improvements in medical care and technology and reductions in traffic-related fatalities in Great Britain. Accid. Anal. Prev. 36 (1), 103–113.
+- Noland, R.B., 2002. Traffic Fatalities and Injuries: The Effect of Changes in Infrastructure and Other Trends (accessed on 1 January 2009) <http://www.cts.cv.ic.ac.uk/staff/wp22-noland.pdf>.
+- Qadeer Memon, A., 2006. Road accident prediction models developed from a national database: Poisson and negative binomial regressions. In: TRB 2006 Annual Meeting CD-ROM.
+
+- Qin, X., Ivan, J.N., Ravishanker, N., 2004. Selecting exposure measures in crash rate prediction for two-lane highway segments. Accid. Anal. Prev. 36 (2), 183–191.
+- Quddus, M.A., 2008. Modelling area-wide count outcomes with spatial correlation and heterogeneity: an analysis of London crash data. Accid. Anal. Prev. 40 (4), 1486–1497.
+- Shankar, V., Milton, J., Mannering, F., 1997. Modeling accident frequencies as zeroaltered probability processes: an empirical inquiry. Accid. Anal. Prev. 29 (6), 829–837.
+- STATA, 2007. STATA Users Manual. StataCorp LP.
+- Washington, S., Schalwyk, I.V., Meyer, M., Dumbaugh, E., Zoll, M., 2006. Incorporating Safety into Long-Range Transportation Planning. NCHRP Report 546, TRB, Washington, DC.
+- Washington, S.P., Karlaftis, M.G., Mannering, F.L., 2003. Statistical and Econometirc Methods for Transportation Data Analysis. Chapman & Hall/CRC.
+- Wier, M., Weintraub, J., Humphreys, E.H., Seto, E., Bhatia, R., 2009. An area-level model of vehicle-pedestrian injury collisions with implications for land use and transportation planning. Accid. Anal. Prev. 41 (1), 137–145.

Zotero/001_artiklid/Road Safety Indicators/navinroadsafetymodelsome1996.md

@@ -0,0 +1,386 @@
+---
+category: literaturenote
+citekey: navinroadsafetymodelsome1996
+title: "Road Safety Model: Some Fundamental Ideas"
+authors: "Navin, Francis P. D.; Bergan, Arthur; Zhang, Guanyu"
+year: 1996
+date: 1996-00-00 1996
+doi: 10.1177/0361198196156300107
+publication: "Transportation Research Record: Journal of the Transportation Research Board"
+url: "https://doi.org/10.1177/0361198196156300107"
+zotero_key: VTREN8HH
+zotero_storage: 74R7V2SM
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Navin, Francis P. D."
+status: converted
+---
+# **Road Safety Model: Some Fundamental Ideas**
+
+# FRANCIS P. D. NAVIN, ARTHUR BERGAN, AND GUANYU ZHANG
+
+A fundamental relationship has been developed that explains road accident statistics in developed and developing countries. The model uses two variables, traffic hazard measured as deaths per vehicle and motorization measured as vehicles per person, to estimate personal hazard as deaths per person. Special cases of the model are those by Smeed, Trinca et al., and Koornstra. The model of fatalities has two extremes. Early motorization has high traffic hazard and personal safety is low and increasing. Full motorization is characterized by a moderate and falling traffic hazard and a low and decreasing personal safety. Between these extremes, there is a maximum number of fatalities per population. Models for personal injury and total road accidents in developed countries appear to follow a similar trend. Available world data fit the proposed relationships well. The models allow planners and engineers to estimate the future maximum road fatalities for developing countries. The model has been extended to incorporate an automobile ownership model that explains some of the growth in motorization. A traffic hazard model is also outlined, in part on the basis of the ideas developed by Koornstra. The extended models should allow a more detailed analysis of some of the social and engineering factors that contribute to road safety.
+
+More than half a million road deaths occur annually in traffic crashes around the world. Stated in another way, approximately one person in 8000 in the world's population dies from injuries received in traffic crashes according to Trinca et al. (*1*). Traffic accidents constitute a major threat to public health and have been classified as a crisis by the Centers for Disease Control. Road accidents also have a high monetary loss. Since about 1970 developed countries have undergone a downward trend in traffic fatalities, while developing countries are facing significant increases.
+
+The general downward trend of road fatalities is shown in Figure 1. Navin et al. (*2*) used only the fatality-vehicle plan of this figure to illustrate the downward tendency for Canada, the United Kingdom, and the United States. Using data from the beginning of this century, they observed at first a rapid increase in the number of deaths. This peaked in the late 1960s or early 1970s, followed by a steady decline. Available data also support the finding of Navin et al. (*2*) that not only developed countries but also developing countries will go through the same phases.
+
+What insight may be found in such common trends? The study of fatal road accidents must consider major factors related to road safety. Factors such as the economic situation (*3*) or legislative intervention (*4*) play large roles in road safety. Other authorities ascribe change solely to driver behavior. Evans (*5*) has given a hierarchical ordering of road safety factors, ranked as human infrastructure and engineering infrastructure. Evans concluded that the
+
+social norm, an element of human infrastructure, dominates the road safety trend.
+
+More recent findings by Koornstra (*6*) demonstrate that the development of traffic and traffic accidents can be described as a semiclosed system or as a self-organizing social-economic system. The Koornstra system is a gradual learning process of society. In that system road safety benefits by changing the related environment to govern individual experiences.
+
+An empirical model developed by Navin et al. (*2*) explains longrun road accidents in developed and developing countries. This paper extends the original model by incorporating some of Koornstra's ideas to provide additional explanation of the interaction of the variables.
+
+## **LINKING POPULATION, VEHICLES, AND FATALITIES**
+
+Converting the fatality data for Canada, the United States, and the United Kingdom to the rate values shows that there is a relationship among traffic hazard (*T*)—death/vehicle; motorization (*M*)—vehicle/person; and personal hazard (*H*)—death/person as shown in Figure 2. The notation and the idea of using these variables come from the work of Trinca et al. (*1*). The *T*-*M* plane shows that the fatality rate continues to fall regularly as motorization increases.
+
+It is obvious that there is a common tendency of national traffic accidents and that the fatality model has two extremes (*2*). The first extreme occurs during early motorization, which has a high traffic hazard and low and increasing personal safety. The second extreme is at full motorization, which is characterized by a moderate and falling traffic hazard and a low decreasing personal safety. Between these extremes, there is a maximum value of fatalities per capita.
+
+The current decreasing trend in road fatalities in the developed countries has attracted the attention of many researchers. Researchers used several approaches to construct models from the available statistical data to explain the decreasing trends. Smeed (*7*) used regression analysis, and a later work by Minter (*8*) used learning theory or risk theory as used by Koornstra (*6*).
+
+Learning theory in terms of road safety was first introduced by Minter (*8*), who stresses that a collective individual learning is involved in improving road safety. That model suggests that the major feature of the traffic safety system is that drivers learn from accumulated driving experience. Later Koornstra (*6*) rejected Minter's interpretation and developed a risk-adaptation model on the basis of a community learning process.
+
+In 1994, Navin et al. (*2*) investigated a fundamental relationship between traffic hazard, motorization, and personal hazard and built a model to link them. The model fits a wide range of data well and allows planners and engineers to estimate the future maximum road fatalities for developed and developing countries.
+
+F. P. D. Navin, Department of Civil Engineering, University of British Columbia, 2324 Main Mall, Vancouver, B.C., Canada V6T 1Z4. A. Bergan and G. Zhang, Department of Civil Engineering, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 0W0.
+
+![](_page_1_Figure_1.jpeg)
+
+**FIGURE 1 General relationship between fatalities and vehicles over time.**
+
+In this paper, Navin's model has been extended to incorporate a traffic hazard model based in part on the ideas developed by Koornstra (*6*). The extended model should allow a more detailed analysis of some of the social and engineering factors that contribute to road safety.
+
+The proposed revisions to Navin's model are simple empirical ways of summarizing relationships among the elements inside the traffic safety system. Plausible arguments also might be made to use registered vehicles as a measure of exposure. In the absence of accurate data on vehicle mileage over the long term, especially in developing countries where the national reporting system is not adequate, the number of vehicles is a more reliable statistic.
+
+Because Navin's model is a generalization that allows the derivation of previous models, a review of the previous models is made first to help understand this new model.
+
+# **PREVIOUS ROAD SAFETY MODELS**
+
+# **Smeed's Law**
+
+An early, well-known model was developed by Smeed in 1949 (*7*). It was found that traffic hazard was inversely related to motorization. Smeed used motorization as a measure of exposure to the risk of road accident and successfully fitted data to the law that bears his name. Smeed's law is
+
+$$d = 0.0003(vp^2)^{1/3} \tag{1}$$
+
+$$\frac{d}{v} = a \left(\frac{v}{p}\right)^b \tag{2}$$
+
+where
+
+*d* 5 number of deaths,
+
+*v* 5 number of vehicles,
+
+*p* 5 population, and
+
+*a, b* 5 constants.
+
+Many practical models have been established on the basis of Smeed's work. The models provide reasonable methods for predicting fatalities.
+
+![](_page_1_Figure_17.jpeg)
+
+**FIGURE 2 Fundamental relationship among deaths, vehicles, and population over time.**
+
+Smeed's law has critics, such as Andreassen (*9*) and Broughton (*10*). Andreassen pointed out that Smeed analyzed the data for 20 countries for only 1 year. The original work did not have any timeseries data. Andreassen and Broughton showed that each country has a unique set of constants *a* and *b* in Smeed's Law. Broughton went so far as to state, "This paper (his) has demonstrated unequivocally that the relationship detected by Smeed among data from 1938 has no general validity. . . . They are simply empirical relations that may indicate the existence of more fundamental relationships" (*10*). This paper explores some of those more fundamental relationships.
+
+Adams (*11*) was more supportive of Smeed's Law and illustrated the robustness of the law by applying it to a number of countries. Adams considered the model as a national learning curve and was critical of any one group taking credit for the improvements in road safety. Adams cited economic studies by Cooper (*12*) that explained some of the variance in the main models. Adams also used U.S. data after 1966, the year the National Traffic and Motor Vehicle Safety Act and the Highway Safety Act were signed, to show that the same curve fit both data sets.
+
+Fieldwick (*13*) developed an interesting application of Smeed's formula that included speed limits. The equations are
+
+$$D = 0.00097215 (P^{1.02192}Ur^{1.786134}Ru^{1.064236})$$
+
+$$C = 0.0358442 (V^{1.074341} Ur^{1.591740} Ru^{1.394894})$$
+
+where
+
+*D* 5 number of fatalities in 1984,
+
+*C* 5 number of casualties in 1984,
+
+*P* 5 population in 1984 in millions,
+
+*V* 5 vehicles in 1984 in millions,
+
+*Ur* 5 urban speed limit in kilometers per hour, and
+
+*Ru* 5 rural speed limit in kilometers per hour.
+
+Minter (*8*) showed that Smeed's Law is a consequence of Wright's learning curve model. Wright's learning curve model is a double logarithmic that measures performance and is given as
+
+$$t_n = t_1 n^{-b}$$
+
+where
+
+*t<sup>n</sup>* 5 time for *n*th item to be made on *n*th repetition,
+
+*t*<sup>1</sup> 5 time for first repetition,
+
+*n* 5 cumulative number of repetitions, a measure of experience, and
+
+*b* 5 measure of rate of reduction of time per occasion, about 0.4 for United Kingdom.
+
+Minter showed that Smeed's formula is a double-logarithmic model that is analogous to Wright's learning curve model. Minter argues that Smeed's measure of performance is deaths per vehicle and the measure of accumulated experience is vehicles per unit of population. This last measure is close to a measure of accumulated experience, because the number of vehicles grows annually and adds to the travel experience. Also vehicles per head indicates a country's accumulated experience. Minter suggests that because of the underlying learning process road safety will improve in spite of other interventions.
+
+### **Trinca et al. Model (1988)**
+
+Trinca et al. (*1*) showed that there is a relationship among *T, M*, and *P*, and this relationship can reflect a country's motorizing stage. The relationship is
+
+$$P = T \times M \tag{3}$$
+
+where
+
+*P* 5 personal safety measured as number of deaths per 100,000 people,
+
+*T* 5 traffic safety measured as number of deaths per 10,000 motor vehicles, and
+
+*M* 5 motorization measured as number of registered motor vehicles per 1,000 people.
+
+The model is descriptive and graphically showed a relationship among the three variables using data from 33 developed and developing countries. The single data point for each country used information between 1980 and 1986. In general the model showed that at low levels of traffic safety, that is, few deaths per vehicle,
+
+motorization was high and personal safety, or deaths per person, was also high. This situation is representative of developed countries. Developing countries have a high value of traffic safety (deaths/vehicle), low motorization, and low personal safety (deaths/person).
+
+They looked at the time trends from 1970 to 1985. This particular graph provided the inspiration and key ideas used to develop this paper. The time trend shows that countries at the start of motorization will have low values of motorization and personal safety and a high value of traffic safety. The country then, as motorization increases, moves toward a peak value of personal safety, that is, a maximum rate of deaths per population. As motorization increases beyond this maximum value personal safety and traffic safety values decrease.
+
+The research of this paper took this as a starting point and developed mathematical relationships based on time-series data for Canada, the United States, and the United Kingdom.
+
+#### **Towill Learning Curves**
+
+This is an exponential learning curve model that assumes that the number of vehicles in a country grows annually and each year adds to the driving experience (*8*). The performance function is
+
+$$y_x = y_s[1 - \exp(1 - x/t)] + c$$
+ (4)
+
+where
+
+*y<sup>x</sup>* 5 rate of production (measure of performance) at time *x* from start,
+
+*ys* , *c* 5 constants, and
+
+*t* 5 time constant of exponential curve.
+
+The performance measure is the casualty rate defined as casualties per unit distance traveled. Minter (*8*) showed that U.K. road death data starting from 1945 could be fitted successfully to the Towill model. The model has a ceiling in the value of *y<sup>x</sup>* as *x* tends to infinity. The value of *x* that measures experience is the cumulative distance driven. The equation developed by Minter was
+
+$$y_x = a(1 - e^{-x/c}) + c$$
+  
+ $y_x = 2.27 + 119.832(1 - e^{-x/14011.5})$  (5)
+
+where *y<sup>x</sup>* is millions of kilometers per death, and *x* is cumulative distance traveled by year, in kilometers.
+
+The estimate of *x* used an exponential model to estimate first the vehicle population by year and then a linear model to estimate the average annual distance traveled per vehicle. Good results were obtained with an *R*<sup>2</sup> value of 0.75. It was also found that for the United Kingdom an upper limit to *y<sup>x</sup>* is about 156 million km per death and a motor vehicle saturation of about 17.5 million automobiles.
+
+Minter showed that although the models to forecast road casualties are robust there are other variables that overlay the exponential learning curve models. He suggests that these controls arise in social and general state of mind of the road user, and he further states, "The road safety industry would say that the improvement in casualty rate is due only to its efforts over the years. This study suggests otherwise—that things will improve anyway, and left to themselves they might even improve better than with legislative intervention" (*8*).
+
+This work will show that thoughtful intervention into the technology of driving produces beneficial results above those that could be obtained by simple learning curves.
+
+#### **Koornstra's Function (1992)**
+
+Koornstra (*6*) showed that the long-term trends in mobility and safety can be related by an evolutionary learning model of growth and risk adaptation.
+
+Growth of mobility can be described by sigmoid curves with a saturation level (taken from biology or econometrics) and risk in road traffic as the fatality rate which follows a decreasing adaptation curve (taken from mathematical learning theory). A single peaked curve for the road fatalities per years, is a necessary result from saturating growth and steadily decreasing risk adaptation. Such growth and risk adaptation curves are related.
+
+Koornstra used a simple model of the following form:
+
+$$F = \frac{F}{D} \times D \tag{6}$$
+
+where *F* is fatalities per year, and *D* is vehicle-kilometers per year. The mathematical function is
+
+$$F_{t} = zV_{t}^{x}V_{t-k}^{w} \left[ \left( \frac{V_{\text{max}}}{V_{t-k}} \right)^{c} - 1 \right]^{y}$$
+ (7)
+
+where
+
+*F<sup>t</sup>* 5 number of fatalities in year *t*,
+
+*V<sup>t</sup>* 5 vehicle-kilometers in year *t*,
+
+*V*max 5 saturation level of vehicle-kilometers,
+
+*k* 5 time lag in years, and
+
+*w, x, y, z*, and *c* 5 constants.
+
+He showed that this followed Hauer's (*14*) definition of safety:
+
+safety 5 (risk)(exposure)
+
+His model is broader in concept than those proposed by either Broughton (*10*) or Minter (*8*). Minter concentrated on collective individual learning; Koornstra uses a gradual learning process of society that enhances safety through changes in the road, vehicle, operating rules, and individual behavior. He stresses the concept of learning as a function of time, which is better described as adaptation. He also points out that there is a time lag in realizing safety benefits from any improvement.
+
+He went on to successfully apply the model to selected developed countries for both fatalities and casualties and outlined a combined open- and closed-loop system to explain road safety.
+
+#### **Navin Model (1995)**
+
+The fundamental model of road safety developed by Navin et al. (*2*) is shown in Figure 3. It is easily seen that Smeed's Law is on the *T*-*M* plane, but the hyperbola function has been changed to an exponential function of the form
+
+$$T = T_f e^{\frac{-M}{M_o}} \tag{8}$$
+
+where
+
+*M<sup>o</sup>* 5 value of motorization at maximum personal hazard, *P*max, *T<sup>f</sup>* 5 point where exponential curve pierces *T*-axis,
+
+![](_page_3_Figure_24.jpeg)
+
+**FIGURE 3 Four-dimensional view of fundamental relationship over time.**
+
+*T* 5 traffic hazard, deaths per vehicle, and *M* 5 motorization, vehicles per population.
+
+In applying Equation 3, the personal hazard is derived as
+
+$$P = MT_f e^{\frac{-M}{M_o}} \tag{9}$$
+
+Note that this model defines hazard rather than safety. The coordinates of maximum personal hazard are given by
+
+$$P_{\text{max}} = M_o \frac{T_f}{e} \tag{10}$$
+
+$$T_o = \frac{T_f}{e} \tag{11}$$
+
+Equation 8 is analogous to the linear-operational model by Broughton (*10*). Both are shown to correspond with Towill's learning curve in an inverse way and categorized as two special cases. The model is also found in traffic flow theory as proposed by Underwood (*15*).
+
+Navin's model can be built up from basic time-series data as shown in Figure 4. The time-series data of fatalities and vehicles (or any other measure of exposure) as given in Figure 4*a* were used by Koornstra (*3*) as the basis of his model. Smeed (*7*) used a single year time slice from the *F-V* and *V-P* planes of Figures 4 *a* and *b* to obtain the model shown in Figure 4*c*. Adams (*11*) and others have shown that Smeed's formulation may also be applied over time as implied in the diagrams. Navin et al. (*2*) extended the two-dimensional Smeed model to three dimensions.
+
+The basic element of the theory is found, as pointed out by Koornstra, by study of fatalities over time. Fatalities may be considered failures in the system. The system, in this case the technicalsocial road system, is made up of the following components: road, vehicle, driver, operating rules, and devices. The engineering failures of the road vehicle and operating devices may be studied by traditional reliability analysis (*16*). The driver may also be studied with
+
+![](_page_4_Figure_2.jpeg)
+
+**FIGURE 4 Relationship among road fatality models.**
+
+reliability analysis or with some learning curve. Because engineers study failures to learn how to avoid them, the reliability analysis also needs a learning component to reflect the new devices introduced into an existing system. Ideally one would have a set of evolving equations to learn from past failures to forecast future successes. The human-social elements of the system tend to frustrate such an ideal situation, as suggested by Wilde (*17*).
+
+The key to the successful implementation of any of the models to forecast future accident rates is understanding how accidents occurred. It is known that the causes of fatal accidents are 62 percent driver, 5 percent driver-vehicle, and 1 percent driver-road-vehicle (*18*). This implies that the traffic hazard function is
+
+# *T* 5 *f* (driver, road, vehicle, rules, society)
+
+The remainder of this paper deals with this equation and how it may be applied.
+
+# **IMPLICATION AND DEVELOPMENT OF NAVIN ET AL. MODEL**
+
+The first approach is to review the results of data in the postwar period of Canada, the United Kingdom, and the United States on *T*-*M* planes. Attempts were made to fit data to the models starting in 1900. The disruptions caused by the Depression and World War II introduced difficulties that did nothing for the analysis, so only data from after 1949 were considered. These data cover half a century that had an enormous range of changes. Tables 1 and 2 show that Navin's model as given in Equation 8 is more suitable for the data because it has much higher *R*<sup>2</sup> values and fewer standard errors (see Figure 5).
+
+Navin et al. (*2*) indicate that the proposed model can have as the dependent variable the stage of development, or time, or both. This is because the curve is to a large degree linked to the timedependent forces and economic efficiency that somehow make the
+
+**TABLE 1 Regression Result for Smeed's Law**
+
+resulting accident rate change. Partykas (*3*) and Cooper (*12*) have written on the economic aspects of accident rate change.
+
+What roles do those factors play in reducing the traffic hazard? Are there any other factors? Minter (*8*) and Koornstra (*6*) argue that the reduction of fatality rate is the consequence of a community learning process reflected by improved driver attitude to traffic safety. In other words, the key factor is the human learning process in terms of time and other factors that have their own proportional contribution to traffic safety. They provide no guidance for how the other factors may be introduced.
+
+Second, some fluctuation along the curve is found to correspond to economic activity, such as the bump on the curve during the early 1970s and the oil crisis. This finding implies that the curve can be adapted by introducing economic factors as shown by Partykas (*3*) and Cooper (*12*).
+
+The third finding is that erratic deviations along the curve are associated with legislative intervention. This intervention is defined as teaching elements in the learning process. Evans (*5*) considers that such a teaching process is achieved by providing drivers with more frequent feedback to unsafe driving actions.
+
+On the basis of those assumptions, a function of the time-based interventions is imposed on Navin's model in an exponential declining form. The traffic hazard becomes
+
+$$T = e^{a+bx+cy+dz} (12)$$
+
+where
+
+*x* 5 year 2 1949,
+
+*y* 5 economic proxy in terms of personal disposable income,
+
+*z* 5 safety improvements introduced by various legislative interventions, and
+
+*a, b, c, d* 5 constants.
+
+Using the Canadian data set from 1949 to 1991 gives the following relationship:
+
+$$T = e^{-6.97 - 0.02x + 0.01y + (-0.24z_1 - 0.45z_2 - 0.0004z_3 - 0.08z_4)}$$
+(13)
+
+where
+
+*x* 5 year 2 1949,
+
+*y* 5 personal disposable income (year 1949 5 1),
+
+- *z*<sup>1</sup> 5 55 mph dummy variable (1949 to 1973 5 0, 1974 to 1991 5 1),
+- *z*<sup>2</sup> 5 seat-belt percentage at each year,
+- *z*<sup>3</sup> 5 per capita consumption of alcohol at each year, and
+- *z*<sup>4</sup> 5 dummy variable for vehicles not equipped with regulated safety equipment (1949 to 1965 5 1, otherwise 0).
+
+The result is plotted in Figure 6 from the Trinca et al. (*1*) model. The result is found to be satisfactory with an *R*<sup>2</sup> of 0.985 for the regression function and is shown in Figure 7. Similar curves can be found for the United Kingdom and the United States. A general fatality trend on a time series is developed in Figure 8.
+
+The characteristic feature of this model is to evaluate the efficiency of the three main factors related to reduction of fatalities. Figure 8 demonstrates a relationship among the learning process, economic situation, and legislative intervention. It indicates the following:
+
+- 1. Improvement of fatality rate is the outcome of the interaction of these three factors;
+- 2. Learning process dominates the broad trend, which is a curve starting from the original surge line and gradually settling down to the social minimum acceptable level when time tends to infinity;
+- 3. Coefficient of economic activity is significant and positively related to the fatality rate; and
+- 4. Introduction of significant legislative intervention is associated with substantial reduction of fatalities.
+
+These findings appear to violate some previous investigations about the relative weights of the three factors. Among previous authors, one focused only on the learning process (*8*) and another on factors of economic activity or legislative intervention (*4*). However, one key point is missed by these researchers; that is, road transport is performed in a complicated and dynamic way where people, road, environment, and legislative intervention interact. This model considers such interaction.
+
+# **DISCUSSION OF EXTENDED MODEL**
+
+Similarities can be found with the Navin et al. extended model and Leob (*4*) and Garbacz (*19*) cross-sectional models. Garbacz analyzes
+
+**TABLE 2 Regression Result for Navin's Model**
+
+![](_page_6_Figure_0.jpeg)
+
+**FIGURE 5 Fitting Navin's model to data (***2***).**
+
+![](_page_7_Figure_1.jpeg)
+
+**FIGURE 6 Comparison of various fatality models with Canadian data.**
+
+![](_page_7_Figure_3.jpeg)
+
+**FIGURE 7 Fatalities over time forecast by various models.**
+
+![](_page_8_Figure_2.jpeg)
+
+**FIGURE 8 General relationship among fatalities and vehicles over time and various forecasting methods.**
+
+the impact of various safety inspections on the reduction of fatalities by using time-series data. Garbacz (*19*) comments on the coefficient of economic activity using a time-series model.
+
+It is expected that as income rises the value of time rises, and drivers may increase speed in order to cut the cost of travel. Higher speed leads to a higher risk of a traffic accident. Greater speed may lead to more deaths, but this is offset as the value of life rises with rising income. It is expected that drivers with higher income exercise more caution. The predicted sign on the coefficient can therefore only be decided empirically by the net effect of off-setting factors.
+
+However, from analysis of the United States, Japan, Taiwan, and Canada (results of the Navin et al. extended model), the coefficients associated with economic activity tend to be generally positive.
+
+The negative coefficients associated with legislative intervention indicate that intervention makes drivers aware of traffic safety. A great deal of evidence is obtained from the large drop after 1970 in developed counties since the enormous legislative interventions of the late 1960s had taken place. Before this phase, the development of mechanical and electrical engineering on the vehicle fleet and civil engineering in the form of roadway construction contributed to a large degree to accident reduction.
+
+This highlights the point of interaction of three main factors only; details on other factors such as roadway, vehicle, medical care, and population change have been ignored and lumped in the equations constant. These factors might affect the model coefficients but not the broad downward trend.
+
+As increasing accidents and traffic hazards in the world are more closely monitored, the implication of the Navin et al. extended model is available to developing countries; that is, greater care should be placed on road safety as economies undergo extreme change. Effective legislative intervention should be implemented to shape the public attitude to road safety.
+
+# **CONCLUSION**
+
+Smeed's law (*7*), developed in 1949, has been successfully applied to multicountry and country time-series data. The formulation is not without its critics and defenders. Minter (*8*) has shown that it is a
+
+variation of Wright's learning curve. Koornstra (*6*) took a slightly different approach to forecast the change of total road deaths or safety over time as a function of risk and exposure. Risk is defined as deaths per vehicle kilometers per year and exposure is vehicle kilometers per year.
+
+Navin et al. (*2*) extended Smeed's two-dimensional traffic hazard and motorization model to a third dimension, personal hazard. It is shown in this paper how the models are conceptually related and that Koornstra's model is in fact an underlying component in Smeed's model.
+
+By using the start dates of various road programs and indicators of economic activity such as gross national product and alcohol consumption, a time-based model exists for traffic hazard for Canada between 1949 and 1991. Data before 1949 were not used because they made model correlation difficult and were not as well-behaved as after 1949.
+
+The model presented in this paper permits replication of past trends and future forecasts of fatalities. The difficulties fundamental to many regression models are present in this model, that is, how to forecast the results of innovations not represented in the equation variables.
+
+What is needed is a more fundamental mechanistic model of road safety and the effects of new intervention on road safety. The model will require an understanding of driver behavior, vehicle dynamics, consequences of a crash, and response of the emergency support services. To do this probably requires a computer simulation that allows drivers and others to make mistakes as an ideal or look up tables of expected consequences as a minimum. The next major thrust in road safety forecasting model development is to look for an even more fundamental relationship.
+
+#### **ACKNOWLEDGMENTS**
+
+This work has been supported by operating grants from the Natural Science and Engineering Research Council of Canada. The graphics and most of the accident data for Canada, the United States, and the United Kingdom were organized by Connie Leung of the University of British Columbia Department of Civil Engineering.
+
+# **REFERENCES**
+
+- 1. Trinca, G. W., et al. *Reducing Traffic Injury—A Global Challenge*. Royal Australian College of Surgeons, Melbourne, Australia, 1988.
+- 2. Navin, F., A. Bergan, and J. Qi. Fundamental Relationship for Roadway Safety: Model for Global Comparisons. In *Transportation Research Record 1441*, TRB, National Research Council, Washington, D.C., 1994, pp. 53–58.
+- 3. Partykas, S. Simple Models of Fatality Trends Using Employment and Population Data. *Accident Analysis and Prevention*, 1984, pp. 211–222.
+- 4. Leob, P. D. Automobile Safety Inspection: Further Economic Evidence. *Journal of Applied Economics*, Vol. 22, 1990, pp. 1697–1704.
+- 5. Evans, L. *Traffic Safety and the Driver*. Van Nostrand Reinhold, New York, N.Y., 1991.
+- 6. Koornstra, M. J. The Evolution of Traffic and Traffic Safety. *Journal of International Association of Traffic and Safety Sciences*, Vol. 16, No. 2, 1992, pp. 129–148.
+- 7. Smeed, R. J. Some Statistic Aspects of Road Safety Research. *Journal of the Royal Statistical Society*, Series A, Part I, 1949, pp. 1–34.
+- 8. Minter, A. L. Road Casualties—Improvements by Learning Processes. *Journal of Traffic Engineering and Control*, Feb. 1987, pp. 74–79.
+- 9. Andreassen, D. Linking Deaths with Vehicles and Population. *Traffic Engineering and Control*, Vol. 26, No. 11, Nov. 1985, pp. 547–549.
+
+- 10. Broughton, J. Predictive Models of Road Accident Fatalities. *Traffic Engineering and Control*, May 1988, pp. 296–300.
+- 11. Adams, J. Smeed's Law: Some Further Thoughts. *Traffic Engineering and Control*, 1987, pp. 296–299.
+- 12. Cooper, P. J. The Nature and Causes of the 1982 Traffic Accident Casualty Reductions in British Columbia. In *Transportation Research Record 1047*, TRB, National Research Council, Washington, D.C., 1985, pp. 1–9.
+- 13. Fieldwick, R. B. The Effects of Speed Limits on Road Casualties. *Traffic Engineering and Control*, Dec. 1987, pp. 635–640.
+- 14. Hauer, E. Traffic Conflicts and Exposure. *Accident Analysis and Prevention*, Vol. 14, 1982, pp. 352–362.
+- 15. Underwood, R. Traffic Flow and Bunching. *Proc., Australian Road Research*, Vol. 16, No. 6, 1963.
+- 16. Bury, K. *Statistical Models in Applied Science*. John Wiley and Sons, New York, N.Y., 1975.
+- 17. Wilde, G. Theory of Risk Homeostasis: Implications for Safety and Health. *Risk Analysis*, 1982, pp. 209–225 and pp. 249–258.
+- 18. Sayed, T., W. Abdelwahab, and F. Navin. Application of Fuzzy Pattern Recognition to the Identification of Accident Prone Locations. *Journal of Transportation Engineering, ASCE*, Vol. 121, No. 4, 1994, pp. 352–358.
+- 19. Garbacz, C. Traffic Fatalities in Taiwan. *Journal of Transport Economics and Policy*, Vol. 23, No. 3, 1989, pp. 317–327.

Zotero/001_artiklid/Road Safety Indicators/ohaccidentpredictionmodelrailway2006.md

@@ -0,0 +1,436 @@
+---
+category: literaturenote
+citekey: ohaccidentpredictionmodelrailway2006
+title: Accident prediction model for railway-highway interfaces
+authors: "Oh, Jutaek; Washington, Simon P.; Nam, Doohee"
+year: 2006
+date: 2006-00-00 2006
+doi: 10.1016/j.aap.2005.10.004
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2005.10.004"
+zotero_key: NJQEKQHF
+zotero_storage: MVVZAHB3
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Oh, Jutaek"
+status: converted
+---
+![](_page_0_Picture_0.jpeg)
+
+Accident Analysis and Prevention 38 (2006) 346–356
+
+![](_page_0_Picture_2.jpeg)
+
+# Accident prediction model for railway-highway interfaces
+
+Jutaek Oh <sup>a</sup>,∗, Simon P. Washington <sup>b</sup>,1, Doohee Nama,<sup>2</sup>
+
+<sup>a</sup> *The Korea Transport Institute, Ilsan, Koyang-city, Kyeonggi-do 411-701, South Korea* <sup>b</sup> *Department of Civil and Environmental Engineering, Arizona State University, Tempe, AZ 85287-5306, USA*
+
+Received 15 April 2005; received in revised form 29 September 2005; accepted 14 October 2005
+
+## **Abstract**
+
+Considerable past research has explored relationships between vehicle accidents and geometric design and operation of road sections, but relatively little research has examined factors that contribute to accidents at railway-highway crossings. Between 1998 and 2002 in Korea, about 95% of railway accidents occurred at highway-rail grade crossings, resulting in 402 accidents, of which about 20% resulted in fatalities. These statistics suggest that efforts to reduce crashes at these locations may significantly reduce crash costs.
+
+The objective of this paper is to examine factors associated with railroad crossing crashes. Various statistical models are used to examine the relationships between crossing accidents and features of crossings. The paper also compares accident models developed in the United States and the safety effects of crossing elements obtained using Korea data.
+
+Crashes were observed to increase with total traffic volume and average daily train volumes. The proximity of crossings to commercial areas and the distance of the train detector from crossings are associated with larger numbers of accidents, as is the time duration between the activation of warning signals and gates. The unique contributions of the paper are the application of the gamma probability model to deal with underdispersion and the insights obtained regarding railroad crossing related vehicle crashes. © 2005 Elsevier Ltd. All rights reserved.
+
+*Keywords:* Railroad crossings; Poisson; Gamma; Negative binomial; International comparison
+
+## **1. Introduction**
+
+The relationships between vehicle accidents and geometric design of road sections, such as horizontal curvature, vertical grade, lane width, and shoulder width, have been extensively studied using regression models [\(Abbess et al., 1981; Hauer et](#page-9-0) [al., 1988; Persaud and Dzbik, 1993; Kulmala, 1995; Poch and](#page-9-0) [Mannering, 1996; Lord, 2000; Ivan et al., 2000; Lyon et al.,](#page-9-0) [2003; Miaou and Lord, 2003; Oh et al., 2003\).](#page-9-0) A similarly large literature exists for research on roadway intersections. Relatively little research, however, has been carried out to understand and identify factors that contribute to accidents on railway-highway crossings (RHXs). Because of the lack of detailed information on crossing element data and the failure of selecting appropriate tools for analyzing the data, statistical models explaining the relationships between roadway geometry, grade crossing char-
+
+This paper seeks to fill some of the knowledge gaps regarding complex relationships between crashes involving motor vehicles and/or trains through the application of statistical models to crash data. The contrasts and tradeoffs between various probability models are discussed with the main focus to provide the greatest insight into RHX related crashes, especially in Korea.
+
+In Korea, about 95% of railroad accidents are associated with RHXs, and a total of 402 accidents occurred at these locations from 1998 to 2002, of which about 20% resulted in fatalities. These statistics suggest that efforts are needed to develop effective countermeasures to reduce crossing accidents. However, considering the fact that about 1800 crossings are located in Korea, 80 accidents per year suggest that for a period of 5 years most of the crossings observed zero accidents.
+
+Statistical models are used to examine the relationships between crossing accidents and features of crossings. However, many past studies illuminating the numerous problems with linear regression models ([Joshua and Garber, 1990; Miaou](#page-9-0) [and Lum, 1993\)](#page-9-0) have led to the adoption of more appropriate
+
+acteristics, and crossing accident frequencies have rarely been developed. Therefore, research gaps remain regarding the identification of factors associated with crashes at RHXs.
+
+<sup>∗</sup> Corresponding author. Tel.: +82 31 910 3174; fax: +82 31 910 3235. *E-mail addresses:* jutaek@koti.re.kr (J. Oh), Simon.Washington@asu.edu (S.P. Washington), doohee@koti.re.kr (D. Nam).
+
+<sup>1</sup> Tel.: +1 480 965 2220, fax: +1 480 965 0557.
+
+<sup>2</sup> Tel.: +82 31 910 3092, fax: +82 31 910 3235.
+
+regression models such as Poisson regression which is used to model data that are Poisson distributed, and negative binomial (NB) model which is used to model data that have gamma distributed Poisson means across crash sites—allowing for additional dispersion (variance) of the crash data. Although the Poisson and NB regression models possess desirable distributional properties to describe motor vehicle accidents, these models are not without limitations. One problem that often arises with crash data is the problem of 'excess' zeroes (see [Lord](#page-9-0) [et al., 2004; Washington et al., 2003](#page-9-0) for detailed discussions), which often leads to dispersion above that described by even the negative binomial model. 'Excess' does not mean 'too many' in the absolute sense, it is a relative comparison that merely suggests that the Poisson and/or negative binomial distributions predict fewer zeroes than present in the data. As discussed in [Lord et al. \(2004\),](#page-9-0) the observance of a preponderance of zero crashes results from low exposure (i.e. train frequency and/or traffic volumes), high heterogeneity in crashes, observation periods that are relatively small, and/or under-reporting of crashes, and not necessarily a 'dual state' process which underlies the 'zero-inflated' model. Thus, the motivation to fit zero-inflated probability models accounting for excess zeroes often arises from the need to find better fitting models which from a statistical standpoint is justified; unfortunately, however, the zero-inflated model comes also with "excess theoretical baggage" that lacks theoretical appeal (see [Lord et al., 2004\).](#page-9-0)
+
+Another problem not often observed with crash data is underdispersion—where the variance of the data is less than the expected variance under an assumed probability model (e.g. the Poisson). One manifestation might be "too few zeroes", but this is not a formal description. Underdispersion is a phenomenon which has been less convenient to model directly than overdispersion mainly because it is less common observed. Winkelman's gamma probability count model offers an approach for modeling underdispersed (or overdispersed) count data ([Winkelmann and Zimmermann, 1995\),](#page-10-0) and therefore may offer an alternative to the zero-inflated family of models for modeling overdispersed data as well as provide a tool for modeling underdispersion.
+
+The paper also compares the results of accident models developed in the United States and the models developed here. The examination of international differences may illuminate differences in design, driver behavior, or both that may lead to different crash processes and generate ideas for further research.
+
+The remainder of the paper is organized as follows. First, previously estimated RHX accident prediction methods are examined to both understand relationships and to identify potential areas for model improvement. Second, probability models are described. Third, the relative merits of probability models are contrasted and evaluated. Then, the modeling results are compared with modeling results from the US. The final section concludes the study.
+
+## **2. Literature review**
+
+Safety levels at highway-rail crossings continue to be of major concern despite of improved design and application practices. This speaks directly to the need to re-examine both accident prediction methods and application practices at railway-highway crossings. With respect to previously developed accident prediction methods, the Peabody Dimmick Formula, the New Hampshire Index and the National Cooperative Highway Research Program (NCHRP) Hazard Index, all lack descriptive capabilities due to their limited number of explanatory variables. The US Department of Transportation's (USDOT) Accident Prediction Formula, which is most widely used, also has limitations related to the complexity of the three-stage formula and its decline in accident prediction model accuracy over time.
+
+## *2.1. Peabody Dimmick Formula*
+
+The Peabody Dimmick Formula, one of the earliest railwayhighway crossing accident prediction model, was developed in 1941 using accident data from rural railway-highway crossings in 29 states in US. This formula is primarily used for resource allocation at railway-highway crossings. The formula uses AADT (Average Annual Daily Traffic), *P* (0, 1) is the presence of warning devices, and average daily train traffic, and is given by:
+
+$$A_5 = \frac{1.28(V^{0.17} \times T^{0.151})}{P^{0.0171} + K} \tag{1}$$
+
+where *A*<sup>5</sup> is the expected number of accidents in 5 years; *V* is the average annual daily traffic (AADT); *T* is the average daily train traffic; *P* is the protection coefficient indicative of warning devices present; *K* is the additional parameter [\(Federal Highway](#page-9-0) [Administration, 1986\).](#page-9-0)
+
+Clearly, the Peabody Dimmick Formula is derived from a stochastic model where *V* × *T* represents the interaction of motor vehicle traffic and trains, and coefficients were derived from a data fitting procedure. As such, the coefficients are based on data that are 20 years old.
+
+### *2.2. New Hampshire Index*
+
+The next evolutionary step in railway-highway crossing accident prediction models was the New Hampshire Index ([Austin and Carson, 2002\).](#page-9-0) The New Hampshire Index is given as:
+
+$$HI = V \times T \times P_{\rm f} \tag{2}$$
+
+where HI is the hazard Index; *V* is the average annual daily traffic (AADT); *T* is the average daily train traffic; *P*<sup>f</sup> is the protection factor indicative of warning devices present [\(Federal Highway](#page-9-0) [Administration, 1986\).](#page-9-0)
+
+Note the similarity in variables used to predict railwayhighway crossing accident as compared to the Peabody Dimmick Formula. The protection factor varies from state to state, and this has raised concern over its validity in accurately predicting railway-highway crossing accidents.
+
+## <span id="page-2-0"></span>*2.3. NCHRP Hazard Index*
+
+Following the development of the New Hampshire Index, the National Cooperative Highway Research Program (NCHRP) Hazard Index was developed in 1964 with a joint effort between the American Association of State Highway Officials (ASSHTO) and the Association of American Railroads (AAR) [\(Austin and Carson, 2002\).](#page-9-0)
+
+The program provides for the installation of automatic warning devices at crossings on a statewide basis according to a hazard index which is calculated using ADT, number of train movements, latest 5-year accident experience, and a protection factor based on the type of existing warning devices at the crossing. Based on the program, the DOT in the state of Connecticut designs and constructs between five and six crossings per year [\(Federal Railway Administration, 1996\).](#page-9-0) Since the start of the program, 120 crossings have received safety improvements. The NCHRP Hazard Index resembles the basic formula for the New Hampshire Index, and is given by:
+
+$$EA = A \times B \times CTD \tag{3}$$
+
+where EA is the expected accident frequency; *A* is the vehicles per day factor provided in tabular format as a function of vehicles per day; *B* is the protection factor indicative of warning devices present; CTD is the current trains per day [\(Schoppert et al.,](#page-9-0) [1968\).](#page-9-0)
+
+### *2.4. USDOT accident prediction equations*
+
+The US Department of Transportation (USDOT) accident equations were developed in the early 1980s and sought to address earlier model limitations ([Austin and Carson, 2002\).](#page-9-0) The equations were used to assist in the assessment of grade crossing hazards. The USDOT equations consist of a basic equation that predicts the number of crossing collisions and equations that predict the probabilities that collisions will result in fatality or personal injuries. The basic USDOT accident prediction equations consist of an equation for each of the three categories of warning devices: passive devices only, flashing lights, and automatic gates (see Table 1). The three equations generally utilize two highway, three railroad and two combination railwayhighway factors.
+
+Various other collision prediction or hazard index formulas utilize greater or fewer numbers of factors, many that are identical or similar to the USDOT equation factors. It may be noted that the results of the various formulas when applied to a group of crossings generally rank crossings in the same order though the predicted number of collisions or hazard index at a particular grade crossing as calculated by the various formulas differs. The Federal Highway Administration's *Railroad-Highway Grade Crossing Handbook* provides information concerning some of the various formulas [\(Federal Highway Administration,](#page-9-0) [1986\).](#page-9-0)
+
+$$a = K \times EI \times MT \times DT \times HP \times HL \times MS \times HT \tag{4}$$
+
+where *K* is the formula constant; EI is the exposure index; MT is the main tracks; DT is the day through trains; HP is the highway paved; HL is the highway lanes; MS is the max train speed; HT is the highway type.
+
+The USDOT equation factors are based on crossing characteristics that are identified in the national crossing inventory database. The basic two USDOT equation highway factors are based on the number of highway lanes and whether or not the highway is paved (a third highway factor, the type of highway: urban or rural arterial, collector, local road, etc., was removed from the equation when it was updated in 1987). The three railroad factors are based on the number of main tracks, number of daylight through trains per day, and the maximum authorized train speed. The two combination railroad-highway factors are the types of warning device (passive, flashing lights only, or gates), and exposure index. The exposure index in turn is based on the product of the annual average number of highway vehicles per day (AADT) and the average total number of train movements per day.
+
+The full USDOT formula includes an adjustment based on recent collision experience, typically the most recent 5-year
+
+Table 1 1987 USDOT accident prediction equations
+
+| Accident prediction equation factors | Warning device category |                                                      |                        |  |  |  |
+|--------------------------------------|-------------------------|------------------------------------------------------|------------------------|--|--|--|
+|                                      | Passive                 | Lights                                               | Gates                  |  |  |  |
+| K                                    | 0.006938                | 0.0003351                                            | 0.0005745              |  |  |  |
+| EI                                   | ((ct + 0.2)/0.2)0.370   | ((ct + 0.2)/0.2)0.4106                               | ((ct + 0.2)/0.2)0.2942 |  |  |  |
+| MT                                   | 1.0                     | e0.1917mt                                            | e0.1512mt              |  |  |  |
+| DT                                   | ((d + 0.2)/0.2)0.178    | ((d + 0.2)/0.2)0.1131                                | ((d + 0.2)/0.2)0.1781  |  |  |  |
+| HP                                   | e−0.5766*(hp−1)         | 1.0                                                  | 1.0                    |  |  |  |
+| HL                                   | 1.0                     | e−0.1826(h1-1)                                       | e−0.1420(h1−1)         |  |  |  |
+| HT                                   |                         | Factor estimated when equations were revised in 1987 |                        |  |  |  |
+| MS                                   | e0.0077ms               | 1.0                                                  | 1.0                    |  |  |  |
+
+*a* is the annual collisions unadjusted for accident experience; *c* is the average number of highway vehicles per day; *t* is the average total train movements per day; *d* is the average number of through train per day during daylight; *K* is the formula constant factor; MT is the number of main tracks; DT is the day thru trains; HT is the highway pavement (yes = 1, no = 2); HL is the number of highway lanes; MS is the maximum timetable speed (mph); HT is the highway type factor (defined as urban and rural, 1 = interstate, ..., 6 = local) ([Schoppert et al., 1968\).](#page-9-0)
+
+<span id="page-3-0"></span>period for which complete collision information is available. The numeric value of each of these factors is calculated using the relationship given in [Table 1.](#page-2-0)
+
+## **3. Statistical methodology**
+
+Given the variables shown in [Table 2, i](#page-4-0)ncluding the response variable 'crashes', various statistical models are considered for revealing relationships in the data. Because of the random, discrete, non-negative nature of accident data, multiple regression models were not considered to be appropriate. The Poisson regression is usually a good modeling starting point, since crash data often are approximately Poisson distributed. When data are observed with overdispersion (crash mean less than crash variance), some modifications to the standard Poisson regression are available. The most commonly applied (and described in the literature) variations include the negative binomial model and the zero-inflated models including both the zero-inflated Poisson (ZIP) and zero-inflated negative binomial (ZINB). A less common model that can deal with overdispersion and underdispersion (crash mean greater than crash variance) is the gamma probability count model. The general forms of these regression models for count data, their limitations, and brief descriptions of their estimation procedures are now presented. In all models 'subjects' are crossings (RHXs), and events are crashes.
+
+## *3.1. The Poisson model*
+
+Integer count data are often approximated well by the Poisson distribution. In the Poisson regression model, the expected number of crashes follows a Poisson distribution, where the expected crash count for the *i*th crossing ˆyi, *i* = 1, ..., *N* is a function of covariates X*ij*, *i* = 1, ..., *N*, *j* = 1, ..., *M* so that:
+
+$$y_i \sim Poi(\lambda_i)$$
+
+$$\hat{\lambda}_i = \exp(\beta_0 X_{i0} + \beta_1 X_{i1} + \beta_2 X_{i2} + \ldots + \beta_M X_{iM})$$
+
+$$= \exp\left(\sum_{j=1}^{M} \beta_j X_{ij}\right) \tag{5}$$
+
+where the β*j*'s are the estimated regression coefficients across covariates *j* = 1, ..., *M* (for the slope intercept model the first covariate is a vector of 1's) averaged across crossings *i* = 1, ..., *N*.
+
+Because the Poisson regression model is heteroscedastic, the model coefficients for (5) are typically estimated via maximum likelihood methods. The likelihood function for the Poisson regression model (Eq. (5)) is given as (with *j*subscripts removed for simplicity):
+
+$$L(\beta) = \prod_{i} \frac{\exp[-\exp(\beta X_i)][\exp(\beta X_i)]^{y_i}}{y_i!}$$
+ (6)
+
+The maximum value possible of the likelihood for a given data set occurs if the model fits the data exactly, resulting in a value of 0. In practice, however, the value 0 is a theoretical lower bound, since integer outcomes are observed while the model predicts non-integer outcomes.
+
+If the mean of the crash counts is not equal to the variance (after accommodating a reasonable degree of sampling variability), then the data are said to be over- or underdispersed. In practice, overdispersion is the most commonly observed condition (variance mean) with respect to crash data, where the 'extra' variation is thought to arise from unobserved differences across sites ([Washington et al., 2003; Lord et al., 2004\).](#page-10-0)
+
+## *3.2. The negative binomial model*
+
+The negative binomial regression is a commonly applied alternative statistical model to deal with overdispersed data. The negative binomial model takes the relationship between the expected number of accidents occurring at the *i*th element and the *M* parameters, *Xi*1, *Xi*2, ..., *XiM*, as follows:
+
+$$y_i \sim Poi(\lambda_i)$$
+
+$$\hat{\lambda}_i = \exp(\beta_0 X_{i0} + \beta_1 X_{i1} + \beta_2 X_{i2} + \dots + \beta_M X_{iM} + \varepsilon_i)$$
+
+$$= \exp\left(\sum_{j=1}^M \beta_j X_{ij} + \varepsilon_i\right)$$
+(7)
+
+where exp(ε*i*) is distributed as gamma with mean 1 and variance α2. The negative binomial distribution arises as a consequence of gamma heterogeneity in Poisson means, hence the name. The effect of the error term in the negative binomial regression model allows for overdispersion of the variance, such that:
+
+$$Var(y_i) = E(y_i) + \alpha E(y_i)^2$$
+(8)
+
+where α is the overdispersion parameter.
+
+If overdispersion, α, equals 0, the negative binomial reduces to the Poisson model. The larger the value of α, the more variability there is in the data over and above that associated with the mean λˆi. As is the case for the Poisson regression model, the coefficients β*<sup>j</sup>* are estimated by maximizing the log likelihood loge*L*(β).
+
+### *3.3. The gamma model*
+
+Underdispersion – when the crash mean is greater than the crash variance – is a phenomenon which has been much less convenient to model directly. The gamma model proposed by Winkelman (1995) provides an approach for count data with under or overdispersion. More detailed discussion can also be found in [Cameron and Trivedi \(1998\).](#page-9-0)
+
+The gamma probability model for count data is given as:
+
+$$Pr[y_i = j] = Gam(\alpha j, \lambda_i) - Gam(\alpha j + \alpha, \lambda_i)$$
+(9)
+
+<span id="page-4-0"></span>Table 2 Summary statistics of key variables
+
+| Variables                                                                                             | Mean                 | Median           | Minimum       | Maximum           | Frequency                                 |
+|-------------------------------------------------------------------------------------------------------|----------------------|------------------|---------------|-------------------|-------------------------------------------|
+| Number of accidents                                                                                   | 0.33                 | 0                | 0             | 1                 | 162                                       |
+| Number of tracks                                                                                      |                      |                  |               |                   |                                           |
+| 1<br>2                                                                                                |                      |                  |               |                   | 100 (61.73%)<br>62 (38.27%)               |
+| Average daily traffic (ADT) (vehicles per day)<br>Average daily railway traffic (trains per day)      | 4617<br>70           | 671<br>57        | 10<br>32      | 61199<br>203      | 162<br>162                                |
+| Road pavement<br>0 (Paved)<br>1 (Not paved)                                                           |                      |                  |               |                   | 156 (96.30%)<br>6 (3.70%)                 |
+| Number of lanes<br>Road width (ma)<br>Grade crossing width (m)                                        | 1.6<br>5.9<br>5.9    | 1<br>1<br>4.5    | 1<br>2<br>2.5 | 6<br>37.5<br>39.7 | 162<br>162<br>162                         |
+| Presence of adjacent residential area<br>0 (No)<br>1 (Yes)                                            |                      |                  |               |                   | 109 (67.28%)<br>53 (32.72%)               |
+| Presence of adjacent commercial area<br>0 (No)<br>1 (Yes)                                             |                      |                  |               |                   | 13 (8.02%)<br>149 (91.98%)                |
+| Presence of agricultural area<br>0 (No)<br>1 (Yes)                                                    |                      |                  |               |                   | 37 (22.84%)<br>125 (77.16%)               |
+| Presence of adjacent school<br>0 (No)<br>1 (Yes)                                                      |                      |                  |               |                   | 146 (90.12%)<br>16 (9.88%)                |
+| Distance to nearest school (m)                                                                        | 3.2                  | 0                | 0             | 200               | 162                                       |
+| Presence of speed hump<br>0 (No)<br>1 (Yes)                                                           |                      |                  |               |                   | 134 (82.72%)<br>28 (17.28%)               |
+| Presence of rumble strip<br>0 (No)<br>1 (Yes)                                                         |                      |                  |               |                   | 146 (90.12%)<br>16(9.88%)                 |
+| Presence of reflecting mirror<br>0(No)<br>1 (Yes)                                                     |                      |                  |               |                   | 107 (66.05%)<br>55 (33.95%)               |
+| Distance of train detector (m)<br>Warning time difference (s)<br>Crossing angle (◦)                   | 824.5<br>25.5<br>6.6 | 919.5<br>20<br>0 | 0<br>0<br>0   | 1329<br>232<br>70 | 162<br>162<br>162                         |
+| Types of barrier<br>Duplex semaphore barrier<br>Simplex semaphore barrier<br>Vertical-lifting barrier |                      |                  |               |                   | 41 (25.31%)<br>90 (55.56%)<br>31 (19.14%) |
+| Types of crossing gate<br>Pole gate<br>Normal gate                                                    |                      |                  |               |                   | 19 (11.73%)<br>143 (88.27%)               |
+| Types of interception<br>One-way interception<br>Two-way interception                                 |                      |                  |               |                   | 68 (41.98%)<br>94 (58.02%)                |
+| Types of warning device<br>Normal warning device<br>Suspended warning device                          |                      |                  |               |                   | 123 (75.93%)<br>39 (24.07%)               |
+| Presence of train moving direction indicator<br>0 (No)<br>1 (Yes)                                     |                      |                  |               |                   | 103 (63.38%)<br>59 (36.42%)               |
+
+<span id="page-5-0"></span>Table 2 (*Continued* )
+
+| Variables                            | Mean | Median | Minimum | Maximum | Frequency    |
+|--------------------------------------|------|--------|---------|---------|--------------|
+| Control types                        |      |        |         |         |              |
+| Track circuit                        |      |        |         |         | 49 (30.25%)  |
+| Controller                           |      |        |         |         | 30 (18.52%)  |
+| Others                               |      |        |         |         | 81 (50.00%)  |
+| Presence of a guide                  |      |        |         |         |              |
+| 0 No)                                |      |        |         |         | 126 (77.78%) |
+| 1 (Yes)                              |      |        |         |         | 36 (22.22%)  |
+| Presence of adjacent intersection    |      |        |         |         |              |
+| 0 (No)                               |      |        |         |         | 88(54.32%)   |
+| 1 (Yes)                              |      |        |         |         | 74 (45.68%)  |
+| Presence of traffic signal           |      |        |         |         |              |
+| 0 (No)                               |      |        |         |         | 108 (66.67%) |
+| 1 (Yes)                              |      |        |         |         | 54 (33.33%)  |
+| Sight distance of grade crossing (m) |      |        |         |         |              |
+| Left-side                            | 85.6 | 70     | 9       | 997     | 162          |
+| Right-side                           | 82.6 | 65     | 10      | 991     | 162          |
+
+<sup>a</sup> 1 m = 3.28083 feet.
+
+where:
+
+$$\lambda_i = \exp(\beta' X_i)$$
+
+$$\operatorname{Gam}(\alpha j, \lambda_i) = 1, \quad \text{if} \quad j = 0, \quad \text{or} \quad \frac{1}{\Gamma(\alpha j)} \int_0^{\lambda_i} u^{\alpha j - 1} e^{-u} du$$
+
+$$\text{if} \quad j > 0, j = 0, 1, \dots$$
+
+The dispersion parameter is again α; there is underdispersion if α > 1, overdispersion if α < 1, and equidispersion if α = 1, which reduces the gamma probability to the Poisson model. Due to the relative scarcity of the gamma probability count model in the transportation literature, additional model details are provided. The conditional mean function is given by:
+
+$$E[y_i|X_i] = \sum_{j=1}^{\infty} j \operatorname{Gam}(\alpha_j, \lambda_i)$$
+ (10)
+
+and the cumulative distribution function is:
+
+$$F(T|\alpha, \lambda_i) = \int_0^T \frac{\lambda_i^{\alpha j}}{\Gamma(\alpha j)} u^{\alpha j - 1} e^{-\lambda_i u} du, \alpha > 0, \lambda_i > 0$$
+
+$$= \frac{1}{\Gamma(\alpha j)} \int_0^{\lambda_i T} u^{\alpha j - 1} e^{-u} du, j = 0, 1, \dots$$
+
+$$= \operatorname{Gam}(\alpha j, \lambda_i T)$$
+(11)
+
+### *3.4. Zero-inflated Poisson model*
+
+Zero altered count models, such as the zero-inflated Poisson (ZIP) and zero-inflated negative binomial ZINB models have seen recent attention in crash analysis literature ([Miaou, 1994;](#page-9-0) [Shankar et al., 1997\).](#page-9-0) However, as pointed out in [Washington](#page-10-0) [et al. \(2003\)](#page-10-0) and with greater emphasis in [Lord et al. \(2004\),](#page-9-0) zero-inflated models may offer improved fit and perhaps better predictive performance, but these models lack theoretical appeal with respect to crash data in most circumstances. So, if statistical fit is the main objective of modelling, then zero-inflated models can often outperform Poisson and negative binomial models. However, if agreement with underlying model theory is paramount (in addition to statistical fit), then alternatives might be sought.
+
+Zero-inflated models are theorized to account for "excess zeroes"—zeroes observed in the data above and beyond the number of zeroes predicted by Poisson or negative binomial models. A troubling assumption (of the zero-inflated theory with respect to crash data) is that excess zeroes may be present because certain crash locations can be considered to be virtually safe—in a zero accident state ([Lord et al. \(2004\)](#page-9-0) demonstrate that the excess zeroes are not likely to be caused by an underlying zero state but instead by high heterogeneity in crash counts, low exposure, or small spatial or temporal measurement scales). The remaining 'non-zero' locations are theorized to follow a normal count process for accident frequency in which non-negative integers (i.e., including zero) are possible accident frequency outcomes over a specified time period. The zip models can be thought of two stage models, where the first stage is a splitting model (e.g. binomial) between two states (zero or count), and the second stage is the count model—Poisson.
+
+The zero-inflated Poisson (ZIP) assumes that the events, *Y* = (*Y*1, *Y*2, ..., *Yn*), are independent and
+
+$$Y_i = 0$$
+ with probability  $p_i + (1 - p_i)e^{-\lambda}$ 
+
+$$Y_i = y$$
+ with probability $(1 - p_i)e^{-\lambda i} \frac{\lambda_i^y}{y!}$   $y = 1, 2, \dots$  (12)
+
+<span id="page-6-0"></span>To test the goodness of fit of a zero-inflated model, [Vuong](#page-10-0) [\(1989\)](#page-10-0) proposed the following test statistic for non-nested models:
+
+$$m_i = \log\left(\frac{f_1(y_i/x_i)}{f_2(y_i/x_i)}\right)$$
+ (13)
+
+where *f*1(*yi*/*xi*) is the probability density function of the zeroinflated model and *f*2(*yi*/*xi*) is the probability density function of the Poisson or negative binomial distribution. Then Vuong's statistic for testing the non-nested hypothesis of zero-inflated model versus traditional model is [\(Greene, 1997\):](#page-9-0)
+
+$$v = \frac{\sqrt{n} \left[ ((1/n)) \sum_{i=1}^{n} m_i \right]}{\sqrt{((1/n)) \sum_{i=1}^{n} (m_i - \bar{m})^2}} = \frac{\sqrt{n(\bar{m})}}{S_m}$$
+(14)
+
+where ¯m is the mean, *Sm* is standard deviation, and *n* is a sample size.
+
+### **4. Data analysis**
+
+## *4.1. Data description*
+
+With consideration given to variables applied in past models and data availability, data were obtained for estimating RHX crash models. Accident records at a total of 162 crossings, which constitute about 10% of railway-highway crossings in Korea, were first sampled randomly from all crash locations. Detailed records of accidents by approach for the 5-year period from 1998 to 2002 were extracted from the Korean National Railroad accident database.
+
+A total of 56 explanatory variables were explored during model development. Some of the main explanatory variables include *Average Daily Traffic*, *Average Daily Train Volumes*, *number of tracks*, *number of roadway lanes*, *road widths*, *crossing widths*, *traffic control devices*, *presence of flashing lights*, *warning times*,*stop signs*, *crossing warning sings*,*speed humps*, *road grade*, *crossing angles*, *roadway speed limits*, *train detector types*, *train detector distances*, and *sight distances*. These railway-highway crossing data available were collected over a period from December 2003 to February 2004 through site visits and investigations.
+
+Several indicator variables were also developed. An *adjacent intersection* variable indicates the possible effect of approaching and departing traffic from an adjacent intersection and is equal to 1 if an adjacent intersection is within 50 m from the crossing under consideration and 0 otherwise. A *land use* variable is created to account for additional 'distractions' and 'complexity' associated with nearby residents, shopping, employees, and schools related activities. It is expected that introduction of these variables may yield significant results as observed in other studies ([Laffey, 1999; Long, 2003\).](#page-9-0) [Table 2](#page-4-0) summarizes the main variables considered in the study.
+
+### *4.2. Results of the analysis*
+
+The modeling starting point is a Poisson regression model with functional form guided by prior research regarding RHX
+
+Table 3 Poisson estimation of railway-crossing accidents (α = 0.1)
+
+| Variables                            | Coefficient | t-Ratio |  |
+|--------------------------------------|-------------|---------|--|
+| Constant                             | −5.406      | −5.94   |  |
+| ADT                                  | 0.460       | 5.09    |  |
+| Presence of commercial area          | 0.975       | 2.53    |  |
+| Train detector distance              | 0.0016      | 2.61    |  |
+| Presence of track circuit controller | −0.917      | −2.72   |  |
+| Presence of guide                    | −0.613      | −1.56   |  |
+| Presence of speed hump               | −1.063      | −2.50   |  |
+| Log-L                                | −98.35      |         |  |
+| G2                                   | 102.80      |         |  |
+| R2                                   | 0.27        |         |  |
+| R2                                   | 0.30        |         |  |
+
+crashes (and as reflected in the 56 potential explanatory variables). The correlation matrix between variables was estimated to avoid multicollinearity between explanatory variables. A 90% confidence interval was used as an acceptable cutoff for statistical significance. The results of the Poisson model are shown in Table 3. In the Poisson model,*roadway traffic flows, presence of commercial areas near RHX crossings, presence of speed hump*, *track circuit detection system*, *presence of guide at RHX*, *train detector distance from RHX crossings* were found to be statistically significant.
+
+As mentioned in Section [3](#page-3-0) of this paper, the regression-based test for overdispersion can determine the appropriateness of negative binomial regression models over Poisson regression models, and Vuong's test statistic is suitable for testing the appropriateness between the zero-inflation model and the traditional Poisson. This decision rule was adopted in selecting the proper econometric method for the RHX crash models.
+
+Because overdispersion has been observed in prior studies, a negative binomial regression model and zero-inflated Poisson model were estimated and compared to the Poisson. The Poisson model without zero inflation accounts sufficiently well for the number of observed zeros in the data, and in fact underdispersion was observed. The negative binomial regression model and zeroinflated model, therefore, are not well suited for the RHX data.
+
+Suspecting possible underdispersion, a gamma probability count model was fit to the data (see Table 4). The results
+
+Table 4 Gamma estimation of railway-crossing accidents (α = 0.1)
+
+| Variables                                                            | Coefficient | t-Ratio |
+|----------------------------------------------------------------------|-------------|---------|
+| Constant                                                             | −3.438      | −3.41   |
+| ADT                                                                  | 0.230       | 3.01    |
+| Average daily railway traffic                                        | 0.004       | 1.66    |
+| Presence of commercial area                                          | 0.651       | 2.27    |
+| Train detector distance                                              | 0.001       | 2.24    |
+| Time duration between the activation of<br>warning signals and gates | 0.004       | 1.82    |
+| Presence of speed hump                                               | −1.58       | −1.84   |
+| α (Dispersion parameter)                                             | 2.062       | 2.72    |
+| Log-L                                                                | −98.69      |         |
+| Restricted log likelihood                                            | −100.52     |         |
+| Chi-squared                                                          | 3.67        |         |
+
+in [Table 4](#page-6-0) revealed that for any subset of the independent variables, the RHX crash data exhibit underdispersion, where α = 2.062 (refer to [Winkelmann \(1997\)](#page-10-0) and[Cameron and Trivedi](#page-9-0) [\(1998\)](#page-9-0) for detailed discussions for underdispersion) the model with six covariates and a slope intercept term. The log likelihood ratio test, comparing underdispersion versus equidispersion, favors the former. Thus, the gamma probability model for count data appears to offer considerably good statistical fit with both significant variables and accounts for the mild underdispersion.
+
+In the RHX crash model, *road width*, *crossing width*, *number of tracks*, *existence of nearby schools*, *gate control types*, *crossing angle*, *presence of adjacent intersections*, *crossing shape*, *rumble strip*, *reflecting mirror*, etc. were not statistically significant. Although road and crossing width and number of tracks are thought to be important, the effects of these factors are likely to be captured by the correlated variables, traffic flow and train volumes.
+
+Consistent with the concept of exposure to risk, an aggregate relationship between accident frequency and functions of roadway traffic flows (entering and leaving traffic flows) were explored. Expectedly, the total roadway traffic volume was found to increase annual RHX crashes significantly (*t* = 3.01), similar to findings by others [\(Austin and Carson, 2002; Hauer et al.,](#page-9-0) [1988; Lyon et al., 2003; Oh et al., 2003, 2004\).](#page-9-0) Not only do higher roadway traffic volumes increase aggregate exposure to risk, but increased roadway traffic volumes at crossings restrict the mobility of motor vehicle drivers at crossings, raising driver frustration levels, leading to greater risk taking on approaches to crossings.
+
+The model results also indicate that crossing accidents increase (*t* = 1.66) as daily train volumes increase, again as expected. Similar to roadway traffic flows, crash opportunities increases with increasing train volumes.
+
+The presence of commercial areas near RHX crossings is associated with higher accidents (*t* = 2.27), which agrees with expectation because relatively more complicated traffic activities occur in commercial areas and because drivers in commercial areas may be relatively unfamiliar with the crossing (compared with residential and agricultural areas). There are likely to be a larger percentage of trucks in these areas also, perhaps reducing visibility for motorists.
+
+Many past studies ([Berg et al., 1982; Hopkins, 1981; Mather,](#page-9-0) [1991\)](#page-9-0) indicate that grade crossing safety increases with provision of reasonable and consistent warning times. This study also supports these findings. In this study, the distance of train detector from crossings was associated with an increase in accidents (*t* = 2.24) and the time duration between the activation of warning signals and the activation of gates increases the probability of accidents (*t* = 1.82). Warning times refer to the time between device activation and arrival of a train at the crossing and are usually controlled by the train detection device. In Korea, less than 5% of crossings are using constant warning time systems. In other words, whenever a train is detected in front of the crossing, warning signals are activated regardless of speeds of the train resulting in inconsistent warning times and potentially excessive delays. Excessively long warning times also have negative impacts on crossing safety and traffic operations because frequent users of a crossings become aware that signals flash 'too long' in advance of a train's arrival and thus proceed through the crossing when the warning device is activated. The drivers' expectation, then, is that a train crossing is not imminent. By eliminating unnecessarily long and inconsistent warning times, the rate of disobedience towards crossing signals would be expected to decline.
+
+Installation of an appropriate train detection system may also improve crossing safety. Train detection types are important for the crossing safety. [Table 3](#page-6-0) shows that the track circuit detection system is associated with a decrease in accident probability (*t* = −2.72), while the controller detection system is associated with greater likelihood of crashes. This is explained by the fact that the track circuit system uses lower frequency for train detection than the controller system. Thus, the track circuit system is cheaper to use, while the controller system provides efficient operations if crossings are closely spaced (i.e. 2–3 crossings within 1 km), which is common in Korea. However, the controller system sometimes suffers from detection errors when higher speed trains come because the detection length of the system is relatively short. Therefore, installing an appropriate train detection system should be considered carefully because there are tradeoffs in cost, performance, and safety of various systems. Other train detection systems were found to have insignificant effects RHX crossing safety.
+
+The presence of speed hump decreases the probability of crossing accidents (*t* = −1.84), which agreed with expectations. Because speed humps force motorists to reduce speeds prior to RHX crossings, for a given reaction time (e.g. to the presence of a train) shorter stopping distances are required. Furthermore, perhaps motorists are discouraged from 'racing' to beat a train, resulting in fewer illegal crossings.
+
+## *4.3. Model comparison*
+
+[Table 5](#page-8-0) compares the findings of this study with those from the previous studies. As discussed in the literature review, the Peabody Dimmick Formula, the New Hampshire Index, and the NCHRP model are simple to use, but they lack descriptive capabilities and some their coefficients are derived from fairly dated data. Though much more comprehensive and descriptive than the other models discussed, the USDOT accident formula is also not without limitation. As discussed by [Austin and Carson \(2002\),](#page-9-0) the complexity of three-stage formula makes it difficult to identify design improvement activities that will be most effective from improving crossing safety.
+
+It is difficult to compare directly the findings (regarding RHX factors) in this study to the USDOT accident formula, mainly because all crossings in Korea are equipped with gates, flashing lights, and warning bells. Looking for commonalities, however, both studies observed that *number of trains* and *ADT* are significant factors associated with RHX crossing accident frequencies.
+
+Surprisingly, crossing and roadway factors found to be significant predictors are different across models. In the USDOT
+
+<span id="page-8-0"></span>Table 5 Accident prediction model comparison
+
+| Independent variable                                                                                                                                                                                                     | Peabody Dimmick<br>Formula | New Hampshire<br>Index | NCHRP<br>Hazard Index | USDOT prediction<br>formula | Gamma<br>model |
+|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|----------------------------|------------------------|-----------------------|-----------------------------|----------------|
+| Constant                                                                                                                                                                                                                 |                            |                        |                       |                             |                |
+| Traffic characteristics<br>Number of day through trains<br>Total number of trains<br>Maximum timetable speed<br>Number of main tracks<br>Number of traffic lanes<br>AADT in both directions                              | <br>                       | <br>                   | <br>                  | <br><br><br><br><br>        | <br>           |
+| Commercial Area<br>Railway Characteristics<br>Highway paved or gravel<br>Highway type factor<br>Crossing Characteristics                                                                                                 |                            |                        |                       | <br><br>                    |                |
+| Pavement marking<br>Probability of a stop sign<br>Presence of Speed humps<br>Probability of a gate/gates<br>Probability of flashing lights<br>Time between warning signal and gate activation<br>Train detector distance |                            |                        |                       | <br><br>                    | <br><br>       |
+| Protection coefficient/warning devices                                                                                                                                                                                   |                            |                        |                       |                             |                |
+
+accident formula, *type of highway surface*, *presence of stop signs*, and *pavement markings* were found to be significant factors affecting accident frequency, while they were insignificant in the gamma probability model. In contrast, the gamma probability count model includes the *presence of speed humps*, *train detector distance*, the *time duration between warning signal activation and the gate activation*, and *proximity to commercial area* as significant factors in the model.
+
+The differences between models might indicate several things. First, it might indicate that unobserved underlying factors affecting safety are proxied by various easily measured factors, such as *presence of speed humps*,*stop signs*, and *pavement markings*. Second, it may indicate that true differences exist between Korean and US RHX crash experience, perhaps due to differences in driver behavior, design factors, or both. Third, it may indicate that that random fluctuation in crashes in combination with relatively low crash counts pose significant modeling challenges.
+
+## **5. Summary and conclusion**
+
+This paper presents an empirical analysis of RHX accidents in Korea. Various statistical models were estimated to illuminate structure in data with attention paid to appropriate modeling techniques. The modeling results are compared and contrasted to prior US research findings. The study intended to address and explore three main issues:
+
+1. Safety levels at RHX crossings are a major concern for transportation system operators, emphasized by a high proportion
+
+- of fatalities and serious injuries. Relatively little research has been carried out to understand and identify factors that contribute to accidents on the railway-highway crossings, and only few attempts have been made to quantify the safety effects of geometric factors on crossing safety.
+- 2. The nature of crashes between trains and motor vehicles is fundamentally different than crashes between motor vehicles, since trains operate on fixed guideways, trains have virtually no stopping ability, RHX crossings are relatively rare for motorists (compared with potential vehicle to vehicle conflicts), traffic control measures are unique at RHX crossings, and crashes that result are generally more severe. As a consequence of these differences, existing models for intersection or road segment accidents are not reliable for providing sufficient insight.
+- 3. Due to sometimes stark differences in driving behavior, compliance to traffic laws, and design and operational differences across countries, potential exists for substantially different relationships between RHX crossing features and safety. As a result, different countermeasures may have differing effectiveness across countries.
+
+After careful application and assessment of statistical models, accompanied by a detailed examination of RHX crossing accident models between Korea and the US, the following general conclusions are drawn.
+
+The gamma probability model was the most appropriate statistical model (of the ones fitted) for the RHX crossing accident data, which exhibited slight underdispersion with respect to Poisson distributed counts. The gamma probability count model <span id="page-9-0"></span>is a flexible model that can accommodate under and overdispersion, and is relatively new in the transportation safety literature. Significant explanatory factors (and their direction of association) include average *daily traffic volume* (+), *daily train volumes* (+), *proximity of commercial area* (+), *distance of train detector from crossing* (+), *time duration between the activation of warning signals* and t*he activation of gates* (+), *and presence of speed hump* (−) (see [Table 4\).](#page-6-0)
+
+Examination of the modeling results suggests that driver warning and operations associated with passing trains are associated with crash risk. Thus, fruitful research (and crash mitigation investments) may include examination (or application) of intelligent transportation system technologies associated with driver warning devices, such as devices that detect and warn approaching vehicles, trains, or both. More careful research is needed into these and other technologies and devices that offer focus on warning of impending conflicts between motor vehicles and trains.
+
+Of the four railway-highway crossing accident prediction models examined, the Peabody Dimmick Formula, the New Hampshire Index, and the NCHRP model are simple to apply, but generally lack adequate descriptive capabilities. The USDOT accident formula is relatively more comprehensive and descriptive, but the complexity of three-stage formula makes it difficult to use. Perhaps more important, the relationships between the RHX crossing and safety may differ between Korea and the US, diminishing the utility of US based models.
+
+With respect to suggestions for future railroad safety research, there is a need to include additional driver, geometric, and operational features in an analysis. For example, congestion levels (driver fatigue, frustration levels, etc.) at the time of the crash may be important, as might be the age and gender (proxy for risk aversion and driving ability) of the motor vehicle driver.
+
+Limitations in observational cross sectional studies cannot be overcome with the conduct of additional 'similar' studies. So, the results of this study should be used to inform the design of before–after studies, which in general will yield more reliable results regarding countermeasure effectiveness.
+
+## **Acknowledgements**
+
+The authors wish to express their gratitude to the Korean National Railroad and Dr. Dong Joo Park of University of Seoul for their assistance in acquiring data.
+
+### **References**
+
+- Abbess, C., Jarett, D., Wright, C.C., 1981. Accidents at Blackspots: estimating the effectiveness of remedial treatment, with special reference to the "regression-to-mean" effect. Traffic Eng. Control 22, 535–542.
+- Austin, R., Carson, J., 2002. An alternative accident prediction model for highway-rail interfaces. Accid. Anal. Prev. 34, 31–42.
+- Berg, W., Knoblauch, K., Hucke, W., 1982. Causal factors in railroad-highway grade crossing accidents. Transport. Res. Record, 847.
+- Cameron, A., Trivedi, P., 1998. The Analysis of Count Data. Cambridge University Press, New York.
+
+- Federal Railroad Administration, 1996. Highway-Rail Grade Crossing Safety Research. Office of Research and Development, Washington, DC.
+- Federal Highway Administration, 1986. Railroad-Highway Grade Crossing Handbook, second ed. FHWA-TS-86-215, Springfield, Virginia: NTIS.
+- Greene, W.H., 1997. Econometric Analysis, third ed. Prentice Hall.
+- Hauer, E., Ng, J.C.N., Lovell, J., 1988. Estimation of safety at signalized intersections. Transport. Res. Record 1185, 48–61.
+- Hopkins, J.B., 1981. Technological Innovations in Grade Crossing Protection Systems. Report TSC-FRA-71-3, FRA, U.S. Department of Transportation, Cambridge, Mass.
+- Ivan, J.N., Wang, C., Bernardo, N.R., 2000. Explaining two-lane highway crash rates using land use and hourly exposure. Accid. Anal. Prev. 32, 787–795.
+- Joshua, S.C., Garber, N.J., 1990. Estimating truck accident rate and involvements using linear and Poisson regression models. Transport. Planning Technol. 15 (1), 41–58.
+- Kulmala, R., 1995. Safety at Rural Three- and Four-Arm Junctions: Development and Applications of Accident Prediction Models. VTT Publications 233, Technical Research Centre of Finland, Espoo.
+- Laffey, S., 1999. Grade Crossings of Northeastern Illinois: An Analysis of the FRA Grade Crossing and Grade Crossing Accident Inventories, and an Analysis of the Potential Impacts from the Horn Sounding Requirement of the Swift Rail Development Act, working paper, Chicago Area Transportation Study, Chicago, IL.
+- Long, G., 2003. Easy-to-apply solution to a persistent safety problem: clearance time for railroad-preempted traffic signals. Transport. Res. Record 1856, 239–247.
+- Lord, D., 2000. The prediction of accidents on digital networks: characteristics and issues related to the application of accident prediction models. Ph.D. Dissertation, Department of Civil Engineering, University of Toronto, Toronto.
+- Lord, D., Washington, S., Ivan, J., 2004. Poisson, Poisson-gamma, and zeroinflated regression models of motor vehicle crashes: balancing statistical fit and theory. Accident Analysis and Prevention, Pergamon Press/Elsevier Science, Forthcoming, 2004.
+- Lyon, C., Oh, J., Persaud, B.N., Washington, S.P., Bared, J., 2003. Empirical investigation of the IHSDM accident prediction algorithm for rural intersections. Transport. Res. Record 1840, 78–86.
+- Mather, R.A., 1991. Seven years of illumination at railroad-highway crossings. Transport. Res. Record 1316, 54–58.
+- Miaou, S.P., 1994. The relationship between truck accidents and geometric design of road sections: Poisson versus negative binomial regressions. Accid. Anal. Prev. 26, 471–482.
+- Miaou, S.P., Lum, H., 1993. Modeling vehicle accidents and highway geometric design relationships. Accid. Anal. Prev. 25, 689– 709.
+- Miaou, S.-P., Lord, D., 2003. Modeling traffic crash-flow relationships for intersections: dispersion parameter, functional form, and Bayes' versus empirical Bayes'. Transport. Res. Record 1840, 31–40.
+- Oh, J., Lyon, C., Washington, S.P., Persaud, B.N., Bared, J., 2003. Validation of the FHWA crash models for rural intersections: lessons learned. Transport. Res. Record 1840, 41–49.
+- Oh, J., Washington, S.P., Choi, K., 2004. Development of accident prediction models for rural highway intersections. In: Proceedings of the Transportation Research Board Annual Meeting, Washington, DC.
+- Persaud, B.N., Dzbik, L., 1993. Accident prediction models for freeways. Transport. Res. Record 1401, 55–60.
+- Poch, M., Mannering, F.L., 1996. Negative binomial analysis of intersectionaccident frequency. J. Transport. Eng. 122 (2), 105–113.
+- Schoppert, D.W., Dan, W., Hoyt, Alan M. Voorhees and Associates, 1968. Factors influencing safety at highway-rail grade crossings (NCHRP Report 50). NAS-NRC Publication.
+- Shankar, V., Milton, J., Mannering, F., 1997. Modeling accident frequencies as zero-altered probability processes: an empirical inquiry. Accid. Anal. Prev. 29 (6), 829–837.
+
+- <span id="page-10-0"></span>Vuong, Q.H., 1989. Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica 57 (2), 307–333.
+- Washington, S., Karlaftis, M., Mannering, F., 2003. Statistical and Econometric Methods for Transportation Data Analysis. Chapman Hall/CRC, Boca Raton, FL.
+- Winkelmann, R., 1997. Economic Analysis of Counter Data, second ed. Springer-Verlag, Heidelberg.
+- Winkelmann, R., Zimmermann, K., 1995. Recent developments in count data modeling: theory and applications. J. Econ. Surveys 9, 1–24.

Zotero/001_artiklid/Road Safety Indicators/olufowobisafetyeffectivenessforwardcollision2025.md

@@ -0,0 +1,381 @@
+---
+category: literaturenote
+citekey: olufowobisafetyeffectivenessforwardcollision2025
+title: "Safety effectiveness of forward collision warning systems in the vehicle fleet: A driving simulation study"
+authors: "Olufowobi, Oluwaseun; Ivan, John; Wang, Kai; Eluru, Naveen"
+year: 2025
+date: 2025-00-00 2025
+doi: 10.1016/j.aap.2025.108078
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2025.108078"
+zotero_key: PU33VT7Q
+zotero_storage: M4N3UNWT
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Olufowobi, Oluwaseun"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+Contents lists available at [ScienceDirect](www.sciencedirect.com/science/journal/00014575)
+
+# Accident Analysis and Prevention
+
+journal homepage: [www.elsevier.com/locate/aap](https://www.elsevier.com/locate/aap)
+
+![](_page_0_Picture_5.jpeg)
+
+![](_page_0_Picture_6.jpeg)
+
+# Safety effectiveness of forward collision warning systems in the vehicle fleet: A driving simulation study☆
+
+Oluwaseun Olufowobi a,\* , John Ivan <sup>a</sup> , Kai Wang <sup>b</sup> , Naveen Eluru <sup>c</sup>
+
+- <sup>a</sup> *Department of Civil and Environmental Engineering, University of Connecticut, Storrs CT, USA*
+- <sup>b</sup> *Connecticut Transportation Safety Research Center, University of Connecticut, Storrs CT, USA*
+- <sup>c</sup> *Department of Civil, Environmental and Construction Engineering, University of Central Florida, Orlando, FL, USA*
+
+ARTICLE INFO
+
+*Keywords:* Rear-end crashes Forward Collision Warning Driving Simulation Traffic Conflict
+
+#### ABSTRACT
+
+The increasing rate of crashes globally has prompted the development of strategies like the Advanced Driver Assistance systems (ADAS) to improve safety. These systems range from subtle speed adjustment alerts to automatic emergency braking. One such system is the forward collision warning (FCW), which aims to mitigate collisions, particularly rear-end crashes, by providing visual, auditory, or tactile alerts of impending collisions. However, there has not been enough attention given to how drivers might react to FCW without unintentionally distracting them, while also using serious conflicts to quantify safety. This driving simulation study aimed to assess the effectiveness of the FCW by examining serious conflict as a surrogate measure for actual crashes. The goal was to estimate a Crash Modification Factor (CMF) for FCW systems within vehicle fleet, considering varying market penetration rates ranging from 10 percent to 50 percent in increments of 10 percent. Scenarios were created where drivers encountered different road and traffic conditions to evaluate their responses to unexpected events. A total of 133 participants completed a between subject design in which half of them drove the course with the FCW programmed into the simulation, and the other half did not have the FCW. The evaluation of serious conflicts utilized the Swedish Traffic Conflict Technique, employing two key indicators: Time to Accident (TA) and Conflicting Speed. Conflict severity was measured by considering two variables − the lane position and speed of the participants which was used to compute the TA. As expected, drivers in each scenario experienced fewer serious conflicts when assisted by FCW systems compared to the drivers without them. The resulting CMFs from the study can be integrated into crash prediction models to support efforts to keep crash prediction models up to date by accounting for the effects of increasing numbers of vehicles having FCW systems.
+
+#### **1. Introduction**
+
+One of the significant socio-economic challenges worldwide is the prevalence of traffic crashes. According to a recent report by the National Highway Traffic Safety Administration (NHTSA), 39,221 Americans lost their lives in car crashes in 2022 [\(National Highway Traffic](#page-8-0)  [Safety Administration, 2023\)](#page-8-0), with rear-end collisions comprising nearly one-third of these incidents. Driver inattention was identified as a leading cause in approximately 91 percent of these rear-end collisions. Factors contributing to driver inattention include distractions, fatigue, and environmental conditions like fog or sun glare, which can hinder a driver's ability to detect obstacles ahead [\(National Transportation Safety](#page-8-0)  [Board, 2002\)](#page-8-0). In response to these challenges, various strategies have been developed to assist drivers in preventing collisions from such factors, ranging from subtle speed adjustment alerts to systems that automatically engage emergency braking.
+
+With the advancement of these safety strategies, vehicles equipped with these safety systems are expected to experience a decrease in total crashes. One such strategy is the forward collision warning (FCW) system, which aims to assist drivers in mitigating or preventing collisions, particularly rear-end crashes, by providing visual, auditory, or tactile alerts of potential collisions [\(Forkenbrock, 2009](#page-8-0)). According to a 2002 study by Daimler-Chrysler, approximately 60 percent of rear-end collisions could potentially be avoided if drivers had an additional 0.5 s of
+
+<sup>☆</sup> This article is part of a special issue entitled: 'RSS 2024' published in Accident Analysis and Prevention.
+
+<sup>\*</sup> Corresponding author.
+
+*E-mail addresses:* [oluwaseun.olufowobi@uconn.edu](mailto:oluwaseun.olufowobi@uconn.edu) (O. Olufowobi), [john.ivan@uconn.edu](mailto:john.ivan@uconn.edu) (J. Ivan), [kai.wang@uconn.edu](mailto:kai.wang@uconn.edu) (K. Wang), [Naveen.Eluru@ucf.edu](mailto:Naveen.Eluru@ucf.edu) (N. Eluru).
+
+warning, while giving drivers an extra second of warning could potentially prevent about 90 percent of such collisions [\(Ankrum, 1992](#page-7-0)). This places FCW high on the list of solutions that can contribute significantly to reducing the number and the severity of crashes.
+
+The significance of studying FCW systems lies in their potential to enhance traffic safety by mitigating high-risk interactions and preventing crashes before they occur. These systems, by providing timely alerts, can help drivers respond more effectively to imminent hazards, particularly in complex urban and rural environments where the variability of driving conditions poses unique challenges. Using methods such as analyzing traffic conflicts and near-misses as surrogates for crashes provides a more comprehensive understanding of FCW system performance. This approach not only addresses the limitations of traditional crash data analysis but also provides valuable insights into driver behavior and the effectiveness of advanced safety systems, supporting the development of safer roads and more impactful interventions.
+
+This paper describes an effort to quantify the effectiveness of FCW systems in reducing serious traffic conflicts by comparing driver performance in vehicles equipped with FCW systems to those without such systems in a driving simulation experiment. Participants were exposed to road and traffic conditions deliberately designed to assess their responses to unexpected forward obstacles when they get warnings from an FCW system or when there is no system. Unlike previous study such as Lee et al. [\(Lee et al., 2002\)](#page-8-0); which focused on the value of a FCW system in a scenario where the driver is intentionally distracted as part of the experiment, and the hazard was a stopped lead vehicle. Here we explore the value of the FCW system for drivers who may or may not be distracted and where the hazard could be a vehicle that stops or crosses the travel path suddenly, or a non-motorized user crossing the street unexpectedly. By exploring these broader scenarios, this study seeks to provide a more comprehensive evaluation of FCW systems across diverse driving conditions.
+
+## **2. Literature review**
+
+#### *2.1. Forward collision warning systems*
+
+FCW systems, which are part of many vehicles' Collision Avoidance Systems (CAS), assist drivers by keeping their attention on the road ahead, helping them to anticipate and respond to potential hazards more effectively. Several studies ([Najim et al., 2007; Rakha et al., 2010;](#page-8-0)  [Kusano and Gabler, 2012; Seaman et al., 2022](#page-8-0)) emphasize the safety advantages of these systems to significantly lower the risks of serious and fatal injuries. For example, Lee et al., ([Lee et al., 2002](#page-8-0)) exposed driving simulator users to a scenario with a stopped lead vehicle where the driver was intentionally distracted and found that the FCW system reduced the number of collisions for that scenario by 80.7 percent. Similarly, Teoh [\(Teoh, 2021\)](#page-8-0) estimated the effectiveness of FCW using detailed data from exposure measures extracted from video footage and concluded that FCW was associated with a statistically significant 22 percent reduction in the rate of police-reportable crashes per vehicle miles traveled. These findings complement estimates from Jermakian ([Jermakian, 2011\)](#page-8-0) that FCW systems could prevent up to 1.2 million crashes annually in the U.S., or 20 percent of all police-reported crashes, and Kusano and Gabler [\(Kusano and Gabler, 2012](#page-8-0)) indicating dramatic reductions in serious and fatal injuries during rear-end collisions using data extracted from the National Automotive Sampling System/Crashworthiness Data System (NASS/CDS). All this research highlights the critical role of FCW systems in improving roadway safety.
+
+The success of FCW systems, however, depends significantly on the timing of their warning, a critical aspect in human factors research. Studies such as McLaughlin et al., ([McLaughlin et al., 2008\)](#page-8-0) and Abe and Richardson [\(Abe and Richardson, 2008](#page-7-0)) noted that providing early warning has an enormous potential to prevent crashes. Moreover, other studies have explored how various driving situations and history of hazard warning can influence drivers' response to warning and
+
+hazardous situations. Failures of the system to produce warnings that drivers find useful (incorrect warning) and understandable may diminish the driver's reliance on and compliance with warnings ([Seaman et al., 2022\)](#page-8-0). To address these challenges, researchers like Kiefer et al. [\(Kiefer et al., 1999\)](#page-8-0) developed FCW alert timing algorithms based on comprehensive human factors research. These algorithms account for the time required for a driver to perceive and react to an alert, as well as the time needed to execute braking or other evasive actions. Consequently, since FCW alerts are designed to provide drivers with sufficient lead time, unlike other CAS, such as Automatic Emergency Braking (AEB), which activate later in the collision sequence when the driver fails to respond ([Kiefer et al., 1999\)](#page-8-0). This proactive approach to alert timing highlights the importance of early intervention in mitigating potential crashes.
+
+Despite its strengths, FCW systems face limitations in certain scenarios. For instance, in the NHTSA FCW performance test procedure, drivers follow a moving vehicle that suddenly changes lanes, exposing a stopped vehicle ahead. Here, the FCW system issues an alert at a time-tocollision (TTC) of 2.1 s to prompt the driver, who may be distracted, to respond and prevent or mitigate a collision. While FCW systems succeed in initiating driver responses like braking or evasive maneuvers, Kiefer et al., [\(Kiefer et al., 1999\)](#page-8-0) noted that AEB systems may not be effective in such scenarios. This limitation arises from factors such as limited target acquisition time, sensor constraints, and delays in algorithm processing and brake activation. These findings highlight the need to optimize FCW timing and reliability to maximize safety outcomes and maintain driver trust. By addressing these limitations and refining system responses, FCW systems can enhance their role in preventing crashes and improving overall road safety.
+
+Research has also explored the role of market penetration rates (MPR) in increasing the safety benefits of advanced technologies like FCW. For example, Xiao et al. ([Xiao et al., 2021](#page-8-0)) conducted a *meta*analysis evaluating crash reduction by MPR. The results indicated that the number of conflicts is exponentially reduced as MPR goes up, safety is enhanced by 4 percent with an MPR of 10 percent, and by 43 percent with 90 percent MPR. Furthermore, Papadoulis et al. [\(Papadoulis et al.,](#page-8-0)  [2019\)](#page-8-0) evaluated the safety impact of connected vehicles (CVs) using traffic simulation. The results showed that CVs bring about compelling benefits to road safety as traffic conflicts significantly reduce even at low MPRs. Specifically, estimated traffic conflicts were reduced by 12–47 percent, 50–80 percent, 82–92 percent and 90–94 percent for 25 percent, 50 percent, 75 percent, and 100 percent CVs penetration rates, respectively.
+
+Together, this body of research illustrates the widespread benefits of FCW systems, from individual crash mitigation to large-scale reductions in crash severity and frequency. Despite the ongoing advancements in other technologies, FCWs continues to play a crucial role in crash avoidance and mitigation. To further evaluate their performance and ability to mitigate collisions, surrogate safety assessment offers a valuable approach by analyzing traffic conflicts, which often precedes crashes..
+
+#### *2.2. Crashes and serious conflicts*
+
+The conventional approach to traffic safety has primarily focused on the occurrence of traffic crashes and their outcomes ([Lai et al., 2014](#page-8-0)). In contrast, surrogate safety assessment offers an alternative method by focusing on the analysis of safety–critical events such as traffic conflicts to evaluate safety. The concept of traffic conflicts was first introduced by ([Klebelsberg, 1964\)](#page-8-0) as cited by Tarko [\(Tarko, 2018](#page-8-0)), who defined them as dangerous traffic interactions. Laureshyn and Varhelyi [\(Laureshyn](#page-8-0)  [and Varhelyi, 2018\)](#page-8-0) defined a traffic conflict as "an observable situation in which two or more road users approach each other in space and time to such an extent that there is a risk of collision if their movements remain unchanged". The proximity of road users to each other could be physically measured in temporal and/or spatial dimensions, and
+
+thresholds could be used to identify conflicts. Perkins and Harris ([Perkins and Harris, 1968](#page-8-0)) expanded on the concept defining traffic conflicts as any situations necessitating evasive actions such as braking. According to this definition, it implies that conflicts and crashes are of a similar nature except for the presence and the success of an evasive action [\(Lai et al., 2014](#page-8-0)). Traffic conflicts offer significant advantages for road safety analysis compared to collisions. They occur more frequently, are easier to observe, and involve lower costs ([Ismail et al., 2011\)](#page-8-0).
+
+Various conflict indicators have been developed to measure the severity of an interaction by quantifying the spatial and temporal proximity of two or more road users ([Arun et al., 2021\)](#page-7-0). These indicators are classified into two categories: Spatial/temporal proximities and evasive action indicators. Commonly used measures of conflict severity include conflicting speed, TTC, Post-Encroachment Time (PET), and the severity index ([Lai et al., 2014; Tarko, 2018](#page-8-0)). Brown ([Brown, 1994\)](#page-7-0) and Tarko et al., [\(Tarko, 2021\)](#page-8-0) provide comprehensive summaries of these indicators. The TTC indicator, used in this study, is defined as "…the time that remains until a collision between two vehicles would have occurred if the collision course and speed difference are maintained" ([Tarko, 2021\)](#page-8-0). The main advantage of conflict indicators is their ability to capture the severity of an interaction in an objective and quantitative way. Numerous studies have used traffic conflicts together with crashes for safety analysis and have shown that conflicts can adequately serve as a proxy for crashes ([Lai et al., 2014; Tarko et al., 2009; Autey et al.,](#page-8-0)  [2012; Essa and Sayed, 2020; Charly and Mathew, 2019](#page-8-0)).
+
+Although previous studies have contributed to examining the effectiveness of the FCW, few studies have explored estimating serious conflict with the aim of providing information for crash modification factors (CMF) for the FCW systems. Near crashes, also known as serious conflicts, will be used as a surrogate for crashes to represent quantitative safety, because crashes are expected to be extremely rare given the number of drivers that will be observed in the experiment and primarily because accurately identifying crashes which are considered the gold standard epidemiological measure cannot be identified within the constraint of a driving simulator.
+
+## **3. Materials and method**
+
+#### *3.1. Participants*
+
+In total, 142 individuals took part in the study, with 64 females, 77 males, and one person who chose not to disclose their gender. Nine participants were unable to complete the experiment due to simulator sickness. Of the 133 who finished, 58 (41 percent) were between 18–29 years old, 55 (41 percent) were between 30–64, and 20 (15 percent) were 65 or older. A power analysis based on previous research [\(Gaspar](#page-8-0)  [et al., 2015\)](#page-8-0) suggested that 36 pairs of participants (72 total) would be needed to achieve adequate statistical power (β = 0.8). However, considering potential dropouts due to simulator sickness, the estimate increased to 57 pairs (114 participants) if the dropout risk was 36 percent, and up to 71 pairs (142 participants) if the dropout rate reached 51 percent.
+
+To ensure a balanced demographic, a representative sample of demographic were recruited which were deliberately distributed among the scenarios. All participants held valid driver's licenses and were in good physical health. Recruitment was carried out by posting flyers on University of Connecticut (UConn) bulletin boards, adhering to Institutional Review Board (IRB) standards, and distributing flyers at senior and community centers to attract older drivers and residents not affiliated with the university.
+
+## *3.2. Equipment*
+
+The study utilized the Realtime Technologies RDS-2000 Full Cab with Segmented Screen Driving Simulator, along with multiple cameras and four projectors (rear, front, right side, and left side projections) to create a highly realistic virtual environment. This simulator is housed and maintained by the Connecticut Transportation Safety Research Center at UConn. Some of the features of the driving simulator include the Internet Scene Assembler®, used for modification of the virtual environment; and SimObserver®, integrated with the virtual environment and used for data and video synchronization, video capture and after-action review. [Fig. 1](#page-3-0) shows the picture of the UConn driving simulator. The first computer on the left side was used to run the simulator and edit the model, while the computer on the right-hand side was used to extract the data and video clips of the main driving experiment.
+
+## *3.3. Procedure and experimental design*
+
+Upon arrival, participants were provided with an informed consent form to read and sign, which also included their consent for video recording by initialing the document. Following this, participants filled out a questionnaire containing demographic inquiries such as gender, age, race/ethnicity, and years of driving experience. Subsequently, each participant underwent a driving simulator training session lasting approximately five (5) minutes to familiarize themselves with the simulator. An additional five-minute training session was available upon request.
+
+After the training session, half of the participants drove the experiment with an FCW system integrated into the simulator, which provided them with both an audible warning and a visual indicator (on the dashboard) about potential collisions. The remaining participants did not have the FCW system and received no warnings during their drive. Prior to commencing the driving session, participants using the FCW system were informed that their vehicle would produce an audible sound and/or display a flashing indicator on the dashboard to alert them of impending forward collisions. Each participant then engaged in one of four predefined driving scenarios encompassing either urban or rural driving, with the scenario lasting approximately 15 min. Participants were instructed to drive the course as they would in a real-life situation and were told that the purpose of the experiment is to observe their reactions to the scenario they drive.
+
+During the experiment, participants were asked to follow the designated route as a recorded message that was broadcast through speakers in the car provided directions. At some point during the scenario, a stimulus was introduced that necessitated a sudden, unexpected braking maneuver by the driver. For instance, this could involve a parked vehicle suddenly pulling in front without warning, or a vehicle, animal, or pedestrian crossing the vehicle's path.
+
+Upon completing their turns in the driving simulator, participants were debriefed by a researcher and given a debriefing form. Furthermore, the researcher verbally explained the study and its objectives. Since the experiment did not involve any deception, this debriefing was purely informational.
+
+#### *3.4. Driving scenarios*
+
+Driving scenarios were created for both urban and rural environments to reflect real-world conditions. The urban environment ([Fig. 2\)](#page-3-0) consisted of an undivided roadway section with two lanes of traffic in both directions. The roadway included all the typical features of an urban road, such as speed limit signs, traffic signals, parked cars/ trucks on the shoulders, and had paved curbs with pedestrian movements on both sides of the road. The rural scenario ([Fig. 3](#page-3-0)) consisted of a two-lane undivided roadway section with traffic simulated into the experiment to make it feel like real-world driving. The rural setting was a mix of countryside and residential areas with the road primarily straight, with a few horizontal curves.
+
+[Table 1](#page-4-0) illustrates the demographic distribution of participants across each scenario. Participants were assigned based on their demographic information, ensuring a balanced distribution of age, gender,
+
+<span id="page-3-0"></span>![](_page_3_Picture_2.jpeg)
+
+**Fig. 1.** UConn Driving Simulator.
+
+![](_page_3_Picture_4.jpeg)
+
+**Fig. 2.** Photo. Section of urban roadway used in experiment.
+
+![](_page_3_Picture_6.jpeg)
+
+**Fig. 3.** Photo. Section of rural roadway used in experiment.
+
+and years of driving experience in each group with and without the FCW system. Efforts were also made to maintain a balanced distribution of race/ethnicity in each group as practical. The study employed a between-subjects design with the driving simulator equipped with an FCW system as mentioned earlier, which operates by continuously monitoring the road ahead using a combination of sensors and cameras. When a potential collision is detected, the system calculates the TTC based on the vehicle's speed and distance relative to the object ahead. If the TTC falls below a certain threshold, the system issues a warning to the driver. These warnings are intended to give the driver sufficient time to react and take action to avoid a collision. Both visual and auditory alerts were triggered when the distance became too short to avoid a crash. The FCW visual display, located on the dashboard in front of the driver, showed a flashing red icon of two cars in proximity when activated, while a flashing yellow pedestrian icon appeared for potential pedestrian collisions. This design was chosen because most FCW systems on the market utilize a combination of visual and auditory alerts ([Forkenbrock, 2009\)](#page-8-0), with the visual display typically located in the instrument cluster near the speedometer.
+
+## *3.5. Data collection*
+
+To evaluate conflict severity, two primary variables were required: time to accident (TA) and conflicting speed (CS). These variables are further detailed in the data analysis section. TA was calculated using the vehicle's speed and the distance to the collision point, with speed directly obtained from the driving simulator as the participant's driving velocity. Additionally, lane position, measured in meters from the center of the road, was used to represent lateral control. Data were recorded at a frequency of 60 Hz using the simulator's proprietary SimObserver software.
+
+Lane position data were captured for each participant in both "with system" and "without system" scenarios, providing the lateral position at the moment the experimental event occurred. This information was essential for determining the distance to the collision point. Speed data represented the participant's velocity immediately before performing evasive actions, such as braking or swerving.
+
+Evasive actions were identified using video footage from SimObserver, cross-referenced with steering and brake inputs captured by the Data Distillery software. A reduction in steering angle or brake application values was used to pinpoint when participants initiated evasive maneuvers.
+
+The distance to a potential collision was calculated as the gap, in meters, between the vehicle's position during the evasive action and the location of the event along the road. This calculation was used to estimate conflict severity. While this study did not focus on the specific avoidance strategies employed (e.g., braking versus swerving), the actions provided critical data for determining the proximity to potential collisions. [Fig. 4](#page-4-0) shows a graphical description of how these variables were collected. This systematic data collection ensured accurate measurement of critical variables, enabling a robust evaluation of conflict severity under both experimental conditions ([Fig. 5\)](#page-4-0).
+
+<span id="page-4-0"></span>**Table 1**  Driving simulator participant statistics by scenario.
+
+| Gender | Age Group | Scenario 1 |         | Scenario 2 |         | Scenario 3 |         | Scenario 4 |         | Total |
+|--------|-----------|------------|---------|------------|---------|------------|---------|------------|---------|-------|
+|        |           | With       | Without | With       | Without | With       | Without | With       | Without |       |
+| Male   | 18–29     | 4          | 4       | 4          | 4       | 2          | 2       | 4          | 4       | 28    |
+|        | 30–64     | 4          | 5       | 4          | 3       | 2          | 3       | 3          | 3       | 27    |
+|        | 65+       | 1          | 1       | 0          | 1       | 3          | 2       | 1          | 0       | 9     |
+| Female | 18–29     | 2          | 3       | 4          | 3       | 4          | 5       | 4          | 4       | 29    |
+|        | 30–64     | 3          | 1       | 5          | 5       | 4          | 3       | 2          | 3       | 26    |
+|        | 65+       | 1          | 1       | 2          | 3       | 2          | 2       | 1          | 1       | 13    |
+| Total  |           | 15         | 15      | 19         | 19      | 17         | 17      | 15         | 15      | 132   |
+
+![](_page_4_Figure_4.jpeg)
+
+**Fig. 4.** Graphical Representation of Scenarios and Events.
+
+![](_page_4_Picture_6.jpeg)
+
+**Fig. 5.** Variables of interest from the simulator.
+
+## **4. Data analysis**
+
+To analyze the FCW systems in mitigating traffic conflicts, data collected from the driving simulator were processed and evaluated using a traffic conflict methodology. Additionally, chi-square analysis was employed to compare serious conflict frequencies between scenarios with and without FCW systems. The results of the chi-square test validate the findings, ensuring their robustness and providing justification for calculating CMFs (Fig. 6).
+
+### *4.1. Statistical analysis*
+
+To statistically evaluate the differences in serious conflict frequencies between the with FCW and without FCW scenarios, a chisquare test of independence was performed prior to calculating the CMF to validate the findings and ensure they justified the CMFs calculations. The analysis was conducted at two levels:
+
+![](_page_4_Figure_12.jpeg)
+
+**Fig. 6.** Conflict severity diagram (Laureshyn, A., & Varhelyi, A., 2018).
+
+**Aggregated Analysis:** Combined data from all scenarios to assess the overall impact of the FCW system.
+
+**Scenario-Specific Analysis**: Evaluated whether the effectiveness of the FCW system varied across the different simulated scenarios.
+
+The steps in the chi-square analysis were as follows:
+
+- Contingency Table Construction: A table was created to record the number of traffic conflicts in each category (e.g., "with FCW" and "without FCW").
+- Chi-Square Statistic Calculation: The chi-square statistic (χ2) was calculated using equation (1):
+
+$$\chi 2 = \sum \frac{(O-E)^2}{E} \tag{1}$$
+
+• Significance Testing: The calculated χ2 value were compared against the critical value from the chi-square distribution table at a significance level of 0.05 to determine whether differences were statistically significant.
+
+This two-level analysis enabled a comprehensive evaluation of the FCW system's effectiveness, both overall and within specific scenarios, providing insight into its impact on traffic conflict severities.
+
+#### *4.2. Traffic conflict methodology*
+
+This study utilized the Swedish Traffic Conflict Technique (TCT) to evaluate traffic conflicts using two indicators: the time to accident (TA) value and the conflicting speed (CS). The TA is the time between when a particular road user successfully performs an evasive action and when the collision would have occurred if they had continued with unchanged speeds and direction, while the CS is the speed of the road user at the moment just before taking the evasive action [\(Laureshyn and Varhelyi,](#page-8-0)  [2018\)](#page-8-0). The TA and CS values are used to define the conflict severity. The conflicting speed affects the outcome of the collision (the resulting pedestrian injury depends on the speed), and a higher speed would require longer stopping time and distance. Thus, a higher CS would indicate a more severe conflict for the same TA ([Laureshyn and Varhelyi,](#page-8-0)  [2018\)](#page-8-0). Because it can be challenging to directly estimate the TA in live traffic, it can be estimated from the vehicle speed and the distance to the collision point using Table 2**.**
+
+The conflict severity is then determined from the TA and the CS according to the graph in [Fig. 2.](#page-3-0)
+
+Conflicts with a severity level above 26 are ranked as serious. For instance, a vehicle approaching 30 mph (48.3 km/hr.) with a TA of 2.8 sec would pass the potential conflict point with too much time remaining for the interaction to be classified as a serious conflict. However, a 45 mph (72.4 km/hr.) approach speed would just exceed that level. These severity levels can be used to understand serious conflicts to estimate the CMF for FCW systems. A CMF is the ratio of the number of crashes expected with a countermeasure to the number of crashes expected without it. In other words, when implementing a countermeasure, multiplying the existing expected number of crashes by the CMF gives an estimate of the expected number of crashes after implementing the countermeasure. CMFs with a value less than 1.0 indicate an expected decrease while a value greater than 1.0 indicates an expected increase in crashes. The FCW system can be said to be a countermeasure installed in vehicles to help mitigate the occurrence of crashes in vehicles that have it installed. A CMF for this "countermeasure" would have to also account for the penetration rate of the FCW system in the vehicle fleet. Estimating a CMF value for this FCW system would be helpful for adjusting the expected number of crashes in future conditions as the proportion of vehicles with such systems increases.
+
+Following are the steps to estimate the CMF:
+
+● Estimate a ratio to account for the safety effect of having a FCW system (RFCW),
+
+$$R_{FCW} = \frac{n_{FCW}}{n_o} \tag{2}$$
+
+where *nFCW* is the number of serious conflicts in vehicles with the FCW and *no* is the number of serious conflicts in vehicles without the FCW.
+
+- Define the proportion of the vehicle fleet with the FCW system (PFCW). Here, five different proportions of the vehicle fleet from 10 to 50 percent in increments of 10 percent were investigated.
+- Calculate the estimated CMF for the FCW system:
+
+$$CMF_{FCW} = 1 + P_{FCW}(R_{FCW} - 1)$$
+(3)
+
+**Table 2**  TA values estimated from vehicle speed and distance to collision point (Laureshyn, A., & Varhelyi, A., 2018).
+
+| Speed    |     | Distance, m |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |     |
+|----------|-----|-------------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
+| Km<br>/h | 0.5 | 1           | 2   | 3   | 4   | 5   | 6   | 7   | 8   | 9   | 10  | 15  | 20  | 25  | 30  | 35  | 40  | 45  | 50  | 55  |
+| 5        | 0.4 | 0.7         | 1.4 | 2.2 | 2.9 | 3.6 | 4.3 | 5.0 | 5.8 | 6.5 | 7.2 |     |     |     |     |     |     |     |     |     |
+| 10       | 0.2 | 0.4         | 0.7 | 1.1 | 1.4 | 1.8 | 2.2 | 2.5 | 2.9 | 3.2 | 3.6 | 5.4 | 7.2 | 9.0 |     |     |     |     |     |     |
+| 15       | 0.1 | 0.2         | 0.5 | 0.7 | 1.0 | 1.2 | 1.4 | 1.7 | 1.9 | 2.2 | 2.4 | 3.6 | 4.8 | 6.0 | 7.2 | 8.4 | 9.6 |     |     |     |
+| 20       | 0.1 | 0.2         | 0.4 | 0.5 | 0.7 | 0.9 | 1.1 | 1.3 | 1.4 | 1.6 | 1.8 | 2.7 | 3.6 | 4.5 | 5.4 | 6.3 | 7.2 | 8.1 | 9.0 | 9.9 |
+| 25       | 0.1 | 0.1         | 0.3 | 0.4 | 0.6 | 0.7 | 0.9 | 1.0 | 1.2 | 1.3 | 1.4 | 2.2 | 2.9 | 3.6 | 4.3 | 5.0 | 5.8 | 6.5 | 7.2 | 7.9 |
+| 30       | 0.1 | 0.1         | 0.2 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 1.0 | 1.1 | 1.2 | 1.8 | 2.4 | 3.0 | 3.6 | 4.2 | 4.8 | 5.4 | 6.0 | 6.6 |
+| 35       | 0.1 | 0.1         | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | 1.5 | 2.1 | 2.6 | 3.1 | 3.6 | 4.1 | 4.6 | 5.1 | 5.7 |
+| 40       | 0.0 | 0.1         | 0.2 | 0.3 | 0.4 | 0.5 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.4 | 1.8 | 2.3 | 2.7 | 3.2 | 3.6 | 4.1 | 4.5 | 5.0 |
+| 45       |     | 0.1         | 0.2 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.6 | 0.7 | 0.8 | 1.2 | 1.6 | 2.0 | 2.4 | 2.8 | 3.2 | 3.6 | 4.0 | 4.4 |
+| 50       |     | 0.1         | 0.1 | 0.2 | 0.3 | 0.4 | 0.4 | 0.5 | 0.6 | 0.6 | 0.7 | 1.1 | 1.4 | 1.8 | 2.2 | 2.5 | 2.9 | 3.2 | 3.6 | 4.0 |
+| 55       |     | 0.1         | 0.1 | 0.2 | 0.3 | 0.3 | 0.4 | 0.5 | 0.5 | 0.6 | 0.7 | 1.0 | 1.3 | 1.6 | 2.0 | 2.3 | 2.6 | 2.9 | 3.3 | 3.6 |
+| 60       |     | 0.1         | 0.1 | 0.2 | 0.2 | 0.3 | 0.4 | 0.4 | 0.5 | 0.5 | 0.6 | 0.9 | 1.2 | 1.5 | 1.8 | 2.1 | 2.4 | 2.7 | 3.0 | 3.3 |
+| 65       |     | 0.1         | 0.1 | 0.2 | 0.2 | 0.3 | 0.3 | 0.4 | 0.4 | 0.5 | 0.6 | 0.8 | 1.1 | 1.4 | 1.7 | 1.9 | 2.2 | 2.5 | 2.8 | 3.0 |
+| 70       |     | 0.1         | 0.1 | 0.2 | 0.2 | 0.3 | 0.3 | 0.4 | 0.4 | 0.5 | 0.5 | 0.8 | 1.0 | 1.3 | 1.5 | 1.8 | 2.1 | 2.3 | 2.6 | 2.8 |
+| 75       |     | 0.0         | 0.1 | 0.1 | 0.2 | 0.2 | 0.3 | 0.3 | 0.4 | 0.4 | 0.5 | 0.7 | 1.0 | 1.2 | 1.4 | 1.7 | 1.9 | 2.2 | 2.4 | 2.6 |
+| 80       |     | 0.0         | 0.1 | 0.1 | 0.2 | 0.2 | 0.3 | 0.3 | 0.4 | 0.4 | 0.5 | 0.7 | 0.9 | 1.1 | 1.4 | 1.6 | 1.8 | 2.0 | 2.3 | 2.5 |
+| 85       |     | 0.0         | 0.1 | 0.1 | 0.2 | 0.2 | 0.3 | 0.3 | 0.3 | 0.4 | 0.4 | 0.6 | 0.8 | 1.1 | 1.3 | 1.5 | 1.7 | 1.9 | 2.1 | 2.3 |
+| 90       |     | 0.0         | 0.1 | 0.1 | 0.2 | 0.2 | 0.2 | 0.3 | 0.3 | 0.4 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0 | 2.2 |
+| 95       |     | 0.0         | 0.1 | 0.1 | 0.2 | 0.2 | 0.2 | 0.3 | 0.3 | 0.3 | 0.4 | 0.6 | 0.8 | 0.9 | 1.1 | 1.3 | 1.5 | 1.7 | 1.9 | 2.1 |
+| 100      |     | 0.0         | 0.1 | 0.1 | 0.1 | 0.2 | 0.2 | 0.3 | 0.3 | 0.3 | 0.4 | 0.5 | 0.7 | 0.9 | 1.1 | 1.3 | 1.4 | 1.6 | 1.8 | 2.0 |
+
+#### **5. Results**
+
+#### *5.1. Chi square analysis result*
+
+#### *5.1.1. Aggregated analysis*
+
+The data were aggregated across all scenarios to assess the overall effect of the FCW system. The contingency table summarizing the aggregated data is shown in Table 5.
+
+The Chi-Square statistic (X-squared) of 8.7576 with a degree of freedom (df) of 1 resulted in a p-value of 0.0030. Since the p-value is less than the significance level of 0.05, indicating that there is a statistically significant difference between the "With FCW" and "No FCW" conditions. This result suggests that the FCW system significantly reduces the occurrence of serious conflicts when considering all scenarios combined.
+
+## *5.1.2. Scenario-Specific analysis*
+
+The analysis was extended to determine whether the FCW system's effectiveness varied across individual scenarios. The contingency in Table 6 below summarizes the serious conflicts by scenario.
+
+The Chi-Square statistic (X-squared) of 0.5516 with a degree of freedom (df) of 3 resulted in a p-value of 0.9074. Since the p-value exceed the threshold of 0.05, indicating that there is no statistically significant difference in the number of serious conflicts between the "With FCW" and "No FCW" conditions when analyzed on a scenario-byscenario basis. The findings indicate that the FCW system's effectiveness is consistent across various scenarios, showing slight variation with specific conditions. This suggests that FCW systems are broadly effective and suitable for diverse driving conditions. However, this result could be influenced by factors such as sample size.
+
+#### *5.2. Safety effectiveness of FCW*
+
+To calculate the safety effect of having the FCW was determined by calculating the ratio of serious conflict (*nFCW*) in the experiment with FCW to the ratio of serious conflict without FCW as discussed earlier. Table 3 shows the raw factor for the different scenarios conducted in the experiment. The raw factor was also calculated for all cases with and without FCW to see if there are substantial differences between individual scenarios and overall cases.
+
+The CMF was calculated based on the various penetration rates of the system and are shown in Table 4**.**
+
+#### **6. Discussion**
+
+The objective of this experiment was to examine the effect of the FCW on drivers with the aim of providing data for CMF related to FCW system and to statistically assess the significance of these effects using chi-square analysis. The chi-square analysis revealed significant differences in the occurrence of serious conflicts showing a significant overall reduction in serious conflicts when FCW systems were used (χ2 = 8.76, p *<* 0.05). This demonstrates that FCW systems consistently contribute to conflict reduction, especially as their penetration rate increases. The scenario-specific chi-square analysis, however, revealed no statistically significant differences in the number of serious conflicts between the "With FCW" and "No FCW" conditions across the individual scenarios (χ2 = 0.55, p *>* 0.05). Since the p-value exceeds 0.05, this indicates that
+
+**Table 3**  Aggregated Chi-Square Result.
+
+| Condition         | Serious Conflicts | No Conflicts | Total Participants |
+|-------------------|-------------------|--------------|--------------------|
+| With FCW          | 24                | 42           | 66                 |
+| No FCW            | 42                | 26           | 68                 |
+| X-squared (χ2)    | 8.7576            |              |                    |
+| Degree of freedom | 1                 |              |                    |
+| p-value           | 0.0030            |              |                    |
+
+**Table 4**  Scenario specific Chi-Square Result.
+
+| Scenario                      | Serious Conflicts (With<br>FCW) | Serious Conflicts (No<br>FCW) |  |  |
+|-------------------------------|---------------------------------|-------------------------------|--|--|
+| 1: Urban with entering<br>car | 6                               | 9                             |  |  |
+| 2: Urban with<br>pedestrian   | 6                               | 13                            |  |  |
+| 3: Rural with animal          | 7                               | 10                            |  |  |
+| 4: Rural with pedestrian      | 5                               | 10                            |  |  |
+| X-squared (χ2)                | 0.5516                          |                               |  |  |
+| Degree of freedom             | 3                               |                               |  |  |
+| p-value                       | 0.9074                          |                               |  |  |
+
+**Table 5**  FCW Ratios by Scenario.
+
+| Scenario                   | nFCW | nO | RFCW = nFCW<br>no |
+|----------------------------|------|----|-------------------|
+| All cases                  | 24   | 42 | 0.57              |
+| 1: Urban with entering car | 6    | 9  | 0.67              |
+| 2: Urban with pedestrian   | 6    | 13 | 0.46              |
+| 3: Rural with animal       | 7    | 10 | 0.70              |
+| 4: Rural with pedestrian   | 5    | 10 | 0.50              |
+
+**Table 6**  CMF Values by FCW Fleet penetration rate.
+
+| Scenario                      | PFCW ¼<br>0.1 | PFCW ¼<br>0.2 | PFCW ¼<br>0.3 | PFCW ¼<br>0.4 | PFCW ¼<br>0.5 |
+|-------------------------------|---------------|---------------|---------------|---------------|---------------|
+| All cases                     | 0.95          | 0.91          | 0.87          | 0.82          | 0.78          |
+| 1: Urban with<br>entering car | 0.96          | 0.93          | 0.89          | 0.86          | 0.83          |
+| 2: Urban with<br>pedestrian   | 0.94          | 0.89          | 0.83          | 0.78          | 0.73          |
+| 3: Rural with<br>animal       | 0.97          | 0.94          | 0.91          | 0.88          | 0.85          |
+| 4: Rural with<br>pedestrian   | 0.95          | 0.90          | 0.85          | 0.80          | 0.75          |
+
+the effectiveness of the FCW system does not vary significantly between scenarios, suggesting consistent performance across different conditions regardless of environmental or contextual differences, such as traffic volume or road layout. While the scenario-specific results do not show statistically significant variation, the CMF calculations provided important insights into crash reductions for individual scenarios.
+
+The resulting CMF was 0.95 which translates to a 5 percent reduction in crashes for all the cases scenario conducted in the experiment when the penetration rate is at 10 percent and increases to 22 percent when the penetration rate rises to 50 percent. For Scenario 1, when the FCW penetration rate is 10 percent, we expect a 4 percent reduction in crashes and a 17 percent reduction at a 50 percent penetration rate. In the case of scenario 2, a CMF 0.94 for a 10 percent penetration rate means we expect a 6 percent reduction in crashes with a 27 percent reduction when the penetration rate of FCW is at 50 percent. While in scenario 3, the event with an animal suddenly crossing into the path of the participant, had a CMF of 0.97 which translates to a 3 percent reduction in crashes when the penetration rate is at 10 percent and 0.85, a 15 percent reduction when the penetration rate is at 50 percent finally in the case of rural with pedestrian had a CMF of 0.95, 5 percent reduction in crashes when the penetration rate is at 10 percent and a CMF of 0.75, 25 percent reduction in crashes when the penetration is at 50 percent. Overall, we can notice that as the penetration rate increases, there is a reduction in crashes. This aligns with some previous studies ([Najim et al., 2007;](#page-8-0)  [Teoh, 2021; Abdulsattar et al., 2018](#page-8-0)) who also reported a similar conclusion of decreased crashes as the penetration rate of these systems increases.
+
+<span id="page-7-0"></span>As expected, drivers in each scenario experienced fewer serious conflicts when assisted by FCW systems compared to the drivers without them, revealing a significant improvement in safety (potential conflict reduction). These warnings benefited drivers by providing timely alerts, enabling quicker reactions to potential collisions, regardless of whether the drivers are distracted or not. Conflicts are reduced more in urban scenarios than in rural ones. This could suggest that the FCW has its highest effectiveness on urban roads compared to rural roads, due to typical aspects of the urban operating environment, such as a greater variety of obstacles (e.g. parked cars and delivery vehicles), higher volume of traffic, frequent traffic signals, frequent stop and go and pedestrian movement, which maximizes the potential for benefits from FCW. Existing literature also supports this finding; for instance, Yue et al., [\(Yue et al., 2020](#page-8-0)) in a *meta*-analysis estimated the practical effectiveness of six ADAS in avoiding crashes across different roadway facilities and reported that FCW had less practical effectiveness in rural roads compared to urban ones. The study also found that the FCW system was more effective for pedestrian-related events, with drivers experiencing fewer conflicts in both urban and rural environment compared to vehicle and animal-related events. This seems intuitive, as drivers are more proactive in avoiding collisions with pedestrians.
+
+#### **7. Conclusions**
+
+This study examined the effectiveness of FCW with the aim of providing information for CMF for the FCW systems for five different proportions of the vehicle fleet from 10 to 50 percent in increments of ten percent. Four scenarios were modeled in urban and rural settings where participants were exposed to conditions deliberately designed to evaluate their responses to unexpected events when they get warnings from an FCW system or when there is no system. To achieve this, two variables, the lane position and speed, were used to measure conflict severity, which was used as a proxy for crashes. The lane position represents the position of the vehicle measured from the center of the road. The conflict severity was then determined from the time to collision and the conflicting speed of the road users. A total of 133 participants completed the experiment.
+
+The Chi-Square analysis revealed a significant overall reduction in serious conflicts when FCW systems were used (χ2 = 8.76, p *<* 0.05). This demonstrates the system's consistent contribution to conflict reduction, particularly as its penetration rate increases. However, scenario-specific Chi-Square results revealed no statistically significant differences between the "With FCW" and "No FCW" conditions in individual scenarios, suggesting consistent performance across varied conditions. The results suggest that the FCW has the potential of reducing the number of crashes in the vehicles that have them installed. The overall CMF for all the cases of scenarios conducted in the experiment was from 0.78 to 0.95 which is about 10 percent to 22 percent reduction in crashes with a R value of 0.67, while the CMF for scenarios 1 and 2 were from 0.83 to 0.96 and 0.73 to 0.94 which is about 4 percent to 17 percent and 6 percent to 27 percent reduction in crashes respectively and had a R value of 0.67 and 0.46. In the rural scenario, the CMF for scenarios 3 and 4 were from 0.85 to 0.97 and 0.75 to 0.95 with an R value of 0.70 and 0.50, respectively. This indicates that the number of crashes reduces by 3–15 percent for scenario 3 and 5–10 percent for scenario 4. The study also reveals that the FCW enhances drivers' response over a range of speeds, strongly suggesting that the FCW can enhance drivers' response in scenarios that could lead to collisions. Conflicts reduced more in urban scenarios than in rural ones suggesting high effectiveness on the urban roads, in addition the FCW system was more effective for pedestrian-related events, with drivers experiencing fewer conflicts in both urban and rural environment compared to vehicle and animal-related events.
+
+This study provides a valuable contribution by estimating CMFs for FCW systems using realistic driving scenarios that substitute serious traffic conflicts for actual crashes. This approach offers an alternative method to assess safety and evaluate the effectiveness of FCW systems allowing for possibility of the resulting CMFs to be incorporated into crash prediction models, to keep these models up to date by accounting for the effects of increasing numbers of vehicles having FCW systems. The study also provides a more holistic assessment of FCW performance by examining a wide range of real-world scenarios including diverse and unexpected road events in both urban and rural settings without deliberately distracting drivers. This broad approach offers a wider understanding of the system's effectiveness across different contexts.
+
+While the findings highlight the substantial benefits of FCW systems, there are still some unanswered questions. Specifically, although the warning enhanced driver's awareness of potential conflict in the simulator, warnings might be perceived as a nuisance in an actual driving situation when they are too sensitive or too early. Further data collection might be required to assess drivers' response to nuisance alarms generated by early warning or if dynamic adjustment of the warning is feasible. Also, a follow up and longer-term study might be helpful to understand the evolution of driver's attitude and to validate the benefits observed in the study with the possible use of naturalistic studies which can provide more insight into the driver's behavior during their everyday trips. Data from these will help validate the benefits of the FCW and assess drivers' attitude to the system compared to the one time the participants are exposed to the FCW system.
+
+#### **CRediT authorship contribution statement**
+
+**Oluwaseun Olufowobi:** Data curation, Methodology, Writing – original draft, Writing – review & editing. **John Ivan:** Conceptualization, Supervision, Writing – review & editing. **Kai Wang:** Visualization. **Naveen Eluru:** Visualization, Writing – review & editing.
+
+#### **Declaration of competing interest**
+
+The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
+
+#### **Acknowledgements**
+
+The authors would like to gratefully acknowledge FHWA's sponsorship in this research under project number 693JJ31950017. The ideas expressed in this work do not necessarily indicate acceptance by FHWA of the findings and conclusions expressed within.
+
+## **Data availability**
+
+The authors do not have permission to share data.
+
+#### **References**
+
+Abdulsattar, H., Mostafizi, A., Wang, H., 2018. Surrogate safety assessment of work zone rear-end collisions in a connected vehicle environment: agent-based modeling framework. J. Transport. Eng. 144 (X). [https://doi.org/10.1061/JTEPBS.0000164.](https://doi.org/10.1061/JTEPBS.0000164) Article 0000164.
+
+Abe, G., Richardson, J., 2008. Alarm timing, trust, and driver expectation for forward collision warning systems. Applied Ergonomics 37 (5), 577–586. [https://doi.org/](https://doi.org/10.1016/j.apergo.2005.11.001) [10.1016/j.apergo.2005.11.001.](https://doi.org/10.1016/j.apergo.2005.11.001)
+
+[Ankrum, D.R., 1992. Smart vehicles, smart roads.](http://refhub.elsevier.com/S0001-4575(25)00164-2/h0015) *Traffic Safety* 92 (3), 6–9.
+
+Arun, A., Haque, S.M.M., Bhaskar, A., Washington, S., Sayed, T., 2021. A systematic mapping review of surrogate safety assessment using traffic conflict techniques. Accid. Anal. Prevent. 153, 106016. [https://doi.org/10.1016/j.aap.2021.106016.](https://doi.org/10.1016/j.aap.2021.106016)
+
+Autey, J., Sayed, T., Zaki, M.H., 2012. Safety evaluation of right-turn smart channels using automated traffic conflict analysis. Accid. Anal. Prevent. 45, 120–130. [https://](https://doi.org/10.1016/j.aap.2011.11.015)  [doi.org/10.1016/j.aap.2011.11.015.](https://doi.org/10.1016/j.aap.2011.11.015)
+
+Brown, G.R., 1994. Traffic conflicts for road user safety studies. Retrieved from Canad. J. Civil Eng. 21 (1), 1–15. <http://www.nrcresearchpress.com/loi/cjce>.
+
+Charly, A., Mathew, T.V., 2019. Estimation of traffic conflicts using precise lateral position and width of vehicles for safety assessment. Accid. Anal. Prevent. 132, 105264. [https://doi.org/10.1016/j.aap.2019.105264.](https://doi.org/10.1016/j.aap.2019.105264)
+
+Essa, M., Sayed, T., 2020. Comparison between surrogate safety assessment model and real-time safety models in predicting field-measured conflicts at signalized
+
+- <span id="page-8-0"></span>intersections. Transport. Res. Record 2674 (3), 100–112. [https://doi.org/10.1177/](https://doi.org/10.1177/0361198120907874) [0361198120907874.](https://doi.org/10.1177/0361198120907874)
+- [Forkenbrock, G.J., 2009. A forward collision warning \(FCW\) performance evaluation.](http://refhub.elsevier.com/S0001-4575(25)00164-2/h0045)  *[National Highway Traffic Safety](http://refhub.elsevier.com/S0001-4575(25)00164-2/h0045)*. Administration.
+- Gaspar, J. G., Carbonari, R., Kaczmarski, H., & Kramer, A. F. (2015). Characterizing the effect of videophone conversations on intersection driving performance. In Proceedings of the Eighth International Driving Symposium on Human Factors in Driver Assessment, Training, and Vehicle Design.
+- Ismail, K., Sayed, T., Saunier, N., 2011. Methodologies for aggregating traffic conflict indicators. Transport. Res. Rec.: J. Transport. Res. Board 2237 (1), 10–19. [https://](https://doi.org/10.3141/2237-02)  [doi.org/10.3141/2237-02.](https://doi.org/10.3141/2237-02)
+- Jermakian, J.S., 2011. Crash avoidance potential of four passenger vehicle technologies. Accid. Anal. Prevent. 43 (3), 732–740. [https://pubmed.ncbi.nlm.nih.gov/](https://pubmed.ncbi.nlm.nih.gov/21376861/)  [21376861/](https://pubmed.ncbi.nlm.nih.gov/21376861/).
+- Kiefer, R., LeBlanc, D., Palmer, M., Salinger, J., Deering, R., & Shulman, M. (1999). Development and validation of functional definitions and evaluation procedures for collision warning/avoidance systems. National Highway Traffic Safety Administration.
+- [Klebelsberg, D., 1964. Derzeitiger sand der verhaltensanalyse des kraftfahrens. Arbeit](http://refhub.elsevier.com/S0001-4575(25)00164-2/h0070)  [und leitsung. Ablt. Arbeitswissenscaft Soziale Betriebspraxis 18, 33](http://refhub.elsevier.com/S0001-4575(25)00164-2/h0070)–37.
+- Kusano, K.D., Gabler, H.C., 2012. Safety benefits of forward collision warning, brake assist, and autonomous braking systems in rear-end collisions. IEEE Trans. Intellig. Transport. Syst. 13 (4), 1546–1555. [https://doi.org/10.1109/TITS.2012.2191542.](https://doi.org/10.1109/TITS.2012.2191542)
+- Lai, Z., Karim, I., Hai, M.X., 2014. Traffic conflict techniques for road safety analysis: Open questions and some insights. Canad. J. Civil Eng. 41 (7), 633–641. [https://doi.](https://doi.org/10.1139/cjce-2013-0558)  [org/10.1139/cjce-2013-0558](https://doi.org/10.1139/cjce-2013-0558).
+- [Laureshyn, A., Varhelyi, A., 2018. The Swedish Traffic Conflict Technique: Observers](http://refhub.elsevier.com/S0001-4575(25)00164-2/h0085)  [manual. Lund University, Lund, Sweden.](http://refhub.elsevier.com/S0001-4575(25)00164-2/h0085)
+- Lee, J.D., McGehee, D.V., Brown, T.L., Reyes, M.L., 2002. Collision warning timing, driver distraction, and driver response to imminent rear-end collisions in a highfidelity driving simulator. Human Factors 44 (2), 314–334. [https://doi.org/](https://doi.org/10.1518/0018720024497844)  [10.1518/0018720024497844](https://doi.org/10.1518/0018720024497844).
+- McLaughlin, S.B., Hankey, J.M., Dingus, T.A., 2008. A method for evaluating collision avoidance systems using naturalistic driving data. Accid. Anal. Prevent. 40 (1), 8–16. [https://doi.org/10.1016/j.aap.2007.03.016.](https://doi.org/10.1016/j.aap.2007.03.016)
+- [Najim, W.G., Samuel, T., John, H., 2007. Pre-crash scenario framework for crash](http://refhub.elsevier.com/S0001-4575(25)00164-2/h0100)  [avoidance systems based on vehicle-to-vehicle communications. National Highway](http://refhub.elsevier.com/S0001-4575(25)00164-2/h0100) [Traffic Safety Administration, United States.](http://refhub.elsevier.com/S0001-4575(25)00164-2/h0100)
+
+- National Highway Traffic Safety Administration. (2023). *Traffic safety facts: Early estimates of motor vehicle traffic fatalities and fatality rate by sub-categories in 2022*. NHTSA's National Center for Statistics and Analysis.
+- National Transportation Safety Board. (2002). *Vehicle and infrastructure-based technology for prevention of rear-end collisions* (NTSB/SIR-01/01, PB2001-P17003). National Transportation Safety Board.
+- Papadoulis, A., Quddus, M., Imprialou, M., 2019. Evaluating the safety impact of connected and autonomous vehicles on motorways. Accid. Anal. Prevent. 124, 12–22. <https://doi.org/10.1016/j.aap.2018.12.019>.
+- Perkins, S.R., Harris, J.I., 1968. Traffic conflict characteristics—Accident potential at intersections. Retrieved from Highway Research Record 225, 35–43. [http://onlinepu](http://onlinepubs.trb.org/Onlinepubs/hrr/1968/225/225%e2%88%92004.pdf)  [bs.trb.org/Onlinepubs/hrr/1968/225/225](http://onlinepubs.trb.org/Onlinepubs/hrr/1968/225/225%e2%88%92004.pdf)− 004.pdf.
+- [Rakha, H.A., Fitch, G.M., Arafeh, M., Blanco, M., Hanowski, R.J., 2010. Evaluation of](http://refhub.elsevier.com/S0001-4575(25)00164-2/h0125) [safety benefits from a heavy-vehicle forward collision warning system. Transport.](http://refhub.elsevier.com/S0001-4575(25)00164-2/h0125) [Res. Rec. 2194 \(1\), 44](http://refhub.elsevier.com/S0001-4575(25)00164-2/h0125)–54.
+- Seaman, S., Gershon, P., Angell, L., Mehler, B., Reimer, B., 2022. Evaluating the associations between forward collision warning severity and driving context. Article 5 Safety 8 (1). <https://doi.org/10.3390/safety8010005>.
+- Tarko, A.P., 2018. Estimating the expected number of crashes with traffic conflicts and the Lomax distribution—A theoretical and numerical exploration. Accid. Anal. Prevent. 113, 63–73. <https://doi.org/10.1016/j.aap.2018.01.008>.
+- Tarko, A.P., 2021. A unifying view on traffic conflicts and their connection with crashes. Accid. Anal. Prevent. 158, 106187. [https://doi.org/10.1016/j.aap.2021.106187.](https://doi.org/10.1016/j.aap.2021.106187)
+- [Tarko, A., Davis, G., Saunier, N., Sayed, T., 2009. Surrogate measures of safety. In](http://refhub.elsevier.com/S0001-4575(25)00164-2/h0145)  [Transportation Research Board Annual Meeting Compendium of Papers.](http://refhub.elsevier.com/S0001-4575(25)00164-2/h0145) [Transportation Research Board, Washington, DC](http://refhub.elsevier.com/S0001-4575(25)00164-2/h0145).
+- Teoh, E.R., 2021. Effectiveness of front crash prevention systems in reducing large truck real-world crash rates. Traffic Injury Prevention 22 (4), 284–289. [https://trid.trb.](https://trid.trb.org/view/1850298) [org/view/1850298.](https://trid.trb.org/view/1850298)
+- Xiao, G., Lee, J., Jiang, Q., Huang, H., Abdel-Aty, M., Wang, L., 2021. Safety improvements by intelligent connected vehicle technologies: a meta-analysis considering market penetration rates. Accid. Anal. Prevent. 159, 106234. [https://](https://doi.org/10.1016/j.aap.2021.106234)  [doi.org/10.1016/j.aap.2021.106234](https://doi.org/10.1016/j.aap.2021.106234).
+- Yue, L., Abdel-Aty, M.A., Wu, Y., Farid, A., 2020. The practical effectiveness of advanced driver assistance systems at different roadway facilities: system limitation, adoption, and usage. IEEE Trans. Intellig. Transport. Syst. 21 (9), 3859–3870. [https://doi.org/](https://doi.org/10.1109/TITS.2019.2935195)  [10.1109/TITS.2019.2935195](https://doi.org/10.1109/TITS.2019.2935195).

Zotero/001_artiklid/Road Safety Indicators/onelcinevacuationplanindustrialzone2013.md

@@ -0,0 +1,364 @@
+---
+category: literaturenote
+citekey: onelcinevacuationplanindustrialzone2013
+title: "Evacuation plan of an industrial zone: Case study of a chemical accident in Aliaga, Turkey and the comparison of two different simulation softwares"
+authors: "Onelcin, Pelin; Mutlu, Mehmet Metin; Alver, Yalcin"
+year: 2013
+date: 2013-00-00 2013
+doi: 10.1016/j.ssci.2013.07.006
+publication: Safety Science
+url: "https://doi.org/10.1016/j.ssci.2013.07.006"
+zotero_key: K7ST5TI3
+zotero_storage: 7XQGIVCA
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Onelcin, Pelin"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+Contents lists available at [SciVerse ScienceDirect](http://www.sciencedirect.com/science/journal/09257535)
+
+# Safety Science
+
+journal homepage: [www.elsevier.com/locate/ssci](http://www.elsevier.com/locate/ssci)
+
+![](_page_0_Picture_5.jpeg)
+
+# Evacuation plan of an industrial zone: Case study of a chemical accident in Aliaga, Turkey and the comparison of two different simulation softwares
+
+![](_page_0_Picture_7.jpeg)
+
+Pelin Onelcin ⇑ , Mehmet Metin Mutlu <sup>1</sup> , Yalcin Alver <sup>1</sup>
+
+Department of Civil Engineering, Ege University, 35100 Bornova, Izmir, Turkey
+
+#### article info
+
+Article history: Received 13 May 2013 Received in revised form 3 July 2013 Accepted 7 July 2013 Available online 31 July 2013
+
+Keywords: Evacuation Traffic simulation Evacuation time estimate
+
+#### abstract
+
+Chemical accidents cause great danger for people working in the chemical facilities and residents living close to these facilities where in general evacuation is required in case of an accident. The most efficient way to create evacuation plans is to simulate the transportation network. In this paper, evacuation simulations are carried out for Aliaga, the industrial zone of Izmir, Turkey where petro-chemical companies such as Petkim and Tupras are active. The scenario is based on an explosion in the acrylonitrile (ACN) factory in Petkim. The risk zone is determined considering the features of ACN, the wind speed and the direction of the wind. Total number of evacuees is found 14,472, almost 26% of the population of Aliaga. The second part of this study deals with the simulations executed via mesoscopic simulation software called Cube Avenue and microscopic simulation software called MATSim. The aim of using two simulation softwares is to be able to compare evacuation time estimates and to obtain reliable results from evacuation simulations which are crucial and critical in creating emergency plans. In the final part, a comparison of the two simulation results is presented.
+
+> -2013 Elsevier Ltd. All rights reserved.
+
+### 1. Introduction
+
+The number of man-made disasters has increased in the last decades with the industrial development. One type of these disasters is the chemical accidents which for the most part has an instantaneous characteristic that cannot be estimated beforehand, and appears with explosion and fire.
+
+History witnessed disastrous chemical accidents requiring evacuation such as Seveso, Italy accident in 1976 leading to Seveso Directive fulfill the necessity of a legislation concerning the prevention and control of such accidents developed by European Union in 1982 [\(European Commission, 2012](#page-6-0)).
+
+In 1984, the Bhopal, India accident proved the significant role of an evacuation plan. Since the authorities were caught unprepared for this disaster, the consequences were tragic. Several estimations range from 70,000 to 500,000 evacuees ([Sriramachari, 2005\)](#page-7-0).
+
+The decision of evacuation or shelter-in-place is made concerning factors such as the available shelters and evacuation time estimates (ETE) [\(Sorensen et al., 2004](#page-7-0)). It is apparent that the conditions differ in emergency situations with the entrance of high number of vehicles to the network in a short time leading to an increase in demand. Evacuation time estimate is strongly dependent on the capacity of the evacuation route system being capable of handling the traffic demand [\(Lindell and Prater, 2007](#page-7-0)).
+
+Evacuation time can be used in two meanings, one for the time required to move the persons in the evacuation area to safer places, and the other for the time passed from the moment of the evacuation order to the clearance of the evacuation area ([Barrett et al.,](#page-6-0) [2000](#page-6-0)).
+
+The consideration of the behavior of the households is a crucial point in ETE, but, it is generally ignored. In emergency situations some people would tend to enter the risk area to save their children, etc. Thus, the simulation time could give more optimistic results than reality ([Murray-Tuite and Mahmassani, 2003\)](#page-7-0).
+
+The number of evacuees shows variations comparing the day and nighttime due to the increase in the population for work, education or special facilities (e.g., shopping centers, etc.) during the day. For nighttime evacuation planning, census information on the number of residents could be sufficient. After determining the number of persons to be evacuated, this information should be translated into the number of evacuating vehicles in order to estimate the area clearance rates and the potential traffic delays ([Southworth, 1991\)](#page-7-0). [Goldblatt \(2004\)](#page-7-0) presents a well-developed list involving the steps in order to identify the best evacuation routes and time estimates.
+
+In emergency situations static assignment models do not allow realistic results as the traffic conditions remain constant during
+
+<sup>⇑</sup> Corresponding author. Tel.: +90 232 388 60 26; fax: +90 232 342 56 29. E-mail addresses: [pelin.onelcin@ege.edu.tr](mailto:pelin.onelcin@ege.edu.tr) (P. Onelcin), [mmetinm@gmail.com](mailto:mmetinm@gmail.com) (M.M. Mutlu), [yalcin.alver@ege.edu.tr](mailto:yalcin.alver@ege.edu.tr) (Y. Alver).
+
+<sup>1</sup> Tel.: +90 232 388 60 26; fax: +90 232 342 56 29.
+
+simulation period. Dynamic assignment models consider the changes in traffic conditions and those in the level of service ([Als](#page-6-0)[nih and Stopher, 2004](#page-6-0)).
+
+Traffic simulation models have been an assisting tool for evaluating alternative roadway designs, calculating optimum travel times through route analysis or retiming intersection signaling. However, with increase in number of emergency situations, modeling traffic flow via simulation softwares becomes an essential part of transport emergency management plans ([Moriarty et al.,](#page-7-0) [2007\)](#page-7-0).
+
+There are three types of models used for modeling the traffic network based on the scale of a geographic area and level of detail; macroscopic, microscopic and mesoscopic models [\(Hardy and](#page-7-0) [Wunderlich, 2007](#page-7-0)). Macroscopic models are used to simulate the traffic flow based on speed, flow, and traffic density relationships, and do not model the interactions between individual vehicles whereas microscopic models simulate interactions of individual vehicles [\(Federal Highway Administration, 2004\)](#page-6-0). The earliest evacuation planning models were due to major accidents and were based on macroscopic scale simulations. Microscopic models are also used for evacuation modeling though the requirement for comprehensive data on the network (i.e., roadway geometry) brings limitations to the extensive use of these softwares ([Hardy](#page-7-0) [and Wunderlich, 2007\)](#page-7-0). Mesoscopic models are used to simulate the individual vehicles describing their interactions on aggregate relationships [\(Federal Highway Administration, 2004](#page-6-0)).
+
+Mesoscopic models came into use in the sense, filling the gap between the aggregate level approach of macroscopic models and the individual interactions of the microscopic ones. They are able to describe the traffic entities at a high level of detail but, the interactions among them show a lower level of detail which is ideal for prediction applications where a detailed modeling of route choice is sufficient ([Burghout, 2005](#page-6-0)).
+
+There are five mesoscopic simulation models which are being currently used for evacuation modeling: Simulex, TRANSIMS, TransModeler, DYNSMART-P and Cube Avenue ([Hardy and Wun](#page-7-0)[derlich, 2007](#page-7-0)).
+
+This study deals with evacuation of Aliaga, Izmir, where chemical facilities densely take place and presents the evacuation process via simulation models executed in Cube Avenue and MATSim softwares. The need of this study emerged from the lack of an evacuation plan for such an important area in terms of risk being held and the residential areas being located in the vicinity of the chemical facilities conducted by Petkim and Tupras. The factory where the accident would occur was selected as a first step and the evacuation area was determined by taking the risk zones given by Aloha software into consideration. The estimation of the evacuation time was the primary goal of this study. In order to have reliable results, two simulation softwares were used, being MATSim a microscopic simulation model and Cube Avenue a mesoscopic simulation model.
+
+#### 2. Case study: Aliaga
+
+Aliaga is one of the 30 districts of Izmir Province in the Aegean Region of Turkey. The town is situated in about 50 km north of Izmir and has crucial contributions to Turkey's economy. Since Aliaga has one of the greatest petrochemical enterprises of Turkey, called Petkim, the study focuses on this region. In 2011 Petkim was ranked as the 12th greatest enterprise among 500 top industrial enterprises in Turkey ([Istanbul Chamber of Industry, 2011\)](#page-7-0). There are 14 plants and seven auxiliary units belonging to Petkim. As the authors live in Izmir city, Aliaga was a perfect target to operate the evacuation studies.
+
+Turkish Statistical Institute announces the population of Aliaga as 68,432 among which 56,440 live in the central neighborhoods and 11,992 live in the villages in year 2011 ([Turkish Statistical](#page-7-0) [Institute, 2011](#page-7-0)).
+
+#### 2.1. Evacuation zone and zone division
+
+[Southworth \(1991\)](#page-7-0) indicated that applying an evacuation planning model begins with the selection of the evacuation zone boundary. It is believed that outside this zone the hazard in question will not be a threat for people.
+
+The evacuation zone is determined based on a scenario developed for a chemical accident in one of the factories of Petkim. Features of the chemical substances and the National Fire Protection Association (NFPA 704) ratings ranging from 1 to 4, given for flammability, health and reactivity are compared. The NFPA values are shown in Table 1. It is apparent that the most dangerous substance is acrylonitrile (ACN) with the ratings 3, 4, 2 for flammability, health and reactivity, respectively. This is confirmed by the authorities in Petkim ([Petkim Petrochemical Holding Seminar, 2011\)](#page-7-0). Thus ACN is decided as the location of the impact. Vinyl chloride monomer, propylene or ethylene could also have been chosen, but, risk zones are smaller and thus lower evacuation times would be obtained.
+
+The radius of the risk zones are found via Aloha software developed by Office of Emergency Management and Emergency Response Division. The software divided the risk area into three zones based on the type of the chemical substance, wind speed and wind direction. The wind data obtained from Aliaga wind measurement station shows that the maximum wind speed is 17 m/s ([WolframAlpha, 2012](#page-7-0)) and the prevailing wind direction is W-NW. The wind blowing from west is the most dangerous one for Aliaga, because it would carry the smoke over residential areas which would increase the number of persons to be evacuated. Aloha software shows threat zones as circles and in the estimation of the evacuation times the clearance time of these circles are used. In this regard not only the west wind was assumed but all wind directions were considered. The risk zones had a radius of 1.7, 2.4 and 3.7 km for the first, second, and the third risk zone, respectively. In [Fig. 1](#page-2-0), the evacuation zone boundary is shown. All the factories of Petkim and Tupras, a part of the residential buildings, and a part of the E87 (D550) Highway remain within the boundary.
+
+The definition of transportation analysis zone (TAZ) is given as ''spatial entities of at least a few blocks to much larger areas, like a neighborhood'' by [Church and Sexton \(2002\).](#page-6-0) An origin–destination matrix is assumed between a set of TAZs [\(Church and Sexton,](#page-6-0) [2002\)](#page-6-0).
+
+The evacuation zone is divided into 19 TAZs. The trips that are generated from these zones are directed to six destination TAZs of which three of them are health care centers and three of them are gathering-places. Petkim area is divided into six zones where the first zone is in the impact area. Tupras area remains in the
+
+Table 1 NFPA hazard potential ratings.
+
+|                        | NFPA hazard potential ratings |        |            |  |  |  |
+|------------------------|-------------------------------|--------|------------|--|--|--|
+|                        | Flammability                  | Health | Reactivity |  |  |  |
+| Substances             |                               |        |            |  |  |  |
+| Ethylene glycol        | 1                             | 1      | 0          |  |  |  |
+| Polyvinyl chloride     | 1                             | 1      | 0          |  |  |  |
+| Ethylene               | 4                             | 1      | 2          |  |  |  |
+| Polyethylene           | 1                             | 0      | 0          |  |  |  |
+| Acrylonitrile          | 3                             | 4      | 2          |  |  |  |
+| Vinyl chloride monomer | 4                             | 2      | 1          |  |  |  |
+| Propylene              | 4                             | 1      | 1          |  |  |  |
+
+<span id="page-2-0"></span>![](_page_2_Picture_2.jpeg)
+
+Fig. 1. Evacuation planning zone.
+
+second risk area and is divided into two zones due to remarkable vehicle density difference of the area.
+
+#### 2.2. Number of evacuees
+
+The estimation of the evacuation demand requires the assessment of the size of the population which can be achieved through the subdivision of the emergency planning zones (EPZ) population into three groups with permanent residents, transients and special facility population ([Urbanik, 2000](#page-7-0)). In this study, number of evacuees is calculated considering the permanent residents and employees being as transients. The scenario is prepared by taking into account the following conditions:
+
+- - The explosion occurs in the evening when there are no students in schools and people are awake.
+- - The employees in the first risk zone and partially in the second risk zone are taken to the Aliaga state hospital in zone 30, and to the other health care centers in zones 26 and 27. The first risk zone is the most vulnerable zone, thus, persons who need medical intervention in this area should be taken to hospital. The typical behavior of persons in Turkey is to flock to hospitals in emergency situations. In generation of the scenarios this behavior is considered and not only in the first risk zone but also in the second risk zone health care centers are determined as destination zones.
+- - People living in the residential zones self-evacuate. Since Aliaga is a small town, public transportation service is weak. Moreover, in the evening the frequency of public transportation is low. Therefore, public transportation is not considered in the present study.
+- - The employees in Petkim and in Tupras work in three shifts. The factories are active for 24 h and there always are employees present in the factories.
+
+The total number of the employees is 2586 for Petkim, and 1297 for Tupras. The number of employees to be evacuated from the factories is computed by means of the following assumptions, which are:
+
+- - The total number of the employees is divided into three as they work in three shifts.
+- - The time of the explosion does not correspond to the time of the shift change.
+
+The number of evacuees from the residential buildings is calculated following these steps:
+
+- - The number of the persons living in the neighborhood belonging to the evacuation zone is divided into the number of the buildings in that neighborhood which give the mean number of the persons living in one building.
+- - The number of the buildings that remains in the evacuation zone is multiplied by the mean number of persons in one building.
+
+Based on the above assumptions the number of employees present at Petkim is 862 and at Tupras is 432 at the time of the accident. Total number of the persons that need to be evacuated from Petkim and Tupras is 1294, and from the residential area are 13,178.
+
+#### 2.3. Number of evacuating vehicles
+
+In literature, number of evacuating vehicles is given on the basis of household (EVHH) for permanent residents. [Cova and Johnson](#page-6-0) [\(2002\)](#page-6-0) use a Poisson distribution in order to compute EVHH. Surveys are a guiding light in identifying the evacuating vehicle per residential household (EVHH), as well, because they take into account the behavior of the persons ([Lindell and Prater,](#page-7-0) [2007\)](#page-7-0).
+
+In emergency situations people are not expected to behave as they do under normal conditions. Some people would tend to escape on foot. However, considering the risk zones, it would make sense to hypothesize that people would leave by vehicles. Different from vehicles, pedestrians' flexibility for route choice is greater ([Federal Highway Administration, 2007\)](#page-7-0). They would use pavements, roads, any space that seems available at that moment, and it is very difficult to model their behaviors since it is not clear which route they would choose. Besides, the available pedestrian behavior simulation softwares in general deal with closed area evacuations ([Santos and Aguirre, 2004\)](#page-7-0), and there do not exist many softwares capable of simulating the transportation network under the emergency situations. Pedestrian behavior could be the subject of another study.
+
+Here, to estimate the number of evacuation vehicles, car occupancy ratio rate is used. This rate is 1.57 in normal situations ([Istanbul Metropolitan Municipality Directorate of Transportation](#page-7-0) [Planning, 2011](#page-7-0)) however in emergency situation it is expected to be higher. In this study the car occupancy ratio rate is taken as two. Since the number of evacuees is 14,472, the number of vehicles is computed as 7236.
+
+#### 2.4. Network
+
+The network of Aliaga has been taken from Open Street Map (OSM) and has been converted to a shape file and MATSim network file with NetworkGenerator.class of MATSim. Zones used in generating synthetic population for MATSim were created in Quantum GIS (QGIS). For the simulation executed in Cube Avenue the network was saved as a shape file in QGIS and exported to ArcGIS v.9.3. In Fig. 2, the network in OSM format and the network that is used for simulations are shown.
+
+Functional road classes are used to introduce the transport network. According to a definition given by FHWA ''functional road classification is the process by which streets and highways are grouped into classes, or systems, according to the character of service they are intended to provide''. In urbanized areas there are four functional systems, namely; principal arterials, minor arterial streets, collector streets, and local streets [\(Federal Highway](#page-7-0) [Administration, 2008](#page-7-0)).
+
+The transport network that is used for evacuation simulation in this study constitute of a principal arterial, collectors and local streets. The speed limit varies according to the type of the road. For the local streets it is 50 km/h, for the collectors it is 70 km/h, and for the E87 highway (principal arterial) it is 100 km/h.
+
+The capacity values of the roads are a modification of Highway Capacity Manual (HCM) 2010 values. HCM is a widely used source involving methods for analyzing the operations of streets, highways, pedestrian and bicycle facilities. HCM gives the theoretical capacity values per lane. Driver behavior, weather conditions, lane width are all factors that could change the capacity values, hence the local capacity value of the E87 highway is 1050 veh/h/ln, that of the collectors is 450 veh/h/ln, and of the local streets is taken as 300 veh/h/ln in this study.
+
+#### 2.5. Scenarios
+
+Scenarios are generated once the TAZs, the number of evacuating vehicles and the network characteristics are ready to be processed. Three different scenarios have been identified for the evacuation simulation. In [Table 2,](#page-4-0) OD matrices for each scenario can be seen.
+
+The selection of these three scenarios is mainly based on the location of the destination zones and traffic demand. Free spaces which are close to the risk zone are designated as gathering-areas. The time required for the evacuees to reach health care centers is of significant importance in an emergency situation such as in this case. The traffic demand generated for the health care centers is distributed between zone 26, 27 and 30 in the scenarios though the first risk zone is always directed to Aliaga State Hospital for it has the greatest capacity among the health care centers and severely injured persons should be transferred to this state hospital. Evacuating vehicles departing from the second risk zone are directed to health care centers in zone 26 and 27. Changing the number of evacuating vehicles between these zones result in different evacuation times. Thus, these different scenarios could allow the authors to observe the effect of traffic demand on traffic and if this leads to any reduction in evacuation time.
+
+The aim is to provide the arrival of evacuees to health care centers and gathering areas with a minimum time. The simulations based on these scenarios allow to choose the most useful scenario to be implemented in real case.
+
+![](_page_3_Figure_14.jpeg)
+
+Fig. 2. Aliaga network files (a) network used in simulations and (b) OSM network.
+
+<span id="page-4-0"></span>Table 2 OD matrices for evacuation scenarios.
+
+|          | Scenario 1 |     |     |     |     |     | Scenario 2 |     |     |     |     |     | Scenario 3 |     |     |     |     |     |
+|----------|------------|-----|-----|-----|-----|-----|------------|-----|-----|-----|-----|-----|------------|-----|-----|-----|-----|-----|
+|          | 26         | 27  | 30  | 32  | 33  | 34  | 26         | 27  | 30  | 32  | 33  | 34  | 26         | 27  | 30  | 32  | 33  | 34  |
+| R.Z. 1   |            |     |     |     |     |     |            |     |     |     |     |     |            |     |     |     |     |     |
+| 1        |            |     | 33  |     |     |     |            |     | 33  |     |     |     |            |     | 33  |     |     |     |
+| 3        |            |     | 17  |     |     |     |            |     | 17  |     |     |     |            |     | 17  |     |     |     |
+| 4        |            |     | 33  |     |     |     |            |     | 33  |     |     |     |            |     | 33  |     |     |     |
+| 6        |            |     | 83  |     |     |     |            |     | 83  |     |     |     |            |     | 83  |     |     |     |
+| 7        |            |     | 50  |     |     |     |            |     | 50  |     |     |     |            |     | 50  |     |     |     |
+| R.Z. 1–2 |            |     |     |     |     |     |            |     |     |     |     |     |            |     |     |     |     |     |
+| 10       |            |     | 221 |     |     |     |            |     | 221 |     |     |     |            |     | 221 |     |     |     |
+| 13       | 16         |     |     |     |     |     | 16         |     |     |     |     |     | 16         |     |     |     |     |     |
+| 14       | 90         |     |     |     |     |     | 50         |     | 40  |     |     |     | 50         |     | 40  |     |     |     |
+| R.Z. 2–3 |            |     |     |     |     |     |            |     |     |     |     |     |            |     |     |     |     |     |
+| 11       |            |     |     | 837 |     |     |            |     |     | 837 |     |     |            |     |     | 837 |     |     |
+| 15       |            | 320 |     |     |     |     | 120        | 200 |     |     |     |     |            | 120 |     |     | 200 |     |
+| 17       |            |     |     | 53  |     |     |            |     |     | 53  |     |     |            |     |     | 53  |     |     |
+| R.Z. 3   |            |     |     |     |     |     |            |     |     |     |     |     |            |     |     |     |     |     |
+| 16       |            |     |     |     |     | 660 |            |     |     |     | 660 |     |            |     |     |     |     | 660 |
+| 18       |            |     |     |     | 151 |     |            |     |     |     | 151 |     |            |     |     |     | 151 |     |
+| 19       |            |     |     |     | 158 |     |            |     |     |     | 158 |     |            |     |     |     | 158 |     |
+| 20       |            |     |     |     | 216 |     |            |     |     |     | 216 |     |            |     |     |     | 216 |     |
+| 21       |            |     |     |     | 137 |     |            |     |     |     | 137 |     |            |     |     |     | 137 |     |
+| 22       |            |     |     |     | 175 | 280 |            |     |     |     |     | 280 |            |     |     |     |     | 280 |
+| 23       |            |     |     |     |     |     |            |     |     |     | 175 |     |            |     |     |     | 175 |     |
+| 24       |            |     |     |     |     | 88  |            |     |     |     |     | 88  |            |     |     |     |     | 88  |
+
+## 2.5.1. Scenario 1
+
+In the first scenario, 437 vehicles from evacuation zones 1, 3, 4, 6, 7 and 10 in the first risk zone depart for the Aliaga State Hospital in zone 30. Apart from Aliaga State Hospital, there are two more health care centers outside the boundary of the third risk zone which will accept evacuees from the second and the third risk zones. There are three gathering-areas designated as a priori for the evacuees in case of emergency situations and the individuals are assumed to be informed about their own evacuation destinations. The gathering-areas are designated as zone 32, 33 and 34 which 890, 837 and 1028 vehicles will reach, respectively.
+
+#### 2.5.2. Scenario 2
+
+The second scenario differs from the first scenario in number of persons that will reach health care centers. In this case, 477 vehicles head for Aliaga State Hospital, 186 for zone 26 and 200 for zone 27. The number of vehicles going to gathering-areas remains the same as in the first scenario.
+
+## 2.5.3. Scenario 3
+
+Destination zones of 200 vehicles in risk zone 3 which were sent to the health care centers in the first two scenarios were changed in this case, and their destination was assigned to zone 33. The aim is to observe if the number of vehicles lead to a decrease in evacuation time.
+
+The number of scenarios can be increased; indeed numerous scenarios can be created. However the scenarios should reflect the possible behavior of the evacuees. It is proved for many times that in Turkey the tendency is to try to reach the health care centers.
+
+In industrial areas like Aliaga, it is important to have previous knowledge about the emergency situations and what should be done under those circumstances. In the present study, persons are assumed to be informed about the destination zones, thus they know where the health care centers are located.
+
+The less risky zones can be diverted to the gathering areas that are identified beforehand. Besides, the health care centers gathering areas are also determined as destination zones.
+
+# 3. Methodology
+
+A simulation of a transportation network helps to observe various alternatives before implementing them in real world [\(Frick,](#page-7-0) [2011](#page-7-0)).
+
+Simulation plays a significant role in traffic management research in evacuation [\(Duanmu et al., 2012\)](#page-6-0). In the past decades, for the purpose of evacuation modeling, transportation engineers have become involved in the application of modeling and simulation systems. [\(Murray-Tuite and Wolshon, 2013](#page-7-0)).
+
+The use of simulation-based dynamic assignment model provides two fundamental advantages over analytical model. One is the modeling and observation of vehicle-to-vehicle interactions and the other is evaluation of operational strategies at the individual vehicle level ([Ben-Akiva et al., 2012](#page-6-0)).
+
+Simulations were performed using Citilabs Cube Avenue and MATSim softwares. Brief information about the operation principles of both softwares are given as below.
+
+#### 3.1. Citilabs Cube Avenue
+
+In 2005 after Hurricane Rita, Houston-Galveston Area Council (Texas) was asked to prepare a tool for evacuation planning [\(White](#page-7-0) [and Chanchareon, 2008](#page-7-0)). Cube Avenue was first used to help the local planners and decision makers during emergency evacuations in Houston- Galveston area to be prepared for incidents such as Hurricane Rita and to come up with alternate solutions for evacuations ([Hardy and Wunderlich, 2007\)](#page-7-0).
+
+Avenue is an optional add-onto Cube Voyager that allows planners to study dynamic traffic assignment with mesoscopic simulation [\(Brown, 2011\)](#page-6-0).
+
+Vehicles are analyzed as ''packets'' and simulated as they move through the network. Packets can be either moving or waiting on a link. A vehicle may have to wait due to capacity constraints; cars leaving a link exceed its exit flow capacity or cars entering a link exceed its entrance flow capacity, or due to storage constraints; no room for a vehicle on the next link. The resulting traffic stream on each link is evaluated using aggregate macroscopic speed-flow relationships.
+
+<span id="page-5-0"></span>Table 3 Risk zones evacuation times.
+
+| MATSim simulation results |                |                |                | Citilabs Cube Avenue simulation<br>results |                |                |                |  |
+|---------------------------|----------------|----------------|----------------|--------------------------------------------|----------------|----------------|----------------|--|
+|                           | Risk<br>zone 1 | Risk<br>zone 2 | Risk<br>zone 3 |                                            | Risk<br>zone 1 | Risk<br>zone 2 | Risk<br>zone 3 |  |
+| Scenario<br>1             | 45             | 83             | 86             | Scenario<br>1                              | 43             | 64             | 93             |  |
+| Scenario<br>2             | 44             | 82             | 91             | Scenario<br>2                              | 42             | 65             | 106            |  |
+| Scenario<br>3             | 47             | 86             | 88             | Scenario<br>3                              | 43             | 65             | 80             |  |
+
+As an input, Avenue requires a network file in Cube format, OD matrix, storage which is the number of vehicles that can fill the link, distance, capacity and number of lanes.
+
+The storage of the road links are calculated as the multiplication of the number of lanes, vehicle per distance and the distance of the link.
+
+Cube Avenue uses a generalized cost function as the disutility function in the assignment that can include any variable with its own weight. The assignment in Avenue is user equilibrium that utilizes method of successive averages method.
+
+The centroids are automatically assigned and the trips are generated from the centroid of the origin zone and reach to the centroid of the destination zone. The simulations are performed considering the abovementioned scenarios.
+
+#### 3.2. MATSim
+
+MATSim is an agent-based, activity-based, easily extendable, open-source multiagent dynamic simulation toolkit implemented in Java in a modular structure that can be modified or replaced according to various conditions to handle different scenarios [\(Hor](#page-7-0)[ni et al., 2009](#page-7-0)). [Ferber \(1999\)](#page-7-0) describes an agent as an autonomous entity (real or virtual) with its own abilities, resources and goals that can act on and perceive its environment network; has some internal representation of that environment; and can possibly communicate with other agents [\(Raney, 2005](#page-7-0)).
+
+The initial demand of plans of activities generated for each agent and is executed in a mobility simulation. Demand is the movement of agent, which is a driver on transportation network in this case, from a location to another on a route. Plan of each agent is scored after simulation iterations, and on the basis of the score, agents adapt their plans according to the conditions and events in the simulation, such as queues in intersections, and link capacities ([Bekhor et al., 2011\)](#page-6-0).
+
+The best plan for each agent (i.e., optimum demand for transportation system) is achieved as whole transportation system converges to a relaxed state after adequate iterations (200 iterations for this project) are made using initial demand.
+
+Projecting daily plan of an agent, including mode choice and facilities (e.g., shopping center, hospital, etc.) used depending on agents characteristics (e.g., financial situation, age, car owning, etc.), is possible using MATSim, however, for an evacuation plan, following data is sufficient:
+
+- -Initial demand (i.e., OD matrix).
+- -Road network.
+
+Initial demand refers to synthetic population derived from numbers and locations of evacuees to be transferred to presumed health care centers or gathering-areas according to distance. DemandGenerator module of MATSim is used to generate initial demand, which generates given zone. Zones are modified according to density of population that actually exists in given zone and road links that evacuees will possibly start their trip at a possible chemical accident moment to generate a relatively realistic scenario. MATSim assigns the trip start to closest node to coordinates of agent activity (i.e., home or work) for each agent, but not to the centroid of defined zone, in contrast to Citilabs Cube Avenue.
+
+#### 4. Comparison of the two simulation tools
+
+The clearance time of three risk zones and total arrival time for three different scenarios are listed in Table 3. It can be seen that evacuation times calculated with both softwares are close, especially for the first risk zone.
+
+Cube Avenue distributes the vehicles from the centroids of the origin zones. 10 centroid connectors for each zone are assigned. Trips are generated from the centroids and through the connectors they are assigned to the links inside the zone. Once the trips start, vehicles move towards the centroids of the destination zones.
+
+MATSim performs in a different way. The population zones are set for a group of origin zones which are assigned for a predefined destination. For each agent, random activity coordinates are generated such as home, work, and leisure, or in this case evacuation
+
+![](_page_5_Figure_21.jpeg)
+
+![](_page_5_Figure_22.jpeg)
+
+Fig. 3. Simulation snapshots (a) Citilabs Cube Avenue simulation snapshot and (b) MATSim simulation snapshot.
+
+<span id="page-6-0"></span>Table 4 Evacuation time averages.
+
+| Evacuation time averages |             |             |             |  |  |  |  |
+|--------------------------|-------------|-------------|-------------|--|--|--|--|
+|                          | Risk zone 1 | Risk zone 2 | Risk zone 3 |  |  |  |  |
+| Scenario 1               | 44          | 74          | 90          |  |  |  |  |
+| Scenario 2               | 43          | 74          | 99          |  |  |  |  |
+| Scenario 3               | 45          | 76          | 84          |  |  |  |  |
+
+zones and hospitals-gathering areas. Departures and arrivals of the agents take place on the nodes which are closest to the activity coordinates, and in the first iteration shortest path is calculated for route choice.
+
+Simulation results of MATSim are analyzed by via - visualization and analysis tool for MATSim by Senozon ([via|Senezon, 2008\)](#page-7-0).
+
+In [Fig. 3](#page-5-0) two snapshots from the simulation scenario 3 in 03:16th min are shown.
+
+### 5. Conclusion
+
+Regions where chemical facilities are active might confront accidents in any time. Thus, evacuation plans are important in identifying the measures to be taken during emergencies. An evacuation plan needs a well-defined scenario. Once the scenario is selected, the evacuation planning zone can be defined and subsequently the evacuation demand, traffic analysis zones and OD matrices can be evaluated.
+
+The study has been accomplished for a chemical accident in an industrial zone established nearby residential settlements in Aliaga, Izmir, Turkey.
+
+The factory of incident, that is the ACN factory, has been identified based on the features of the chemical substances processed in Petkim and their NFPA ratings. After the destination areas, where the health care centers and available gathering-areas exist, have been designated, the origin and destination zones have been defined. Based on the census and building data, number of evacuees has been calculated and found 14,472, and then has been transformed to the number of vehicles taking the car occupancy rate as two and was found 7236.
+
+The simulation has been performed by Cube Avenue and MAT-Sim softwares. Using two softwares allowed comparing the evacuation times. Emergency plans are important tools in saving lives and preventing losses. Thus, relying only on one simulation might cause inadequate measures. In this study, two approaches – microscopic and mesoscopic – were adopted and compared. Evacuation times of the risk zones calculated with Cube Avenue and MATSim are close, only in the second risk zone Cube Avenue gives lower values which can be a result of distributing the vehicles from 10 centroid connectors and MATSim distributes the vehicles from less number of nodes. Consequently, the averages of the results are taken as the evacuation time and are shown in Table 4.
+
+The final results show that for each scenario the clearance time of the first and second risk zones are close. However, for the first risk zone, 43–45 min clearance times are quite long durations to reach the hospital considering the severity of the event. A corridor with a higher capacity that could be used in emergency situations between Petkim area and Aliaga State Hospital could be designed which could result in lower evacuation times. For the third risk zone, the maximum clearance time is obtained for the second scenario. In this scenario Aliaga State Hospital attracts more vehicles, and not only the health care center in zone 27, but also in zone 26 are assigned as destination zones, and in order to reach zone 26, local streets are used which have lower speed values whereas for zone 27 collectors are used. This difference causes the increase in evacuation time. For zone 26 and 27 it would be proper to place flashing lights along the main roads so that persons could follow these signs to reach health care centers. For gathering areas, the advantage of these lights could be used too.
+
+Overall, the third scenario gives the best results with the minimum clearance time of the entire risk area among the given scenarios. This is a result of diverting 200 vehicles to a gathering area in zone 33 instead of a health care center. This makes sense for the evacuees living in the border of the second risk zone. They may clear the area without being affected by the chemical within a shorter time rather than driving to a health care center.
+
+One significant step of the evacuation plans is defining zones. As the zone dimension and location changes, the demand will change accordingly which will affect the whole transportation system. Especially, in evacuation the network will be used in full capacity which makes the zonal definitions more important. The accessibility of the zones, the capacity of the roads connecting the zones, the traffic demand are all aspects that need to be studied in detail. Roads with low capacities and high traffic demand will result in traffic congestion and high evacuation times which is undesirable. The simulations should be performed in order to determine the location of the destination zones which would avoid the false location choice. An area could be large enough to welcome a high number of persons but if only it is accessible and gives lower arrival times among the alternatives it could be accepted as a destination zone. In the process of preparing simulation data, in this study, it was observed that small modifications in zones and network could lead to great differences in simulation results. Thus, the initial step is the most important step of simulation which is data preparation that must be done in great detail.
+
+The number of scenarios can be increased, considering the time of the day, the impact area, etc. In the present study, the scenarios are generated for a nighttime event and all evacuees are assumed to leave the risk zones by private vehicles with a occupancy rate of two.
+
+Evacuation plans should be prepared and persons should be informed about the plan in peace time. During the time of the emergency, evacuees should already know where to escape and this is only accomplished by a well-functioning public information system. An evacuation plan for Aliaga, Izmir did not exist thus the main goal of this study was to create a basic evacuation plan for further improvements.
+
+# References
+
+[Alsnih, R., Stopher, P.R., 2004. Review of procedures associated with devising](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0005) [emergency evacuation plans. Transportation Research Record: Journal of](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0005) [Transportation Research Board 1865, 89–97](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0005).
+
+[Barrett, B., Ran, B., Pillai, R., 2000. Developing a dynamic traffic management](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0010) [modeling framework for hurricane evacuation. Transportation Research](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0010) [Record: Journal of the Transportation Research Board 1733, 115–121.](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0010)
+
+[Bekhor, S., Dobler, C., Axhausen, K.W., 2011. Integration pf activity-based and agent](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0015)[based models: case of Tel Aviv, Israel. Transportation Research Record: Journal](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0015) [of Transportation Research Board 2255, 38–47](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0015).
+
+[Ben-Akiva, M.E., Gao, S., Wei, Z., Wen, Y., 2012. A dynamic traffic assignment model](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0020) [for highly congested urban networks. Transportation Research Part C, 62–82.](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0020)
+
+Brown, C.M., 2011.PTP, Citilabs. Dynamic Traffic Assignment and Mesoscopic Simulation with Cube Avenue. SEFL FSUTMS User Group July 8 Meeting. Traffic Analysis Tools and Specialized Applications.
+
+[Burghout, W., 2005. Mesoscopic Simulation Models for Short-Term Prediction.](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0025) [Institute of Technology Linköpings Universitet](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0025).
+
+[Church, R.L., Sexton, R., 2002. Modeling Small Area Evacuation: Can Existing](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0030) [Transportation Infrastructure Impede Public Safety? California Department of](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0030) [Transportation](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0030).
+
+[Cova, T.J., Johnson, J.P., 2002. Microsimulation of neighborhood evacuations in](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0035) [the urban – wildland interface. Environment and Planning A 34 \(12\), 2211–](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0035) [2229.](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0035)
+
+[Duanmu, J., Chowdhury, M., Taaffe, K., Jordan, C., 2012. Buffering in evacuation](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0040) [management for optimal traffic demand distribution. Transportation Research](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0040) [Part E: Logistics and Transportation Review, 684–700](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0040).
+
+European Commission, 2012. European Commission Environment. <[http://](http://www.ec.europa.eu/environment/seveso/index.htm) [www.ec.europa.eu/environment/seveso/index.htm>](http://www.ec.europa.eu/environment/seveso/index.htm), (accessed 15.07.12).
+
+[Federal Highway Administration, 2004. Traffic Analysis Toolbox Volume III:](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0045) [Guidelines for Applying Traffic Microsimulation Modeling Software. US](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0045) [Department of Transportation.](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0045)
+
+- <span id="page-7-0"></span>Federal Highway Administration (2007) Managing Pedestrians During Evacuation of Metropolitan Areas. 2007. Final Report. Publication No. FHWA-HOP-07-066. Prepared by Battelle The Business of Innovation. Center for Human Performance and Safety.
+- Federal Highway Administration, 2008. Transportation, US Department of, ed. <[http://](http://www.fhwa.dot.gov/policy/ohpi/hpms/fchguidance.cfm) [www.fhwa.dot.gov/policy/ohpi/hpms/fchguidance.cfm>](http://www.fhwa.dot.gov/policy/ohpi/hpms/fchguidance.cfm), (accessed 16.08.12).
+- Ferber, J., 1999. Multi-Agent System: An Introduction to Distributed Artificial Intelligence. Harlow: Addison Wesley Longman. ISBN: 0-201- 36048-9.
+- Frick, R., 2011. Simulation of transportation networks. In: SCSC'11 Proceedings of the 2011 Summer Computer Simulation Conference, pp. 188–193.
+- [Goldblatt, P.R., 2004. Evacuation Planning: A Key Part of Emergency Planning.](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0050) [Transportation Research Board, Washington, DC.](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0050)
+- [Hardy, M., Wunderlich, K., 2007. Evacuation Management Operations \(EMO\)](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0055) [Modeling Assessment: Transportation Modeling Inventory. US Department of](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0055) [Transportation](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0055).
+- [Horni, A., Scott, D.M., Balmer, M., Axhausen, K.W., 2009. Location choice modeling](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0060) [for shopping and leisure activities with MATSim: combining microsimulation](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0060) [and time geography. Transportation Research Record: Journal of the](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0060) [Transportation Research Board 2135, 87–95](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0060).
+- Istanbul Chamber of Industry, 2011. [<http://www.iso.org.tr/en/iso-500/EN-Turkiye](http://www.iso.org.tr/en/iso-500/EN-Turkiye-nin-500-Buyuk-Sanayi-Kurulusu--ISO-500-raporunun-sonuclari.html)[nin-500-Buyuk-Sanayi-Kurulusu–ISO-500-raporunun-sonuclari.html>](http://www.iso.org.tr/en/iso-500/EN-Turkiye-nin-500-Buyuk-Sanayi-Kurulusu--ISO-500-raporunun-sonuclari.html), (accessed 03.07.12).
+- Istanbul Metropolitan Municipality Directorate of Transportation Planning, 2011. Istanbul Metropoliten Alanı Kentsel Ulasim Plani [Urban Transportation Plan of Metropolitan Area of Istanbul].
+- [Lindell, M.K., Prater, C.S., 2007. Critical behavioral assumptions in evacuation time](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0065) [estimate analysis for private vehicles: examples from hurricane research and](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0065) [planning. Journal of Urban Planning and Development 133](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0065).
+- [Moriarty, K.D., Daiheng, N., Collura, J., 2007. Modeling traffic flow under emergency](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0070) [evacuation situations: current practice and future directions. Transportation](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0070) [Research Board](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0070).
+
+- [Murray-Tuite, P.M., Mahmassani, H.S., 2003. Model of household trip-Chain](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0075) [sequencing in emergency evacuation. Transportation Research Record: Journal](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0075) [of the Transportation Research Board 1831, 21–29](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0075).
+- [Murray-Tuite, P., Wolshon, B., 2013. Evacuation transportation modeling: an](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0080) [overview of research, development, and practice. Transportation Research](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0080) [Part C: Emerging Technologies, 25–46](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0080).
+- Petkim Petrochemical Holding Seminar, 2011. Azerkimya Company, Baku, Azerbaijan.
+- [Raney, B.K., 2005. Learning Framework for Large-Scale Multi-Agent Simulations.](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0085) [Swiss Federal Institute of Technology, Zurich](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0085).
+- Santos, G., Aguirre, B.E., 2004. A critical review of emergency evacuation simulation models. In: NIST Workshop on Building Occupant Movement during Fire Emergencies, Disaster Research Center, University of Delaware.
+- [Sorensen, J., Shumpert, B., Vogt, B., 2004. Planning for protective action decision](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0090) [making:evacuate or shelter-in-place. Journal of Hazardous Materials A 109, 1–](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0090) [11](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0090).
+- [Southworth, F., 1991. mRegional Evacuation Modeling: A State-of-the-Art Review.](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0095) [Oak Ridge National Laboratory.](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0095)
+- [Sriramachari, S., 2005. Bhopal gas tragedy:scientific challenges and lessons for](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0100) [future. Journal of Loss Prevention in the Process Industries 18 \(4–6\),](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0100) [264–267](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0100).
+- Turkish Statistical Institute, 2011. [<http://www.turkstat.gov.tr/Start.do;jsessionid=](http://www.turkstat.gov.tr/Start.do;jsessionid=BQyyQpcW1QLblPmpRnNTnlhC0fbyKQwx1k61hlPs1761Zp8lQbPV!-1063374806) [BQyyQpcW1QLblPmpRnNTnlhC0fbyKQwx1k61hlPs1761Zp8lQbPV!-106337](http://www.turkstat.gov.tr/Start.do;jsessionid=BQyyQpcW1QLblPmpRnNTnlhC0fbyKQwx1k61hlPs1761Zp8lQbPV!-1063374806) [4806>](http://www.turkstat.gov.tr/Start.do;jsessionid=BQyyQpcW1QLblPmpRnNTnlhC0fbyKQwx1k61hlPs1761Zp8lQbPV!-1063374806).
+- [Urbanik, T., 2000. Evacuation time estimates for nuclear power plants. Journal of](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0105) [Hazardous Materials 75, 165–180](http://refhub.elsevier.com/S0925-7535(13)00156-2/h0105).
+- via|Senezon, 2012. <<http://www.senozon.com/products/via>>, (accessed 10.07.12). White, W.L., Chanchareon, N., 2008. Use of Mesoscopic Simulation for Hurricane and other Emergency Evacuation Planning with Cube Avenue. [<http://](http://www.sfbayite.org/technicalprograms/2008_03-28/Cube-Avenue_SFBayITE_Modeling_2008_03-28.pdf) [www.sfbayite.org/technicalprograms/2008\\_03-28/Cube-Avenue\\_SFBayITE\\_](http://www.sfbayite.org/technicalprograms/2008_03-28/Cube-Avenue_SFBayITE_Modeling_2008_03-28.pdf) [Modeling\\_2008\\_03-28.pdf](http://www.sfbayite.org/technicalprograms/2008_03-28/Cube-Avenue_SFBayITE_Modeling_2008_03-28.pdf)>, (accessed 19.08.12).
+- WolframAlpha, 2012. [<http://www.wolframalpha.com>](http://www.wolframalpha.com), (accessed 02.08.12).

Zotero/001_artiklid/Road Safety Indicators/paganyspatiotemporalriskpredictionwildlife2022.md

@@ -0,0 +1,454 @@
+---
+category: literaturenote
+citekey: paganyspatiotemporalriskpredictionwildlife2022
+title: A spatiotemporal risk prediction of wildlife-vehicle collisions using machine learning for dynamic warnings
+authors: "Pagany, Raphaela"
+year: 2022
+date: 2022-00-00 2022
+doi: 10.1016/j.jsr.2022.09.001
+publication: Journal of Safety Research
+url: "https://doi.org/10.1016/j.jsr.2022.09.001"
+zotero_key: 3X2N5HHA
+zotero_storage: WFJTHIB6
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Pagany, Raphaela"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+## Journal of Safety Research
+
+journal homepage: www.el [sevier.com/locate/jsr](http://www.elsevier.com/locate/jsr)
+
+![](_page_0_Picture_5.jpeg)
+
+# A spatiotemporal risk prediction of wildlife-vehicle collisions using machine learning for dynamic warnings
+
+![](_page_0_Picture_7.jpeg)
+
+## Raphaela Pagany
+
+Institute for Applied Informatics, Deggendorf Institute of Technology, Freyung, Germany Interfaculty Department of Geoinformatics, University of Salzburg, Salzburg, Austria
+
+#### article info
+
+Article history: Received 12 September 2021 Received in revised form 13 May 2022 Accepted 1 September 2022 Available online 16 September 2022
+
+Keywords: Wildlife-vehicle collision Machine learning Spatiotemporal prediction Random forest Neural networks Support vector machine classifier
+
+#### abstract
+
+Introduction: The technology in the automotive industry is becoming increasingly safer in the age of automated driving, but the number of accidents is still high, especially in wildlife-vehicle collisions (WVCs). To better avoid these accidents, patterns involved in these accidents must be detected. Method: This paper presents a spatiotemporal risk prediction of WVCs, including various road and environmental characteristics. A process of data preparation using GIS automated by Python scripts was developed to enable a spatiotemporal link of diverse features for the subsequent predictive data analysis. Different machine learning (ML) approaches were applied- random forest (RF), feedforward neural networks (FNN), and support vector machine classifier (SVM) - including automated ML to predict the risk of WVCs. Therefore, a dataset of approximately 731,000 accidents reported to the police in Bavaria over a period of 10 years (2010–2019) was used. In addition, non-accidents were randomly generated in Python over time and space for the supervised ML processes. As the actual risk probability for WVCs and non-WVCs is not entirely known, the impact of different training ratios between accidents and nonaccidents was tested on the risk prediction quality (RPQ) (25%, 50%, 75%, 90% WVCs) of the doubleweighted sensitivity and single-weighted specificity rate. Results: The test yielded high mean values of RPQ as an indicator for a suitable WVC prediction. Both RF (86.6%) and FNN (86.7%) were identified as suitable choices for WVC risk prediction in terms of RPQ. The SVM yielded a lower prediction quality, even though acceptable results could be achieved within a shorter runtime. Practical Applications: Spatial transferability was verified since the algorithm was trained on the dataset of Bavaria (excluding Upper Bavaria) and successfully tested in Upper Bavaria. WVC forecasts were also proven through training with datasets from 2010-2017 and in prediction for 2018 and 2019.
+
+2022 National Safety Council and Elsevier Ltd. All rights reserved.
+
+## 1. Introduction
+
+Injuries and fatalities due to traffic accidents with wildlife have a great impact on the society and wildlife ecology. Vehicle damages cause significant financial costs, even when the actual numbers of wildlife-vehicle collisions (WVCs) are often unknown. German vehicle insurers registered nearly 300,000 accidents with wildlife in 2019 - more than ever before [\(GDV, 2020\)](#page-11-0). In total, WVCs cost German insurers around 885 million euros. In other countries, the WVC rates are also estimated to be significantly high, for instance, four million in Belgium ([Morelle, Lehaire, & Lejeune,](#page-11-0) [2013\)](#page-11-0), 1.5 million collisions with deer in the United States ([Sullivan, 2011\)](#page-12-0), and five million with amphibians and reptiles in Australia per year [\(Duellman, 2001\)](#page-11-0). But while the collisions are mostly counted in small regions, nationwide or international numbers are estimated to be several times higher than the reported collision numbers ([Gkritza, Baird, & Hans, 2010; Steiner, Leisch, &](#page-11-0) [Hackländer, 2014\)](#page-11-0).
+
+Predictive analysis seems to be a useful approach to analyze WVC risk and its influential conditions in a spatiotemporal manner aiming to transfer the risk prediction model globally, for superregional mitigation possibilities. Subsequently, the transferability of such a model would decimate the need for WVC data as a prerequisite for adequate measures in specific areas. By studying the environmental characteristics of the roadway, topography or weather, patterns can be identified to determine if there is a risk of an accident. By using these patterns, predictive analyses enlarge the knowledge about the accident risk, even if little is known about the accident or non-accident locations.
+
+This study applies three ML algorithms - random forest, feedforward neural networks, and support vector machine classifier - to achieve a spatiotemporal WVC risk prediction model and, hence,
+
+E-mail address: [raphaela.pagany@th-deg.de](mailto:raphaela.pagany@th-deg.de)
+
+a basis for greater road safety and wildlife conservation. By using GIS tooling automated with Python, heterogeneously structured data regarding the time and the environmental conditions can be brought to one key reference. In this feature engineering process, different spatiotemporal resolutions were harmonized to obtain a pre-processed dataset for further analyzing steps. Road, infrastructure, land use, weather, topographical and further geo-data were identified as potentially influential. A spatiotemporal resolution of one hour and approximately 25-metres road sections was chosen for a detailed WVC analysis. The predictive quality of the algorithms, and their spatial and temporal transferability were examined. For an improved prediction, automated ML (AutoML) was applied. This study addresses three research questions as follows: (1) Are ML methods suitable for an appropriate risk prediction WVCs? (2) How must the training data be structured to enable a high quality of WVC risk classification? (3) Can the approach be transferred temporally and spatially?
+
+The paper is organized as follows: After the literature review (Section 2), the input data for the prediction model and the data pre-processing are described, before the case study region is presented [\(Section 3\)](#page-2-0). [Section 4](#page-4-0) describes the analyzing structure including the selection of algorithms and parameters, and evaluation metrics. In [Section 5,](#page-6-0) the results are presented, while in [Sec](#page-8-0)[tion 6](#page-8-0), the results and advantages of the findings for ecology and traffic management are proposed. Finally, an outlook for future work is given in [Section 7.](#page-10-0)
+
+#### 2. Literature review
+
+Several studies have been employed to address the problem of low data availability since years, either by monitoring carcasses along the road networks [\(Grilo, Ferreira, & Revilla, 2015;](#page-11-0) [Lutterschmidt, Weidler, & Schalk, 2019\)](#page-11-0), by voluntary citizen science approaches ([Heigl et al., 2016; Keken, Sedoník, Kušta,](#page-11-0) [Andrášik, & Bíl, 2019\)](#page-11-0), or by obliging the driver to report the accident to the police or to the local hunter ([Snow et al., 2018](#page-11-0)). Especially in countries with dense road networks, national parks, or high wildlife populations, diverse research teams collected data to study WVCs and their environment ([Arevalo, Honda, Arce-](#page-10-0)[Arias, & Häger, 2017; Cherry et al., 2019; Pagany, 2020\)](#page-10-0).
+
+Over the last few decades, various studies constitute that the risk for WVCs is influenced by the drivers' and animals' behavior due to diverse environmental conditions. The temporal factor determines the daily or annual behavior and, thereby, correlates with land use, weather, or light conditions ([Garriga, Franch,](#page-11-0) [Santos, Montori, & Llorente, 2017; Kämmerle et al., 2017; Steiner](#page-11-0) [et al., 2014\)](#page-11-0). These dynamical patterns are completed by statistical patterns which together affect the accidents in a spatiotemporal manner. For instance, road conditions, such as the street width, a gentle topography, and the proximity to forest influence the WVC risk, according to a comprehensive review of WVC studies worldwide ([Pagany, 2020\)](#page-11-0). In contrast, traffic volume, the distance to urban areas, or road accompanying infrastructure are not clearly assignable influencing or non-influencing factors.
+
+However, most of the studies were undertaken in small areas or considering road sections of only a few kilometers ([Gunson,](#page-11-0) [Mountrakis, & Quackenbush, 2011; van der Ree, Jaeger, van der](#page-11-0) [Grift, & Clevenger, 2011](#page-11-0)), which means that these studies lack information required to mitigate WVCs on a higher level. In addition, the relationship between space and time in WVC is also only sparsely studied. However, this investigation would be essential for dynamically adapted measures in the event of a high WVC risk at a specific time at a specific location, as [Pagany \(2020\)](#page-11-0) discussed. The review also shows that most studies only analyze a few influencing parameters.
+
+Until now, only a few studies developed a prediction model for WVCs. For instance, [Santos, Mota-Ferreira, Aguiar, and Ascensão](#page-11-0) [\(2018\)](#page-11-0) developed a Bayesian hierarchical occupancy model that estimated WVC risk, especially in agricultural, open habitats, and within four-lane road sections. Another example is the study of [Visintin, van der Ree, and McCarthy \(2016\),](#page-12-0) which predicted WVCs with the main determinants being traffic volume, traffic speed, and species occurrence. Both studies explicitly named the advantage of predictive studies, as the analysis decimate the need for a broad data collection. Besides, classic statistics and geo-statistical analyses are chosen in previous studies. They applied hot spot analyses such as the kernel density distribution, as presented in [Bíl,](#page-10-0) [Andrášik, Svoboda, and Sedoník \(2016\),](#page-10-0) and calculated the density of accidents along certain road section lengths. As further statistical methods for WVC distribution, [Malo, Suárez, and Díez \(2004\)](#page-11-0) used a Poisson distribution, [Valero, Picos, and Álvarez \(2015\)](#page-12-0) applied a nearest-neighbor hierarchical clustering, and [Tanner,](#page-12-0) [Leroux, and Saunders \(2017\)](#page-12-0) used a generalized linear mixed model. Furthermore, [Liu, Nieuwenhuis, and McCullagh \(2018\)](#page-11-0) [and Seo, Thorne, Choi, Kwon, and Park \(2015\)](#page-11-0) applied regression analysis to determine the influence of environmental factors such as the roadside land use or seasonal effects.
+
+Studies about other types of traffic accidents used data mining techniques such as machine learning (ML) for identifying the accident risk. Datasets with georeferenced accident occurrences combined with environmental characteristics are often included in the risk analysis. [Yang, Chen, and Brown \(1999\)](#page-12-0) investigated the severity of road accident injuries in Alabama, United States. A backpropagation neural network was trained including variables such as light conditions and traffic speed to detect safer driving patterns, which may reduce fatalities and injuries by up to 40%. [Komol et al. \(2021\)](#page-11-0) chose ML-based classification approaches for modeling injury severity of the vulnerable road users: pedestrian, bicyclist, and motorcyclist using k-nearest neighbor, support vector machine, and random forest. They found that motorcyclists have an especially high crash severity. Another study about injury severity is [Chen, Song, and Ma \(2019\)](#page-11-0). Using a random parameters bivariate ordered probit model, they showed correlations between two drivers' injuries such as driver age, gender, vehicle, airbag or seat belt use or traffic flow. [Abdelwahab and Abdel-Aty \(2001\)](#page-10-0) applied multilayer perceptron (MLP) and fuzzy adaptive resonance theory (ART) neural networks and included parameters such as the vehicle types and the sex of the driver to predict the severity of the driver's injury in Florida, United States.
+
+In addition, [Deublein, Schubert, Bryan, and de Soto Boria \(2015\)](#page-11-0) developed an accident risk model using Bayesian Probabilistic Networks including road-related variables, such as the number of lanes, curvature, and traffic characteristics. With a correct prediction of 86.53% of the road segments with a tolerance of 25%, the model could be applied to predict spatial ''black spot" locations regarding the accidental injuries on the Swiss highway network. [Beshah and Hill \(2010\)](#page-10-0) used a rule mining approach to classify road traffic accidents using adaptive regression trees for a case study in Ethiopia. The study included diverse data about the drivers' age and driving experience, the vehicle age, the road surface, as well as light and weather conditions. Besides, recent approaches can be mentioned that are characterized by similar pre-processing steps of GIS data like for accident risk prediction (e.g., the prediction of connected vehicle movement at intersections). Trajectory movement labelling for predicting driver movements and for warning at intersections ensure advanced transportation safety using ML-based approaches to implement these technologies ([Komol et al., 2021; Shrivastava, Verma, & Jain, 2021\)](#page-11-0).
+
+Other studies compared ML techniques to test their quality or predictive accuracy. [Chang and Chen \(2005\),](#page-11-0) for instance, compared a classification and regression tree (CART) with a negative <span id="page-2-0"></span>binomial regression model to analyze the relationship between traffic accidents and highway geometric variables, traffic characteristics, and environmental factors in Taiwan. They found that the CART model is an appropriate method for analyzing the frequency of freeway accidents. Precipitation variables and daily traffic volume are the key determinants for the accident risk. [Chong,](#page-11-0) [Abraham, and Paprzycki \(2005\)](#page-11-0) compared the performance of neural networks, support vector machines, decision trees, and a concurrent hybrid model involving decision trees and neural networks. The severity of an injury resulting from a traffic accident was predicted. The hybrid approach of decision tree and neural networks outperformed the individual approaches.
+
+The applications and the comparisons of diverse ML techniques to model traffic accidents show that these approaches are suitable for accident risk prediction. Several studies are undertaken to understand the characteristics of driver behavior, road conditions, environmental and weather conditions that are related to the accident probability. However, for a risk prediction, it is not decisive to know the impact of an individual factor, as analyzed by classical statistics, but to develop a decision model considering the environmental factors and the learning characteristics of ML techniques. Thus, ML may also enlarge the knowledge about the accidents with wildlife, as already applied for other road accidents types. A supervised ML classification seems to enable this WVC risk prediction.
+
+#### 3. Material and data preparation
+
+#### 3.1. Data description
+
+For this study, a dataset of 730,867 WVCs that were officially registered by the Police between 2010 and 2019 in Bavaria, Germany was used ([StMI & Bavarian State Police, 2020](#page-12-0)). Each of these registered WVCs from the last 10 years contains a georeference and a timestamp. Additional features were used that are identified in previous studies as influential characteristics for road traffic accidents with wildlife ([Pagany, 2020](#page-11-0)) (Table 1): road-related features, such as road class, road width, infrastructure elements, (e.g., middle barriers, road barriers, and additional lanes), and topographic features (e.g., ditches, or slopes aside the road network). In addition, weather data were used, such as air temperature, precipitation, and cloud coverage, as well as land use data in the prediction model. The land use data are classified in 12 classes as follows: artificial land, cropland seasonal, cropland perennial, forest broadleaved, forest coniferous, forest mixed, shrubland, grassland, bare land, water, wetland, and snow and ice areas. Based on the accident data, time-related features were derived, namely the day of the year, the weekday, the hour of a day, and the solar altitude.
+
+The road information was obtained from the database BAYSIS (Bavarian road infrastructure system) provided by the Bavarian Ministry of the Interior, for Sport and Integration [\(StMI, 2017\)](#page-11-0) and the Official Topographical-Cartographic Information System Data - Digital Landscape Models (ATKIS-DLM data) by the Surveying Authorities of the Federal States of Germany [\(Surveying](#page-12-0) [Authorities of the Federal States of Germany, 2015\)](#page-12-0). Weather data from the German weather service [\(DWD, 2019](#page-11-0)) were used. The DWD obtains its data from COSMO-REA6 - a regional reanalysis dataset (product of a numerical data model that assimilates terrestrial and remotely sensed weather data) that covers the EUROCOR-DEX area with a spatial resolution of 6 6 km and uses the ERA-INTERIM global reanalysis dataset for boundary conditions. For the land use classification, PANGAEA land use data with a spatial resolution of 30 30 m based on remote sensing Landsat and LUCAS data [\(Pflugmacher, Rabe, Peters, & Hostert, 2018\)](#page-11-0) from the year 2018 were used in the prediction model.
+
+### 3.2. Data pre-processing
+
+The different datasets required data preparation and cleaning for the later data mining process. Nearly all data required a trans-
+
+Table 1 Data and derived features as input for the ML predictive analysis (not normalized values).
+
+|                                                        | Data Source                                                           | Feature/Label       | Unit     | Range          |
+|--------------------------------------------------------|-----------------------------------------------------------------------|---------------------|----------|----------------|
+| Wildlife-vehicle collisions (WVCs)<br>(spatiotemporal) | StMI & Bavarian State Police, 2020                                    | WVC label           | -        | 1              |
+| Non-WVCs<br>(spatiotemporal)                           | Randomly calculated                                                   |                     | -        | 0              |
+| Road-related                                           | ATKIS (mainly)                                                        | Road class          |          | 1–5            |
+| (spatial)                                              | (Surveying Authorities of the Federal States of Germany (2015), 2015) | Road width          | metre    | 2 – 22         |
+|                                                        | BAYSIS data (StMI, 2017)                                              | Middle barrier      | -        | 0–1            |
+|                                                        |                                                                       | Road barriers       | -        | 0–1            |
+|                                                        |                                                                       | Additional lanes    | -        | 0–1            |
+|                                                        |                                                                       | Ditch               | -        | 0–1            |
+|                                                        |                                                                       | Slope               | -        | 0–1            |
+| Weather                                                | Weather data (DWD, 2019)                                              | Ambient temperature | C        | 22.20 to 39.07 |
+| (spatiotemporal)                                       |                                                                       | Precipitation       | litre/m2 | 0–72.95        |
+|                                                        |                                                                       | Cloud coverage      | %        | 0–100          |
+| Land use                                               | PANGAEA land use data (Pflugmacher et al., 2018)                      | Artificial land     | %        | 0–100          |
+| (spatial)                                              |                                                                       | Cropland, seasonal  | %        | 0–100          |
+|                                                        |                                                                       | Cropland, perennial | %        | 0–100          |
+|                                                        |                                                                       | Forest, broadleaved | %        | 0–100          |
+|                                                        |                                                                       | Forest, coniferous  | %        | 0–100          |
+|                                                        |                                                                       | Forest, mixed       | %        | 0–100          |
+|                                                        |                                                                       | Shrubland           | %        | 0–100          |
+|                                                        |                                                                       | Grassland           | %        | 0–100          |
+|                                                        |                                                                       | Bare land           | %        | 0–100          |
+|                                                        |                                                                       | Water               | %        | 0–100          |
+|                                                        |                                                                       | Wetland             | %        | 0–100          |
+|                                                        |                                                                       | Snow and ice        | %        | -              |
+| Time-related                                           | WVC data (StMI & Bavarian State Police, 2020)                         | Day of year         | -        | 1–366          |
+| (temporal)                                             |                                                                       | Weekday             | -        | 1–7            |
+|                                                        |                                                                       | Hour of the day     | -        | 0–23           |
+|                                                        |                                                                       | Solar altitude      |          | 64.50–64.50    |
+
+formation of geographical projections and data pre-processing to obtain a homogenous key component, with which the features WVCs, and non-WVCs (as label) could be correlated. GIS methods automated by Python scripts were used, namely GDAL and OGR from the OSGeo library ([OSGeo, 2019\)](#page-11-0) for interacting with geodata file formats, and shapely ([Toblerity, 2020](#page-12-0)) for geometric processing. The road shapefile ver01\_l of the ATKIS-DLM data were divided into point vectors with a distance of 25-metres along the road network to define the set of locations to be examined (Fig. 1). These locations were used as a basis layer to extract the feature values from several other datasets.
+
+From the ATKIS-DLM data, the information about the road class and road width was obtained. Missing values of the road classes were derived from the road names, for example, the federal road class (German: B = Bundesstraße) from the street name ''B 12," while the missing road width were derived from the road classes using standardized values for road construction in Germany (11.5 m for highways, 7.5 m for federal roads, 6.5 m for state roads, and 5.5 m for municipal road ([FGSV, 1996\)](#page-11-0). To avoid duplicate information, road axes of highways were deleted as the lane axes are also registered for highways that run parallel to the road axis. The road information was augmented by the street data of BAYSIS. According to this road database, the road network was classified in sections with middle barriers, additional lanes, ditches, and slopes ([Table 1\)](#page-2-0). Details in the attribute column legend of the BAYSIS shapefile Straßenbestand.Deckschichtalter were used for the classification (e.g., hard shoulders or bus bays were classified as roads with an additional lane, while protective stripes with crash barriers were classified as sections with barriers). To obtain all infrastructure elements at both sides of the road, vertical lines at the 25 metre distance points from the ATKIS data were constructed. Along these vertical lines, all elements of the BAYSIS shapefile were identified, and the information were written into the 25-metre point layer as key elements for the later data mining.
+
+The land use classes of the PANGAEA data were identified based on the 25-metre points. The land use types were included on both sides of the road within three different radii to test the impact of the environmental land use of an accident within near, medium, and long distances (Fig. 1, semicircles in the right image). The relative area share of each of the 12 different land use classes was calculated separately for both sides of the street within semicircles of 100, 500, and 200 meter radii. Thus, six different buffers were used for each land use type to model the impact of the land use structure at each potential accident location for ML analysis.
+
+Furthermore, the CSV dataset of WVCs was stored as a shapefile and joined to the nearest of the 25-metre points to create a spatial reference. For the temporal reference, the timestamp of the WVCs were discretised to an hourly resolution. In total, 730,867 WVCs were registered but only 567,233 WVCs (78%) were included in the predictive analysis, since the remaining accidents do not contain any geographic references. A set of non-WVC locations was generated in addition to the WVC registrations that are required for the ML approach. A number of 657 billion spatiotemporal points for potential accident occurrences could be obtained by multiplying the 7.5 million 25-meter road points and 87,600 hours of the 10 year study period, but only 12 million non-WVCs are necessary for the latter analysis (see chapter 4). Hence, spatiotemporal points were randomly generated in an hourly resolution based on the 25 meter distance points of the ATKIS road network using the random library in Python based on the Mersenne Twister algorithm, while spatiotemporal locations with the locations and timestamps of the WVCs were excluded to obtain non-WVCs for supervised ML training. For each of the accidents and non-accidents, the timestamp was used to obtain the features weekday, day of the year, and hour of the day. In addition, the solar altitude was calculated for each spatiotemporal point using the solar position algorithm [\(Holmgren,](#page-11-0) [Hansen, & Mikofski, 2018\)](#page-11-0). Finally, the features air temperature, precipitation, and cloud coverage were extracted from the weather dataset by determining the fitting tile of the weather data at the specific point and at the specific time.
+
+A SQLite database was used to combine the different generated datasets in one dataset with a label (WVC/non-WVC) as input for the ML process. After the spatiotemporal points of non-WVCs and the time-based features were generated, they were stored directly in tables in the database. The same was applied for the weather data, which was selected according to the spatiotemporal locations. The road infrastructure, land use, andWVC data were stored in shapefiles as an intermediate step and, later, imported into the database as separate tables for each shapefile. Finally, each data row has a reference to one of the 25-m points from the ATKIS dataset using WGS 84 UTM system (World Geographical System 84, Universal Transverse Mercator) as geographical reference, and an hourly resolution (the hour of the day and day of the year) as temporal dimension, which are included in all tables as key elements. The normalization of the data was done as part of the SQL statement that joined the several tables to one source table for ML processing. In this table, each spatiotemporal point is characterized by 88 features, whereby-six classes were deleted due to the nonexistence of snow and ice areas (at both roadsides within three radii).
+
+#### 3.3. Case study area
+
+The data were applied for WVC analysis in Bavaria. The Free State of Bavaria in the southeast of Germany covers 7,548 km2
+
+![](_page_3_Figure_10.jpeg)
+
+![](_page_3_Figure_11.jpeg)
+
+Fig. 1. Data preparation using spatiotemporal points of 25-metres road sections (left image), including, e.g., land use environment within 100-, 500-, and 2000-metres radii (right image).
+
+<span id="page-4-0"></span>and seven administrative districts: Lower Bavaria, Upper Bavaria, Lower Franconia, Middle Franconia, Upper Franconia, Upper Palatinate, and Swabia (Fig. 2; [Table 2\)](#page-5-0).
+
+Bavaria borders the Czech Republic in the east and Austria in the south-east and south. While a significant share of the around 13 million inhabitants in Bavaria live in a few metropolitan areas (such as Munich or Nuremberg-Fuerth-Erlangen), large parts of Bavaria, such as the Upper Palatinate and Lower Bavaria, are sparsely populated. The northern and eastern parts with the densely forested Bavarian Forest belong to the German low mountain range, while the mostly flat glacial foothills extend from the Alps to the Danube basin. Almost 35% of the Bavarian area is forested. Because of the altitude above sea level and the distance from the sea, Bavaria's climate is rougher and more continental than in most other German federal states. WVCs mainly occur in rural areas, for example, in the county Rottal-Inn in the southeast of Lower Bavaria ([Fig. 3](#page-5-0)). There is a low rate of accidents in the Alpine and forested regions, such as the national parks Bavarian Forest and Berchtesgadener Land, as the road network is not as dense as in other areas. Despite the dense road network, there is a low density of WVCs in urban areas. The WVC density is the highest on state and county roads in relation to the road length ([Table 3](#page-5-0)).
+
+#### 4. Analysis
+
+#### 4.1. Selection of algorithms and parameters
+
+The prediction analysis determines the collision risk in order to warn drivers of WVCs only when necessary. The risk for WVCs was calculated using ML algorithms in Python. The following algorithms were used for the analysis: random forest [\(Breiman,](#page-11-0) [2001](#page-11-0)), feedforward neural networks [\(McCulloch & Pitts, 1943\)](#page-11-0), and support vector machine classifier ([Vapnik, 1963](#page-12-0)). Since there are no previous studies for risk prediction of WVCs with ML available, several variants of ML algorithms were tested. The algorithms are chosen to test the prediction capability - in contrast to the commonly used statistical hotspot and impact analysis in WVC research ([Pagany, 2020](#page-11-0)). Research on traffic accidents in general already applies the chosen ML algorithms, while there is still a research gap for WVC prediction using such methods. Especially the temporal forecast and the WVC risk prediction in areas without historical datasets about WVCs are an advantage of the ML techniques. In addition, these methods enable a steady training adjustment of the algorithms due to changed environmental conditions. Therewith, dynamic real-time warnings to the driver via a mobile application [\(wuidi, 2020](#page-12-0)) is applicable in order to avoid accidents. Hence, the paper focuses on the generation of ML models resulting in high prediction rates through testing.
+
+The impact of different training ratios between accidents and non-accidents on the prediction quality was tested, as the actual risk probability of WVCs and non-WVCs is not entirely known. Only the locations and times with a registered WVC are an indicator for spatiotemporal risk zones, while the remaining road sections and times are assumed to be low risk. While the probability is high that only one, a vehicle, or an animal, or neither of them will cross a certain road at the same time, there is also a possibility that both a vehicle and an animal cross at the same point at the same time, but that the individual reaction of the driver or the animal is such that a collision can be avoided. As the WVC data do not allow any information about these near misses, but the risk is higher than the actual WVC data suggest, higher, and also extreme ratios of WVCs were used for ML training (25%, 50%, 75%, 90% WVCs) [\(Fig. 4](#page-6-0)). Afterwards, the models were tested with a uniform ratio of 5% WVCs and 95% non-WVCs to obtain comparable results. Hence, the ML training were conducted with the Bavarian WVC data and different sample sizes of non-WVCs for the predictive risk analysis.
+
+To test the temporal and spatial transferability, WVC data were split in three ways, and the training and test data were enriched with non-WVCs to obtain the abovementioned ratios for training
+
+![](_page_4_Figure_10.jpeg)
+
+Fig. 2. Case study area of Bavaria (map on the left side) and districts (map on the right side).
+
+<span id="page-5-0"></span>Table 2 Population, area, and road length per districts based on ATKIS data and [Destatis, \(2020\).](#page-11-0)
+
+| District         | Population |         | Area   |         |             | Road length |  |
+|------------------|------------|---------|--------|---------|-------------|-------------|--|
+|                  |            |         | km2    | %       | km          | %           |  |
+| Lower Bavaria    | 1,192,641  | 9.48%   | 10,330 | 14.6%   | 44,436,083  | 25.96%      |  |
+| Upper Bavaria    | 4,418,828  | 35.12%  | 17,530 | 24.8%   | 27,839,851  | 16.26%      |  |
+| Lower Franconia  | 1,315,882  | 10.46%  | 8,531  | 12.1%   | 21,935,317  | 12.82%      |  |
+| Middle Franconia | 1,717,670  | 13.65%  | 7,245  | 10.3%   | 17,027,147  | 9.95%       |  |
+| Upper Franconia  | 1,067,988  | 8.49%   | 7,231  | 10.2%   | 18,851,596  | 11.01%      |  |
+| Upper Palatinate | 1,081,800  | 8.60%   | 9,691  | 13.7%   | 17,124,381  | 10.00%      |  |
+| Swabia           | 1,788,729  | 14.21%  | 9,992  | 14.2%   | 23,950,389  | 13.99%      |  |
+| Total            | 12,583,538 | 100.00% | 45,476 | 100.00% | 171,164,764 | 100.00%     |  |
+
+![](_page_5_Figure_4.jpeg)
+
+Fig. 3. Heatmap of WVCs in Bavaria.
+
+Table 3 Distribution of WVCs by road class.
+
+| Road class      | WVC share (%) |
+|-----------------|---------------|
+| Municipal roads | 17.09         |
+| Country roads   | 31.43         |
+| State roads     | 33.90         |
+| Federal roads   | 14.88         |
+| Highways        | 2.70          |
+
+and testing. The random split in a ratio of 80% training data and 20% test data were used as a basis for comparison. Furthermore, the predictive analyses were carried out with a training sample containing WVCs and non-WCVs from between 2010 and 2017, while data from 2018 and 2019 were used for testing (timebased split). Also, the algorithm that was trained on the dataset from Bavaria except Upper Bavaria, was then tested in Upper Bavaria (geographical split) [\(Fig. 4\)](#page-6-0).
+
+<span id="page-6-0"></span>![](_page_6_Figure_2.jpeg)
+
+Fig. 4. Analyzing structure for the WVC risk prediction.
+
+From the class of neural networks, the feedforward neural networks performed best in a pre-test, where convolutional, recurrent, and feedforward neural networks were analyzed. Thus, convolutional networks and recurrent networks, which are typically used for image classification and temporal processes, respectively, can be excluded. With support vector machine classifier, a very fundamental algorithm was added. In addition, random forest was chosen as third option to have an algorithm, which yielded good results in previous studies and enables impurity calculation for the input features.
+
+Furthermore, the hyperparameter tuning was done in an automated way using automated ML (AutoML) systems. The StructuredDataClassifier from AutoKeras was used, which is based on the TensorFlow library for the feedforward neural networks ([Jin,](#page-11-0) [Song, & Hu, 2019](#page-11-0)). Moreover, auto-sklearn, which is an AutoML toolkit for the scikit-learn library, was applied for the random forest and the support vector machine classifier ([Feurer, Klein,](#page-11-0) [Eggensperger, Springenberg, Blum, & Hutter, 2015\)](#page-11-0). Even though the hyperparameters are chosen by AutoML systems, some specifications needed to be delimited.
+
+In AutoKeras, it is necessary to specify the count of trials and the epochs for the model training. Ten trails and 10 epochs were chosen, as tests with 5 trails were insufficient for adequate results and the training with 15 trails did not improve the results significantly but took an inappropriately long runtime. To determine the number of epochs, several trainings were run with 20 epochs and examined at which number of epochs the accuracy and loss functions did not improve significantly. Using Auto-sklearn, the run time of the entire process and of a single model training was specified, which indirectly defines the number of different hyperparameter settings. In addition, a number of 10 cores were continually used for all model generations. For the further parameters, default values were used.
+
+## 4.2. Evaluation metrics
+
+To evaluate the results, confusion matrices (Table 4) and further indicators were generated to show the numbers of correct and incorrect matches of the classification. With the accuracy rate, all correctly classified matches are related to the overall number of matches of the confusion matrix (Eq. (1)). The sensitivity, also called recall, is an indicator that shows how often WVCs are correctly classified in relation to the real number of WVCs (Eq. (2)). Specificity shows how often non-WVCs are correctly classified in relation to the real number of non-WVCs (Eq. (3)). While the false negative (FN) rate is the inverse value of the sensitivity rate, the false positive (FP) rate is the inverse of the specificity rate.
+
+Table 4 Confusion Matrix.
+
+|       | WVC (1) predicted                                                             | Non-WVC (0) predicted                                                                  |  |  |
+|-------|-------------------------------------------------------------------------------|----------------------------------------------------------------------------------------|--|--|
+| WVC   | actual                                                                        | Actual/true positive (TP),<br>meaning WVC occurred, and<br>is predicted to be positive |  |  |
+| False | negative (FN), meaning WVC<br>occurred, and is predicted to be<br>negative    |                                                                                        |  |  |
+| Non-  | WVC actual                                                                    | False positive (FP), meaning<br>no WVC occurred, but is<br>predicted to be positive    |  |  |
+| True  | negative (TN), meaning no WVC<br>occurred, and is predicted to be<br>negative |                                                                                        |  |  |
+
+accuracy rate = 
+$$\frac{TP + TN}{TP + TN + FP + FN}$$
+ (1)
+
+$$sensitivity = \frac{TP}{TP + FN} \tag{2}$$
+
+$$specificity = \frac{TN}{TN + FP} \tag{3}$$
+
+Especially high specificity and sensitivity rates are useful indicators for evaluating the quality of a WVC risk prediction model concerning its application as warning service. Therefore, an own measure, the risk prediction quality, is defined as the follows: the mean value of the double-weighted sensitivity and singleweighted specificity rate (Eq. (4)).
+
+$$\textit{risk predictio nquality}(\textit{RPQ}) = \frac{2*\textit{sensitivity} + \textit{specificity}}{3} \tag{4}$$
+
+With the double-weighted sensitivity, the emphasis is laid on the potential to correctly classified WVC risk in contrast to false classified high WVC risk. This sensitivity value is the basis for a warning in the case of WVC risk and only a few non-warnings even if the risk would be high. Too many warnings due to low specificity rates would annoy the drivers without a reason for an accident risk.
+
+## 5. Results
+
+With accuracy rates between 87.36% and 92.00%, all three ML algorithms assign most observations correctly as WVCs or non-WVCs (bold values in [Table 5\)](#page-7-0), using a ratio of 25% WVCs and 75% non-WVCs in the training dataset. With RPQ values of 86.7% and 86.6%, neural networks and the random forest, respectively, performed slightly better than the support vector machine classifier with its highest value of 84.6% ([Fig. 5](#page-7-0), black line) ([Table 6;](#page-8-0) bold values > 75th quantile). The training ratio of 25% WVCs might achieve the best accuracy rates, but it is not the best choice according to the risk prediction quality measure (see [Section 4\)](#page-4-0). While sensitivity rates of under 70% are calculated, more than 30% of the WVCs are predicted as no risk cases, inversely. Using the measure of RPQ, the ratio of 75% WVCs and 25% non-WVCs has been proven as the best training ratio. The training and test run of the different case studies show that it is possible to forecast WVCs in a temporal and in a spatial manner (bold values > 75th quantile; modes: rnd = random, geo = geographical, and time = time-based; Fig. 4). Hence, the applicability of predictive analysis for WVCs is shown.
+
+Based on the random forest, the importance of the individual factors is calculated. The results also show that temporal, environmental and roadside conditions affect the WVC risk ([Fig. 6](#page-9-0)). Espe-
+
+<span id="page-7-0"></span>Table 5 Confusion matrix of ML results due to WVC ratio, algorithm (neural network = NN, support vector machine classifier = SVM, random forest = RF), and mode (geographical split = geo; randomly split = rnd; time-based split = time).
+
+|           |           |      | WVC                                  |                              | non WVC                              |                              |         |
+|-----------|-----------|------|--------------------------------------|------------------------------|--------------------------------------|------------------------------|---------|
+| WVC ratio | Algorithm | Mode | pred.<br>Non-WVC<br>= No Warn.<br>FN | pred. WVC<br>= Warning<br>TP | pred.<br>Non-WVC<br>= No Warn.<br>TN | pred. WVC<br>= Warning<br>FP | Accur % |
+| 25%       | NN        | Geo  | 45,027                               | 88,600                       | 2,358,856                            | 180,057                      | 91.58   |
+|           |           | Rnd  | 34,067                               | 79,379                       | 1,977,663                            | 177,811                      | 90.66   |
+|           |           | Time | 39,280                               | 84,946                       | 1,902,819                            | 161,669                      | 90.82   |
+|           | SVC       | Geo  | 47,420                               | 86,207                       | 2,294,631                            | 244,282                      | 89.09   |
+|           |           | Rnd  | 39,785                               | 73,661                       | 1,919,229                            | 236,245                      | 87.83   |
+|           |           | Time | 42,538                               | 81,688                       | 1,830,269                            | 234,219                      | 87.36   |
+|           | RF        | Geo  | 61,119                               | 72,508                       | 2,386,359                            | 152,554                      | 92.00   |
+|           |           | Rnd  | 39,622                               | 73,824                       | 1,953,723                            | 201,751                      | 89.36   |
+|           |           | Time | 41,418                               | 82,808                       | 1,859,100                            | 205,388                      | 88.72   |
+| 50%       | NN        | Geo  | 19,563                               | 114,064                      | 2,128,044                            | 410,869                      | 83.89   |
+|           |           | Rnd  | 14,355                               | 99,091                       | 1,743,486                            | 411,988                      | 81.21   |
+|           |           | Time | 16,158                               | 108,068                      | 1,691,907                            | 372,581                      | 82.24   |
+|           | SVC       | Geo  | 21,475                               | 112,152                      | 2,047,878                            | 491,035                      | 80.82   |
+|           |           | Rnd  | 16,102                               | 97,344                       | 1,666,100                            | 489,374                      | 77.72   |
+|           |           | Time | 18,361                               | 105,865                      | 1,603,721                            | 460,767                      | 78.11   |
+|           | RF        | Geo  | 19,356                               | 114,271                      | 2,087,965                            | 450,948                      | 82.40   |
+|           |           | Rnd  | 12,040                               | 101,406                      | 1,638,286                            | 517,188                      | 76.67   |
+|           |           | Time | 14,051                               | 110,175                      | 1,622,192                            | 442,296                      | 79.15   |
+| 75%       | NN        | Geo  | 7489                                 | 126,138                      | 1,812,158                            | 726,755                      | 72.53   |
+|           |           | Rnd  | 4867                                 | 108,579                      | 1,434,604                            | 720,870                      | 68.01   |
+|           |           | Time | 5503                                 | 118,723                      | 1,385,820                            | 678,668                      | 68.74   |
+|           | SVC       | Geo  | 7867                                 | 125,760                      | 1,666,913                            | 872,000                      | 67.08   |
+|           |           |      |                                      |                              |                                      |                              |         |
+|           |           | Rnd  | 5766                                 | 107,680                      | 1,332,631                            | 822,843                      | 63.48   |
+|           |           | Time | 6116                                 | 118,110                      | 1,283,013                            | 781,475                      | 64.02   |
+|           | RF        | Geo  | 8546                                 | 125,081                      | 1,839,873                            | 699,040                      | 73.52   |
+|           |           | Rnd  | 4821                                 | 108,625                      | 1,445,009                            | 710,465                      | 68.47   |
+| 90%       |           | Time | 5122                                 | 119,104                      | 1,388,963                            | 675,525                      | 68.90   |
+|           | NN        | Geo  | 1931                                 | 131,696                      | 1,305,375                            | 1,233,538                    | 53.77   |
+|           |           | Rnd  | 2099                                 | 111,347                      | 1,173,934                            | 981,540                      | 56.65   |
+|           |           | Time | 1945                                 | 122,281                      | 1,098,623                            | 965,865                      | 55.78   |
+|           | SVC       | Geo  | 1506                                 | 132,121                      | 1,055,130                            | 1,483,783                    | 44.42   |
+|           |           | Rnd  | 1231                                 | 112,215                      | 847,635                              | 1,307,839                    | 42.30   |
+|           |           | Time | 1129                                 | 123,097                      | 793,250                              | 1,271,238                    | 41.87   |
+|           | RF        | Geo  | 2251                                 | 131,376                      | 1,348,305                            | 1,190,608                    | 55.37   |
+|           |           | Rnd  | 1753                                 | 111,693                      | 1,091,939                            | 1,063,535                    | 53.05   |
+|           |           | Time | 1763                                 | 122,463                      | 1,053,540                            | 1,010,948                    | 53.73   |
+
+![](_page_7_Figure_4.jpeg)
+
+Fig. 5. Sensitivity and specificity rates and their mean values as indicator for WVC risk prediction.
+
+<span id="page-8-0"></span>Table 6 Metrics of WVC risk prediction due to WVC ratio, algorithm (neural network = NN, support vector machine classifier = SVM, random forest = RF), and mode (geographical split = geo; randomly split = rnd; time-based split = time).
+
+| WVC ratio | Algo-rithm | Mode | WVC         |                 | Non-WVC     |            | sensitivity    | FN       | sensitiv:            | FN  2         |
+|-----------|------------|------|-------------|-----------------|-------------|------------|----------------|----------|----------------------|---------------|
+|           |            |      | Sensitivity | FN<br>rate<br>% | Specificity | FP<br>rate | þspecificity 2 | þFP<br>2 | 2 þ specific: 3<br>% | þFP<br>3<br>% |
+|           |            |      | %           |                 | %           | %          | %              | %        |                      |               |
+| 25%       | NN         | geo  | 66.30       | 33.70           | 92.91       | 7.09       | 79.61          | 20.39    | 75.17                | 24.83         |
+|           |            | rnd  | 69.97       | 30.03           | 91.75       | 8.25       | 80.86          | 19.14    | 77.23                | 22.77         |
+|           |            | time | 68.38       | 31.62           | 92.17       | 7.83       | 80.27          | 19.73    | 76.31                | 23.69         |
+|           | SVC        | geo  | 64.51       | 35.49           | 90.38       | 9.62       | 77.45          | 22.55    | 73.13                | 26.87         |
+|           |            | rnd  | 64.93       | 35.07           | 89.04       | 10.96      | 76.99          | 23.01    | 72.97                | 27.03         |
+|           |            | time | 65.76       | 34.24           | 88.65       | 11.35      | 77.21          | 22.79    | 73.39                | 26.61         |
+|           | RF         | geo  | 54.26       | 45.74           | 93.99       | 6.01       | 74.13          | 25.87    | 67.50                | 32.50         |
+|           |            | rnd  | 65.07       | 34.93           | 90.64       | 9.36       | 77.86          | 22.14    | 73.60                | 26.40         |
+|           |            | time | 66.66       | 33.34           | 90.05       | 9.95       | 78.36          | 21.64    | 74.46                | 25.54         |
+| 50%       | NN         | geo  | 85.36       | 14.64           | 83.82       | 16.18      | 84.59          | 15.41    | 84.85                | 15.15         |
+|           |            | rnd  | 87.35       | 12.65           | 80.89       | 19.11      | 84.12          | 15.88    | 85.19                | 14.81         |
+|           |            | time | 86.99       | 13.01           | 81.95       | 18.05      | 84.47          | 15.53    | 85.31                | 14.69         |
+|           | SVC        | geo  | 83.93       | 16.07           | 80.66       | 19.34      | 82.29          | 17.71    | 82.84                | 17.16         |
+|           |            | rnd  | 85.81       | 14.19           | 77.30       | 22.70      | 81.55          | 18.45    | 82.97                | 17.03         |
+|           |            | time | 85.22       | 14.78           | 77.68       | 22.32      | 81.45          | 18.55    | 82.71                | 17.29         |
+|           | RF         | geo  | 85.51       | 14.49           | 82.24       | 17.76      | 83.88          | 16.12    | 84.42                | 15.58         |
+|           |            | rnd  | 89.39       | 10.61           | 76.01       | 23.99      | 82.70          | 17.30    | 84.93                | 15.07         |
+|           |            | time | 88.69       | 11.31           | 78.58       | 21.42      | 83.63          | 16.37    | 85.32                | 14.68         |
+| 75%       | NN         | geo  | 94.40       | 5.60            | 71.38       | 28.62      | 82.89          | 17.11    | 86.72                | 13.28         |
+|           |            | rnd  | 95.71       | 4.29            | 66.56       | 33.44      | 81.13          | 18.87    | 85.99                | 14.01         |
+|           |            | time | 95.57       | 4.43            | 67.13       | 32.87      | 81.35          | 18.65    | 86.09                | 13.91         |
+|           | SVC        | geo  | 94.11       | 5.89            | 65.65       | 34.35      | 79.88          | 20.12    | 84.63                | 15.37         |
+|           |            | rnd  | 94.92       | 5.08            | 61.83       | 38.17      | 78.37          | 21.63    | 83.89                | 16.11         |
+|           |            | time | 95.08       | 4.92            | 62.15       | 37.85      | 78.61          | 21.39    | 84.10                | 15.90         |
+|           | RF         | geo  | 93.60       | 6.40            | 72.47       | 27.53      | 83.04          | 16.96    | 86.56                | 13.44         |
+|           |            | rnd  | 95.75       | 4.25            | 67.04       | 32.96      | 81.39          | 18.61    | 86.18                | 13.82         |
+|           |            | time | 95.88       | 4.12            | 67.28       | 32.72      | 81.58          | 18.42    | 86.34                | 13.66         |
+| 90%       | NN         | geo  | 98.55       | 1.45            | 51.41       | 48.59      | 74.98          | 25.02    | 82.84                | 17.16         |
+|           |            | rnd  | 98.15       | 1.85            | 54.46       | 45.54      | 76.31          | 23.69    | 83.59                | 16.41         |
+|           |            | time | 98.43       | 1.57            | 53.22       | 46.78      | 75.82          | 24.18    | 83.36                | 16.64         |
+|           | SVC        | geo  | 98.87       | 1.13            | 41.56       | 58.44      | 70.22          | 29.78    | 79.77                | 20.23         |
+|           |            | rnd  | 98.91       | 1.09            | 39.32       | 60.68      | 69.12          | 30.88    | 79.05                | 20.95         |
+|           |            | time | 99.09       | 0.91            | 38.42       | 61.58      | 68.76          | 31.24    | 78.87                | 21.13         |
+|           | RF         | geo  | 98.32       | 1.68            | 53.11       | 46.89      | 75.71          | 24.29    | 83.25                | 16.75         |
+|           |            | rnd  | 98.45       | 1.55            | 50.66       | 49.34      | 74.56          | 25.44    | 82.52                | 17.48         |
+|           |            | time | 98.58       | 1.42            | 51.03       | 48.97      | 74.81          | 25.19    | 82.73                | 17.27         |
+
+cially the road-related factors road width (19%) and road class (14%) are determinants for the WVC prediction. Also, artificial land in the near and middle circle around the potential WVC locations (6% and 4% for each side s1 and s2; see [Fig. 6](#page-9-0)), and time-based factors, such as sun angle (6%), hour (3%), and day of the year (1%), influence the ML prediction. Therein, the importance of the factors should not be understood as individual impact, but as combination of all factors influencing the prediction model. It is also independent of whether the factors increase or reduce the WVC risk. Finally, the ML analysis only determines the collision risk in order to warn drivers of WVCs only when necessary.
+
+#### 6. Discussion
+
+From the large dataset of WVCs, spatiotemporal environmental patterns could be extracted with an ML model that predict the risk for WVCs appropriately. Even when the accident conditions varied case by case, it could be shown that the interaction of different road-. infrastructure-. land use-. weather-. topographic-. and temporal components is a complex cause-effect relationship for actual collisions. With ML and the collected dataset for the last 10 years covering the whole Bavaria, it was possible to predict WVCs spatiotemporally in a resolution and regional extent that had never been achieved before. Especially for dynamic warning systems, highly detailed data mining seems to be a critical advantage for precise warning opportunities. Thus, a detailed space–time-link between the accidents and potential influencing factors was able to be done. To not overlook the broader environment of WVCs, which may have an influence, the proportional shares of each land use type within the near, medium, and long distances, different infrastructure elements, and the topography were considered in the ML analysis.
+
+The case study showed that ML, especially feedforward neural networks and random forests, are suitable for an appropriate risk prediction. Also, the support vector machine classifier achieved a relatively high risk prediction quality (RPQ) – not as high as with the beforementioned algorithms, but within a shorter runtime. As the WVC data does not allow any information about near misses and, hence the actual risk probability is not entirely known, different ratios of WVCs were used for ML training. The models with a training ratio of 75% WVCs and 25% non-WVCs yielded the highest quality of WVC risk classification. AutoML helped to define the best hyperparameters for a high quality of risk prediction.
+
+The results show, on the one hand, the predictability in general, as the RPQ achieved continually values of over 85% using feedforward neural networks and random forests (Table 6). On the other hand, the transferability and applicability of the prediction model is proven, as all modes – randomly split, time-based, and geographical training splits yielded similar high results. The model was successfully tested for the Bavarian region, which shows that a transfer of the risk prediction of WVCs is generally possible.
+
+The developed method can be applied to areas outside the case study region. As WVC data are only available for some areas, having a method that can be applied anywhere may be helpful in tack-
+
+<span id="page-9-0"></span>![](_page_9_Figure_2.jpeg)
+
+Fig. 6. Impurity results of features used in the random forest model (s1/s2: side 1/2 of the road); impurity 1% is shown.
+
+ling the problem on a global level. However, the global transferability must be proven in future research and probably adapted to regional or species-specific characteristics. Since the case studies were only tested within Bavaria, this hypothesis is limited because it has not been proven whether the spatial transferability between different test areas in Bavaria also applies to other European or global regions. Since the prediction was made primarily with deer (around 80% of the total sample), the system may be overfit by learning pattern recognitions for collisions between deer and vehicles. Hence, the risk prediction was not applied for species that were not as frequently registered within the test region (e.g., foxes or badgers), nor for other non-native animal species. Also, the nonexistence of land use classes such as snow and ice areas in the Bavarian case study area may generate a bias for other areas during the data pre-processing stage. A limiting factor remains the data availability and quality for other parts of Europe and around the world [\(Pagany, 2020\)](#page-11-0).
+
+Temporal forecast tests showed that it is possible to apply this prediction analysis for dynamic warnings. As the drivers are more precisely warned when the risk is potentially high and animal movement is more likely, warning applications are implicitly also a mitigation measure in the sense of road ecology and wildlife conservation. Real-time warnings regarding the current place, time, and variable conditions avoid familiarization effect for drivers. For providing real time safety warnings, drivers' reaction time as well as vehicle kinematic reaction must be considered, meaning that the warnings needed to be provided earlier for driving along a road. For WVC warning, the application Wildwarner [\(wuidi,](#page-12-0) [2020](#page-12-0)) is already tested successfully with positive feedback from users [\(Trager, Kalová, Pagany, & Dorner, 2021](#page-12-0)).
+
+The impurity calculations confirm that static and dynamic factors influence the WVC risk, which is also confirmed in previous studies ([Pagany, 2020](#page-11-0)). Road width and road class especially determine the risk prediction. The influence of both factors was confirmed in several other studies, such as [Clevenger, Barrueto,](#page-11-0) [Gunson, Caryl, and Ford \(2015\), Pagany, Valdes, and Dorner](#page-11-0) [\(2020\), Santos, Lourenço, Mira, and Beja \(2013\),](#page-11-0) or [Williams,](#page-12-0) [Collinson, Patterson-Abrolat, Marneweck, and Swanepoel \(2019\).](#page-12-0) The majority of the studies identified more open roads with high visibility through wider lanes, more gentle slopes, and fewer curves as reinforcing for an accident risk. Implicitly, higher road classes such as state and federal roads with a higher number of lanes and broader road constructions are riskier than smaller roads. In Germany, WVCs are the lowest on highways due to the high rate of fenced road sections [\(Pagany, 2020](#page-11-0)).
+
+In addition, this study also identified artificial or rather urban areas as highly influential for WVC risk. However, the influence of urban areas and population density is not clearly assignable in scientific literature [\(Pagany, 2020](#page-11-0)). While some studies analyzed urban areas as riskier for several species ([Visintin, van der Ree, &](#page-12-0) [McCarthy, 2017; Wright, Coomber, Bellamy, Perkins, & Mathews,](#page-12-0) [2020](#page-12-0)), other studies identified road sections near urban areas as less risky for wildlife ([Jakubas, Rys´, & Lazarus, 2018; Kantola,](#page-11-0) [Tracy, Baum, Quinn, & Coulson, 2019; Zuberogoitia et al., 2014\)](#page-11-0). For the WVCs in Bavaria, the WVCs seem to be generally agglomerated in rural areas with high distance to urban areas ([Fig. 3](#page-5-0)).
+
+According to this study, temporal factors - sun angle, hour of the day, and day of the year – also influence the WVC risk. Especially the time factor seems to be related to wildlife's behavior and activities due to migration [\(Coe et al., 2015](#page-11-0)), breeding [\(Grilo, Bissonette,](#page-11-0) [& Santos-Reis, 2009](#page-11-0)), or rutting ([Zuberogoitia et al., 2014\)](#page-12-0). Several studies found daily peaks around dawn and dusk, which is a peak time of movement for animals and for humans (rush hours) ([Bíl,](#page-10-0) [Kubecˇek, Sedoník, & Andrášik, 2017; Laliberté & St-Laurent,](#page-10-0) [2020; Visintin et al., 2016\)](#page-10-0). These peak times also correlates with the changes of light conditions. As examples, [Pagany and Dorner](#page-11-0) [\(2016\)](#page-11-0) found a temporal peak of accidents with wild boar in the night hours in autumn, while [Biggs, Sherwood, Michalak, Hansen,](#page-10-0) [and Bare \(2004\)](#page-10-0) identified an increased risk for deer during the early evening hours in autumn and winter. [Sullivan \(2011\)](#page-12-0) sees the human and animals' visibility, [Rodríguez-Morales, Díaz-](#page-11-0)[Varela, and Marey-Pérez \(2013\)](#page-11-0) the wildlife's activities at twilight as causal factors. According to their study, [Visintin et al. \(2016\)](#page-12-0) found an influence of higher temperature on the WVC risk.
+
+<span id="page-10-0"></span>The presence or absence of forested areas also influence the risk prediction in the own analysis. In previous studies, the proximity of the road network to forests and cropland is assessed as risky for WVCs. [Chen and Wu \(2014\)](#page-11-0), for instance, found that threequarters of their investigated WVCs had forest on both sides of the road, while several studies have proven the same increasing effect for cropland [\(Carvalho-Roel, Iannini-Custódio, & Marçal](#page-11-0) [Júnior, 2019; Cuyckens, Mochi, Vallejos, Perovic, & Biganzoli,](#page-11-0) [2016; Found & Boyce, 2011](#page-11-0)).
+
+The importance of the factors should not be understood as individual impact but in the combination of all factors considered into the prediction model. It is not decisive to know the increasing or decreasing impact of an individual factor on the WVC risk, as analyzed by previous WVC research using classical statistics, but to develop a predictive learning model considering the environmental factors altogether.
+
+Targeted traffic and ecological measures are possible through the spatiotemporal data analysis and prediction, for example, to warn drivers dynamically, or to install protective measures only in times of high risk. As the case studies achieved a high quality in the risk prediction of WVCs, a basis is found to apply appropriate measures, on the one hand, for the management of road safety for humans, and, on the other hand, for biological conservation. For humans, the prediction model is able to provide dynamic warnings due to the results with high resolution in space and time. Therefore, impact studies for individual species are not relevant. However, for ecology management, species-specific risk analysis would be interesting, as different species have different activity times. Consequently, measures such as conscious feeding to change the tracks of the animals, should be adapted to reduce road crossings of a species at risky times and zones. As forest and seasonal cropland on both sides of the road led to high WVC risk, measures should be implemented at these particular road sections. Wide state and federal roads should especially be targeted. The effect of light conditions leads us to conclude that temporal warnings could be a suitable method of risk reduction.
+
+Traffic volume and speed were two factors that were not included into the ML analysis, as the available dataset of floating car data does not provide representative information on the actual speed and volume. If the traffic speed and the volume would be available, they might help to improve the prediction performance. Furthermore, it would be interesting to investigate further ML algorithms to improve the WVC risk prediction and, thus, the road safety for human and wildlife. Even when the prediction study tried to include as many conditions as possible, based on extensive data collection and feature engineering, not all eventualities could be considered for the prediction. Even when the dynamic measures, such as the dynamic warning via a mobile application, may mitigate the collision risk as drivers drove more attentively, the remaining risk cannot be ruled out due to negligent driving behavior or unpredictable movements of, for example, startled animals.
+
+#### 7. Conclusions
+
+The paper studied the WVCs of Bavaria from 2010 to 2019 including their environmental conditions, and investigated the performance of a random forest, feedforward neural networks, and a support vector machine for predicting WVC risk in a spatiotemporal manner. A large number of input data concerning WVCs was pre-processed together with a variety of environmental factors and road conditions using GIS. The pre-processing step enabled the researchers to incorporate all components, including the heterogeneous datasets and spatial and temporal dimensions of the chosen criteria into the WVC model. The risk prediction quality - the mean value of the specificity and the doubleweighted sensitivity rate – was calculated as valuable indicator for assessing risk prediction in the event of road accidents with wildlife. With values of 86.7% and 86.6%, neural networks and random forests are the best choice for WVC prediction. A ratio of 75% WVCs and 25% non-WVCs was found to be the best choice for ML training. Support vector machine classifiers achieved acceptable results in shorter runtimes but at lower prediction rates.
+
+The 36 test runs have proven that the approach can be transferred temporally and spatially. The spatial transferability was verified since the algorithm was successfully tested in Upper Bavaria with prediction values as high as with randomly split training data. The WVC forecast in a temporal manner was also proven through training with datasets from the first eight years (2010–2017), and in the prediction and testing with WVC data of 2018 and 2019. Hence, this study shows that a high quality of risk prediction for the occurrence of WVCs is possible, not only for real-time warnings but also for regions where accident data are not available. In addition to previous studies, the tests have also shown that it is important to predict WVCs spatiotemporally, as a combination of static and dynamic factors determines the predictive analysis. Other studies on WVCs have mainly focused on impact and hotspot analysis. This paper extended the WVC research by applying data mining techniques for the WVC risk prediction.
+
+This developed prediction model for WVC risk makes a valuable contribution to take appropriate action at the actual spatiotemporal accident hotspots, even when accidents with wildlife are rarely registered. The model has the potential to be transferable to areas without WVC data availability and, thus, can help decision makers to define better traffic safety control policies, such as dynamical warnings on a global scale. Nevertheless, the limitations of this study show that more efforts should be made to improve data quality and availability of the necessary input data for a globally applicable risk-learning system for the reduction of traffic accidents.
+
+## Acknowledgments
+
+This research was funded by the German Federal Ministry of Transport and Digital Infrastructure (BMVI) as part of the mFund project ''WilDa – Dynamic Wildlife-vehicle collision warning, using heterogeneous traffic, accident and environmental data as well as big data concepts" grant number 19F2014A. I thank the Bavarian Ministry of the Interior, for Sport and Integration & Bavarian State Police for the data provision.
+
+### References
+
+[Abdelwahab, H. T., & Abdel-Aty, M. A. \(2001\). Development of Artificial Neural](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0005) [Network Models to Predict Driver Injury Severity in Traffic Accidents at](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0005) Signalized Intersections. [Transportation Research Record, 1746](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0005). Paper No. 01-
+
+Arevalo, J. E., Honda, W., Arce-Arias, A., & Häger, A. (2017). Spatiotemporal variation of roadkills show mass mortality events for amphibians in a highly trafficked road adjacent to a national park, Costa Rica. Revista de Biología Tropical, 65(4), 1261–1276. <https://doi.org/10.15517/rbt.v65i4.27903>.
+
+Beshah, T. & Hill, S. (2010). Artificial Intelligence for Development, Papers from the 2010 AAAI Spring Symposium, Technical Report SS-10-01, Stanford, California, USA, March 22-24, 2010. Retrieved March 12, 2020, from [http://www.aaai.org/](http://www.aaai.org/ocs/index.php/SSS/SSS10/paper/view/1173) [ocs/index.php/SSS/SSS10/paper/view/1173](http://www.aaai.org/ocs/index.php/SSS/SSS10/paper/view/1173).
+
+Biggs, J., Sherwood, S., Michalak, S., Hansen, L., & Bare, C. (2004). Animal-related vehicle accidents at the Los Alamos National Laboratory, New Mexico. Southwestern Naturalist, 49(3), 384–394. [https://doi.org/10.1894/0038-4909.](https://doi.org/10.1894/0038-4909)
+
+Bíl, M., Andrášik, R., Svoboda, T., & Sedoník, J. (2016). The KDE+ software: A tool for effective identification and ranking of animal-vehicle collision hotspots along networks. Landscape Ecology, 31(2), 231–237. [https://doi.org/10.1007/s10980-](https://doi.org/10.1007/s10980-015-0265-6) [015-0265-6](https://doi.org/10.1007/s10980-015-0265-6).
+
+Bíl, M., Kubecˇek, J., Sedoník, J., & Andrášik, R. (2017). Srazenazver.cz: A system for evidence of animal-vehicle collisions along transportation networks. Biological Conservation, 213, 167–174. [https://doi.org/10.1016/j.biocon.2017.07.012.](https://doi.org/10.1016/j.biocon.2017.07.012)
+
+- <span id="page-11-0"></span>Breiman, L. (2001). Random Forests. Machine Learning, 45, 5–32. [https://doi.org/](https://doi.org/10.1023/A:1010933404324) [10.1023/A:1010933404324](https://doi.org/10.1023/A:1010933404324).
+- Carvalho-Roel, C. F., Iannini-Custódio, A. E., & Marçal Júnior, O. (2019). Do roadkill aggregations of wild and domestic mammals overlap? Revista de Biologia Tropical, 67(1), 47–60. [https://doi.org/10.15517/RBT.V67I1.33011.](https://doi.org/10.15517/RBT.V67I1.33011)
+- Chang, L. Y., & Chen, W. C. (2005). Data mining of tree-based models to analyze freeway accident frequency. Journal of Safety Research, 36(4), 365–375. [https://](https://doi.org/10.1016/j.jsr.2005.06.013) [doi.org/10.1016/j.jsr.2005.06.013](https://doi.org/10.1016/j.jsr.2005.06.013).
+- Chen, F., Song, M., & Ma, X. (2019). Investigation on the Injury Severity of Drivers in Rear-End Collisions Between Cars Using a Random Parameters Bivariate Ordered Probit Model. International Journal of Environmental Research and Public Health, 16(14). <https://doi.org/10.3390/ijerph16142632>.
+- Chen, X., & Wu, S. (2014). Examining patterns of animal-vehicle collisions in Alabama, USA. Human-Wildlife Interactions, 8(2), 235–244. [https://doi.org/](https://doi.org/10.26077/p18k-1089) [10.26077/p18k-1089.](https://doi.org/10.26077/p18k-1089)
+- Cherry, C. C., Dietz, S., Sauber-Schatz, E., Russell, S., Proctor, J., & Buttke, D. (2019). Characteristics of animal-related motor vehicle crashes in select National Park Service units—United States, 1990–2013. Traffic Injury Prevention, 20(1), 58–63. [https://doi.org/10.1080/15389588.2018.1508835.](https://doi.org/10.1080/15389588.2018.1508835)
+- Chong, M., Abraham, A., & Paprzycki, M. (2005). Traffic Accident Analysis Using Machine Learning Paradigms. 11. Informatica (Slovenia). 29(1), 89–98. Retrieved September 24, 2018, from <http://02.softcomputing.net/isda-mam.pdf>.
+- Clevenger, A. P., Barrueto, M., Gunson, K. E., Caryl, F. M., & Ford, A. T. (2015). Context-dependent effects on spatial variation in deer-vehicle collisions. Ecosphere, 6(4). <https://doi.org/10.1890/ES14-00228.1>.
+- Coe, P. K., Nielson, R. M., Jackson, D. H., Cupples, J. B., Seidel, N. E., Johnson, B. K., et al. (2015). Identifying migration corridors of mule deer threatened by highway development. Wildlife Society Bulletin, 39(2), 256–267. [https://doi.org/](https://doi.org/10.1002/wsb.544) [10.1002/wsb.544](https://doi.org/10.1002/wsb.544).
+- Cuyckens, G. A. E., Mochi, L. S., Vallejos, M., Perovic, P. G., & Biganzoli, F. (2016). Patterns and Composition of Road-Killed Wildlife in Northwest Argentina. Environmental Management, 58(5), 810–820. [https://doi.org/10.1007/s00267-](https://doi.org/10.1007/s00267-016-0755-6) [016-0755-6](https://doi.org/10.1007/s00267-016-0755-6).
+- Destatis (2020). Bevölkerungszahlen Kreise. Retrieved March 16, 2020, from [https://www-genesis.destatis.de/.](https://www-genesis.destatis.de/)
+- Deublein, M., Schubert, M., Bryan, A. T., & de Soto Boria, G. (2015). A Bayesian network model to predict accidents on Swiss highways. Infrastructure Asset Management, 2(4), 145–158. [https://doi.org/10.1680/jinam.15.00008.](https://doi.org/10.1680/jinam.15.00008)
+- Duellman, W. E. (2001). Patterns of Distribution of Amphibians. A Global Perspective. Journal of Animal Ecology, 70(3). [https://doi.org/10.1046/j.1365-](https://doi.org/10.1046/j.1365-2656.2001.00515-1.x) [2656.2001.00515-1.x.](https://doi.org/10.1046/j.1365-2656.2001.00515-1.x) 536–536.
+- DWD (German Weather Service) (2019). Climate and environment data. COSMO-Reanalysis-6 Hourly Data. Retrieved March 24, 2020, from [ftp://opendata.dwd.](http://ftp%3a//opendata.dwd.de/climate_environment/REA/COSMO_REA6/hourly/2D/) [de/climate\\_environment/REA/COSMO\\_REA6/hourly/2D/.](http://ftp%3a//opendata.dwd.de/climate_environment/REA/COSMO_REA6/hourly/2D/)
+- Feurer, M., Klein, A., Eggensperger, K., Springenberg, J., Blum, M., & Hutter, F. (2015). Efficient and Robust Automated Machine Learning. In C. Cortes, N. Lawrence, D. Lee, M. Sugiyama, & R. Garnett (Hrsg.) Advances in Neural Information Processing Systems, 28, 2962–2970, Curran Associates, Inc. [https://proceedings.neurips.](https://proceedings.neurips.cc/paper/2015/file/11d0e6287202fced83f79975ec59a3a6-Paper.pdf) [cc/paper/2015/file/11d0e6287202fced83f79975ec59a3a6-Paper.pdf.](https://proceedings.neurips.cc/paper/2015/file/11d0e6287202fced83f79975ec59a3a6-Paper.pdf)
+- FGSV (Forschungsgesellschaft für Straßen- und Verkehrswesen) (1996). Richtlinien für die Anlage von Straßen, FGSV Verlag, Köln, Germany.
+- Found, R., & Boyce, M. S. (2011). Predicting deer-vehicle collisions in an urban area. Journal of Environmental Management, 92(10), 2486–2493. [https://doi.org/](https://doi.org/10.1016/j.jenvman.2011.05.010) [10.1016/j.jenvman.2011.05.010.](https://doi.org/10.1016/j.jenvman.2011.05.010)
+- Garriga, N., Franch, M., Santos, X., Montori, A., & Llorente, G. A. (2017). Seasonal variation in vertebrate traffic casualties and its implications for mitigation measures. Landscape and Urban Planning, 157, 36–44. [https://doi.org/10.1016/](https://doi.org/10.1016/j.landurbplan.2016.05.029) [j.landurbplan.2016.05.029.](https://doi.org/10.1016/j.landurbplan.2016.05.029)
+- GDV (Gesamtverband der Deutschen Versicherer) (2020). Wildunfälle erreichen Rekordhoch. Retrieved 12 November 2020, URL: [http://www.gdv.de/de/medien/](http://www.gdv.de/de/medien/aktuell/wildunfaelle-erreichen-rekordhoch-63692) [aktuell/wildunfaelle-erreichen-rekordhoch-63692.](http://www.gdv.de/de/medien/aktuell/wildunfaelle-erreichen-rekordhoch-63692)
+- Gkritza, K., Baird, M., & Hans, Z. N. (2010). Deer-vehicle collisions, deer density, and land use in Iowa's urban deer herd management zones. Accident Analysis and Prevention, 42(6), 1916–1925. <https://doi.org/10.1016/j.aap.2010.05.013>.
+- Grilo, C., Bissonette, J. A., & Santos-Reis, T. M. (2009). Spatial-temporal patterns in Mediterranean carnivore road casualties: Consequences for mitigation. Biological Conservation., 142(2), 301–313. [https://doi.org/10.1016/j.](https://doi.org/10.1016/j.biocon.2008.10.026) [biocon.2008.10.026.](https://doi.org/10.1016/j.biocon.2008.10.026)
+- Grilo, C., Ferreira, F. Z., & Revilla, E. (2015). No evidence of a threshold in traffic volume affecting road-kill mortality at a large spatio-temporal scale. Environmental Impact Assessment Review, 55, 54–58. [https://doi.org/10.1016/j.](https://doi.org/10.1016/j.eiar.2015.07.003) [eiar.2015.07.003.](https://doi.org/10.1016/j.eiar.2015.07.003)
+- Gunson, K. E., Mountrakis, G., & Quackenbush, L. J. (2011). Spatial wildlife-vehicle collision models: A review of current work and its application to transportation mitigation projects. Journal of Environmental Management, 92(4), 1074–1082. [https://doi.org/10.1016/j.jenvman.2010.11.027.](https://doi.org/10.1016/j.jenvman.2010.11.027)
+- Heigl, F., Stretz, C. R., Steiner, W., Suppan, F., Bauer, T., Laaha, G., et al. (2016). Comparing road-kill datasets from hunters and citizen scientists in a landscape context. Remote Sensing, 8(10). <https://doi.org/10.3390/rs8100832>.
+- Holmgren, W. F., Hansen, C. W., & Mikofski, M. A. (2018). pvlib python: A python package for modeling solar energy systems. Journal of Open Source Software, 3 (29), 884. <https://doi.org/10.21105/joss.00884>.
+- [Jakubas, D., Rys´, M., & Lazarus, M. \(2018\). Factors affecting wildlife-vehicle collision](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0160) [on the expressway in a suburban area in northern Poland.](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0160) North-Western Journal [of Zoology, 14](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0160)(1), 107–116.
+
+- Jin, H., Song, Q., & Hu, X. (2019). Auto-Keras: An Efficient Neural Architecture Search System. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, Anchorage, AK, USA (pp. 1946–1956). <https://doi.org/10.1145/3292500.3330648>.
+- Kämmerle, J.-L., Brieger, F., Kröschel, M., Hagen, R., Storch, I., & Suchant, R. (2017). Temporal patterns in road crossing behaviour in roe deer (Capreolus capreolus) at sites with wildlife warning reflectors. PLoS ONE, 12(9). [https://doi.org/](https://doi.org/10.1371/journal.pone.0184761) [10.1371/journal.pone.0184761.](https://doi.org/10.1371/journal.pone.0184761)
+- Kantola, T., Tracy, J. L., Baum, K. A., Quinn, M. A., & Coulson, R. N. (2019). Spatial risk assessment of eastern monarch butterfly road mortality during autumn migration within the southern corridor. Biological Conservation., 231, 150–160. [https://doi.org/10.1016/j.biocon.2019.01.008.](https://doi.org/10.1016/j.biocon.2019.01.008)
+- Keken, Z., Sedoník, J., Kušta, T., Andrášik, R., & Bíl, M. (2019). Roadside vegetation influences clustering of ungulate vehicle collisions. Transportation Research Part D: Transport and Environment, 73, 381–390. [https://doi.org/10.1016/j.](https://doi.org/10.1016/j.trd.2019.07.013) [trd.2019.07.013](https://doi.org/10.1016/j.trd.2019.07.013).
+- Komol, M. M. R., Hasan, M. M., Elhenawy, M., Yasmin, S., Masoud, M., & Rakotonirainy, A. (2021). Crash severity analysis of vulnerable road users using machine learning. PLoS ONE, 16(8). [https://doi.org/10.1371/journal.](https://doi.org/10.1371/journal.pone.0255828) [pone.0255828.](https://doi.org/10.1371/journal.pone.0255828)
+- Laliberté, J., & St-Laurent, M.-H. (2020). In the wrong place at the wrong time: Moose and deer movement patterns influence wildlife-vehicle collision risk. Accident Analysis and Prevention, 135. <https://doi.org/10.1016/j.aap.2019.105365>.
+- Liu, Y., Nieuwenhuis, M., & McCullagh, A. (2018). The effect of roadside land-use on the occurrence of deer vehicle collisions. Irish Forestry, 75(1&2), 8-25. Retrieved October 21, 2019, from [https://journal.societyofirishforesters.ie/index.](https://journal.societyofirishforesters.ie/index.php/forestry/article/view/10908) [php/forestry/article/view/10908.](https://journal.societyofirishforesters.ie/index.php/forestry/article/view/10908)
+- Lutterschmidt, W. I., Weidler, J. M., & Schalk, C. M. (2019). Hot moments and hot spots in the bayou: Spatiotemporal patterns of road occurrence in a snake assemblage in Louisiana, USA. Herpetological Conservation and Biology, 14(2), 533–545. Retrieved September 24, 2019, from [http://www.herpconbio.org/](http://www.herpconbio.org/Volume_14/Issue_2/Lutterschmidt_etal_2019.pdf) [Volume\\_14/Issue\\_2/Lutterschmidt\\_etal\\_2019.pdf](http://www.herpconbio.org/Volume_14/Issue_2/Lutterschmidt_etal_2019.pdf).
+- Malo, J. E., Suárez, F., & Díez, A. (2004). Can we mitigate animal-vehicle accidents using predictive models? Journal of Applied Ecology, 41(4), 701–710. [https://doi.](https://doi.org/10.1111/j.0021-8901.2004.00929.x) [org/10.1111/j.0021-8901.2004.00929.x](https://doi.org/10.1111/j.0021-8901.2004.00929.x).
+- [McCulloch, W. S., & Pitts, W. \(1943\). A logical calculus of the ideas immanent in](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0220) nervous activity. [Bulletin of Mathematical Biophysics, 5](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0220), 115–133. ISSN 0007-4985.
+- Morelle, K., Lehaire, F., & Lejeune, P. (2013). Spatio-temporal patterns of wildlifevehicle collisions in a region with a high-density road network. Nature Conservation, 5, 53–73. <https://doi.org/10.3897/natureconservation.5.4634>.
+- OSGeo (2019). GDAL/OGR. Retrieved October 21, 2019, from [https://www.osgeo.org/](https://www.osgeo.org/projects/gdal/)
+- Pagany, R. (2020). Wildlife-vehicle collisions Influencing factors, data collection and research methods. Biological Conservation, 251. [https://doi.org/10.1016/j.](https://doi.org/10.1016/j.biocon.2020.108758) [biocon.2020.108758.](https://doi.org/10.1016/j.biocon.2020.108758)
+- Pagany, R., & Dorner, W. (2016). Spatiotemporal Analysis for Wildlife-Vehicle Collision Based on Accident Statistics of the County Straubing-Bogen in Lower Bavaria. ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial, Information Sciences, XLI-B8, 739–745. [https://doi.org/10.5194/](https://doi.org/10.5194/isprsarchives-XLI-B8-739-2016) [isprsarchives-XLI-B8-739-2016.](https://doi.org/10.5194/isprsarchives-XLI-B8-739-2016)
+- Pagany, R., Valdes, J., & Dorner, W. (2020). Risk Prediction of Wildlife-vehicle Collisions Comparing Machine Learning Methods and Data Use. In 2020 IEEE 10th International Conference on Advanced Computer Information Technologies (ACIT), Deggendorf, Germany, 2020 (pp. 436–440). [https://doi.org/10.1109/](https://doi.org/10.1109/ACIT49673.2020.9208946) [ACIT49673.2020.9208946.](https://doi.org/10.1109/ACIT49673.2020.9208946)
+- Pflugmacher, D., Rabe, A., Peters, M., & Hostert, P. (2018). Pan-European land cover map of 2015 based on Landsat and LUCAS data. PANGAEA. [https://doi.org/](https://doi.org/10.1594/PANGAEA.896282) [10.1594/PANGAEA.896282](https://doi.org/10.1594/PANGAEA.896282).
+- Rodríguez-Morales, B., Díaz-Varela, E. R., & Marey-Pérez, M. F. (2013). Spatiotemporal analysis of vehicle collisions involving wild boar and roe deer in NW Spain. Accident Analysis and Prevention., 60, 121–133. [https://doi.org/](https://doi.org/10.1016/j.aap.2013.07.032) [10.1016/j.aap.2013.07.032](https://doi.org/10.1016/j.aap.2013.07.032).
+- [Santos, S. M., Lourenço, R., Mira, A., & Beja, P. \(2013\). Relative effects of road risk,](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0260) [habitat suitability, and connectivity on wildlife Roadkills: The case of tawny](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0260) [owls \(Strix aluco\).](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0260) PLoS ONE, 8(11), e79967.
+- Santos, R. A. L., Mota-Ferreira, M., Aguiar, L. M. S., & Ascensão, F. (2018). Predicting wildlife road-crossing probability from roadkill data using occupancy-detection models. Science of The Total Environment, 642, 629–637. [https://doi.org/10.1016/](https://doi.org/10.1016/j.scitotenv.2018.06.107) [j.scitotenv.2018.06.107](https://doi.org/10.1016/j.scitotenv.2018.06.107).
+- Seo, C., Thorne, J. H., Choi, T., Kwon, H., & Park, C.-H. (2015). Disentangling roadkill: The influence of landscape and season on cumulative vertebrate mortality in South Korea. Landscape Ecology Engineering, 11(1), 87–99. [https://doi.org/](https://doi.org/10.1007/s11355-013-0239-2) [10.1007/s11355-013-0239-2.](https://doi.org/10.1007/s11355-013-0239-2)
+- Shrivastava, A., Verma, J. P. V., Jain, S., et al. (2021). A deep learning based approach for trajectory estimation using geographically clustered data. SN Appl. Sci., 3, 597. <https://doi.org/10.1007/s42452-021-04556-x>.
+- Snow, N. P., Zhang, Z., Finley, A. O., Rudolph, B. A., Porter, W. F., Williams, D. M., et al. (2018). Regional-based mitigation to reduce wildlife–vehicle collisions. Journal of Wildlife Management, 82(4), 756–765. <https://doi.org/10.1002/jwmg.21420>.
+- Steiner, W., Leisch, F., & Hackländer, K. (2014). A review on the temporal pattern of deer-vehicle accidents: Impact of seasonal, diurnal and lunar effects in cervids. Accident Analysis and Prevention, 66, 168–181. [https://doi.org/10.1016/j.](https://doi.org/10.1016/j.aap.2014.01.020) [aap.2014.01.020](https://doi.org/10.1016/j.aap.2014.01.020).
+- StMI (Bavarian State Ministry of the Interior, for Sport and Integration) (2017). Road data from the BAYSIS database (Bavarian road infrastructure system), Munich, Germany. (not published).
+
+- <span id="page-12-0"></span>StMI (Bavarian State Ministry of the Interior, for Sport and Integration) (2017). Road data from the BAYSIS database (Bavarian road infrastructure system), Munich, Germany. (not published).
+- Sullivan, J. M. (2011). Trends and characteristics of animal-vehicle collisions in the United States. Journal of Safety Research, 42(1), 9–16. [https://doi.org/10.1016/j.](https://doi.org/10.1016/j.jsr.2010.11.002) [jsr.2010.11.002.](https://doi.org/10.1016/j.jsr.2010.11.002)
+- Surveying Authorities of the Federal States of Germany (2015). Official Topographical-Cartographic Information System data – Digital Landscape Models (ATKIS-DLM Data). (not published).
+- Tanner, A. L., Leroux, S. J., & Saunders, P. W. (2017). Road characteristics best predict the probability of vehicle collisions with a non-native ungulate. Ecoscience, 24 (1–2), 1–11. <https://doi.org/10.1080/11956860.2017.1292100>.
+- Toblerity (2020). Shapely. Retrieved February 19, 2020, from [https://github.com/](https://github.com/Toblerity/Shapely) [Toblerity/Shapely.](https://github.com/Toblerity/Shapely)
+- Trager, J., Kalová, L., Pagany, R., & Dorner, W. (2021). Warning Apps for Road Safety: A Technological and Economical Perspective for Autonomous Driving – The Warning Task in the Transition from Human Driver to Automated Driving. International Journal of Human-Computer Interaction, 37(4), 363–377. [https://doi.](https://doi.org/10.1080/10447318.2020.1860545) [org/10.1080/10447318.2020.1860545.](https://doi.org/10.1080/10447318.2020.1860545)
+- Valero, E., Picos, J., & Álvarez, X. (2015). Road and traffic factors correlated to wildlife-vehicle collisions in Galicia (Spain). Wildlife Research, 42(1), 25–34. [https://doi.org/10.1071/WR14060.](https://doi.org/10.1071/WR14060)
+- van der Ree, R., Jaeger, J. A. G., van der Grift, E. A., & Clevenger, A. P. (2011). Effects of roads and traffic on wildlife populations and landscape function: Road ecology is moving toward larger scales. Ecology and Society, 16(1). [https://doi.org/](https://doi.org/10.5751/ES-03982-160148) [10.5751/ES-03982-160148.](https://doi.org/10.5751/ES-03982-160148)
+- [Vapnik, V. \(1963\). Pattern recognition using generalized portrait method.](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0340) [Automation remote control, 24](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0340), 774–780.
+
+- Visintin, C., van der Ree, R., & McCarthy, M. A. (2016). A simple framework for a complex problem? Predicting wildlife–vehicle collisions. Ecology and Evolution, 6(17), 6409–6421. <https://doi.org/10.1002/ece3.2306>.
+- Visintin, C., van der Ree, R., & McCarthy, M. A. (2017). Consistent patterns of vehicle collision risk for six mammal species. J. Environ. Manag., 201, 397–406. [https://](https://doi.org/10.1016/j.jenvman.2017.05.071) [doi.org/10.1016/j.jenvman.2017.05.071](https://doi.org/10.1016/j.jenvman.2017.05.071).
+- [Williams, S. T., Collinson, W., Patterson-Abrolat, C., Marneweck, D. G., & Swanepoel,](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0360) [L. H. \(2019\). Using road patrol data to identify factors associated with carnivore](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0360) [roadkill counts.](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0360) PeerJ, 7, e6650.
+- Wright, P. G. R., Coomber, F. G., Bellamy, C. C., Perkins, S. E., & Mathews, F. (2020). Predicting hedgehog mortality risks on British roads using habitat suitability modelling. PeerJ, 1, 2020. [https://doi.org/10.7717/peerj.8154.](https://doi.org/10.7717/peerj.8154)
+- wuidi (2020). Gemeinsam Wildunfälle vermeiden. Wuidi Die Nr. 1. Wildwarner. Retrieved March 11, 2020, from <https://wuidi.com>.
+- [Yang, W. T., Chen, H. C., & Brown, D. B. \(1999\). Detecting Safer Driving Patterns By A](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0375) Neural Network Approach. In [1999 Proceedings of Smart Engineering System](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0375) [Design Neural Network, Evolutionary Programming, Complex Systems and Data](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0375) [Mining, St. Louis, Missouri, USA](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0375) (pp. 839–844).
+- [Zuberogoitia, I., del Real, J., Torres, J. J., Rodríguez, L., Alonso, M., & Zabala, J. \(2014\).](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0380) [Ungulate vehicle collisions in a peri-urban environment: Consequences of](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0380) [transportation infrastructures planned assuming the absence of ungulates.](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0380) PLoS ONE, 9[\(9\), e107713.](http://refhub.elsevier.com/S0022-4375(22)00137-2/h0380)
+
+Raphaela Pagany did this research as a senior scientist at the Technische Hochschule Deggendorf. Her research is focused on Geoinformatics, spatio-temporal analysis and risk prediction. In the project WilDa, she developed algorithms for the Wildwarner service. In addition, she did her PhD study at the Paris Lodron University in Salzburg. Currently, she is working in transport planning.

Zotero/001_artiklid/Road Safety Indicators/parkapplicationfinitemixturemodels2009.md

@@ -0,0 +1,351 @@
+---
+category: literaturenote
+citekey: parkapplicationfinitemixturemodels2009
+title: Application of finite mixture models for vehicle crash data analysis
+authors: "Park, Byung-Jung; Lord, Dominique"
+year: 2009
+date: 2009-00-00 2009
+doi: 10.1016/j.aap.2009.03.007
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2009.03.007"
+zotero_key: WR9KIBZA
+zotero_storage: P27NPKAP
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Park, Byung-Jung"
+status: converted
+---
+Contents lists available at [ScienceDirect](http://www.sciencedirect.com/science/journal/00014575)
+
+# Accident Analysis and Prevention
+
+journal homepage: [www.elsevier.com/locate/aap](http://www.elsevier.com/locate/aap)
+
+![](_page_0_Picture_5.jpeg)
+
+# Application of finite mixture models for vehicle crash data analysis
+
+Byung-Jung Park 1, Dominique Lord<sup>∗</sup>
+
+*Zachry Department of Civil Engineering, Texas A&M University, 3136 TAMU, College Station, TX 77843-3136, United States*
+
+## article info
+
+*Article history:* Received 16 October 2008 Received in revised form 2 February 2009 Accepted 19 March 2009
+
+*Keywords:* Highway safety Over-dispersion Finite mixture Negative binomial regression model Latent class model
+
+#### abstract
+
+Developing sound or reliable statistical models for analyzing motor vehicle crashes is very important in highway safety studies. However, a significant difficulty associated with the model development is related to the fact that crash data often exhibit over-dispersion. Sources of dispersion can be varied and are usually unknown to the transportation analysts. These sources could potentially affect the development of negative binomial (NB) regression models, which are often the model of choice in highway safety. To help in this endeavor, this paper documents an alternative formulation that could be used for capturing heterogeneity in crash count models through the use of finite mixture regression models. The finite mixtures of Poisson or NB regression models are especially useful where count data were drawn from heterogeneous populations. These models can help determine sub-populations or groups in the data among others. To evaluate these models, Poisson and NB mixture models were estimated using data collected in Toronto, Ontario. These models were compared to standard NB regression model estimated using the same data. The results of this study show that the dataset seemed to be generated from two distinct sub-populations, each having different regression coefficients and degrees of over-dispersion. Although over-dispersion in crash data can be dealt with in a variety of ways, the mixture model can help provide the nature of the over-dispersion in the data. It is therefore recommended that transportation safety analysts use this type of model before the traditional NB model, especially when the data are suspected to belong to different groups.
+
+© 2009 Elsevier Ltd. All rights reserved.
+
+#### **1. Introduction**
+
+Developing sound or reliable statistical models for analyzing motor vehicle crashes is important in highway safety studies. The models can be used for identifying hazardous sites, predicting motor vehicle collisions, and developing accident modification factors ([Lord and Park, 2008\).](#page-7-0) What makes the analysis difficult in modeling crash data is that this kind of data often exhibit overdispersion. The level of complexity becomes more significant when the data are characterized by small sample sizes and low sample mean values, which are commonly observed with crash data ([Lord, 2006\).](#page-7-0) [Hauer \(2001\)](#page-7-0) reported that over-dispersion observed in crash data can be described in terms of "*represented traits*" and *unrepresented traits*"; the root cause of over-dispersion is that entities with the same represented traits have different means because of the unrepresented traits (measured or unmeasured) not included in the model. On the other hand, [Lord et al. \(2005\)](#page-7-0) provided a more fundamental definition in which the over-dispersion arises from the actual nature of the crash process. This process dictates that the over-dispersion is the result of Bernoulli trials with unequal
+
+Since the NB model can effectively approximate this underlying crash process, and reduce the unrepresented uncertainties by introducing a probabilistic error term related to the mean of the Poisson variable, transportation safety analysts have adopted this kind of model for developing crash prediction models. Within the NB regression modeling framework, many studies have been focused on the structure of the inverse dispersion parameter (-, or its inverse - =˛<sup>−</sup>1) of the NB distribution. Up until early 2000s, most of researchers in traffic safety have developed predictive models using a fixed or common dispersion parameter model [\(Hauer,](#page-7-0) [2001\).](#page-7-0) In 2001, [Heydecker and Wu](#page-7-0) suggested that could be modeled as a function of the covariates of the model (which can be defined as the varying dispersion parameter). [Hauer \(2001\)](#page-7-0) argued that the inverse dispersion parameter should be modeled as a function of segment,*<sup>i</sup>* = ı*Li*, to correct for the unequal variance of the NB regression model. Since then, other researchers have investigated various structures of the dispersion parameter, both spatially and temporally [\(Lord and Park, 2008; Miaou and Lord, 2003; Miranda-](#page-7-0)Moreno [et al., 2005; El-Basyouny and Sayed, 2006; Geedipally and](#page-7-0) [Lord, 2008\).](#page-7-0) Very recently, it was found that the structure of the dispersion parameter can greatly depend on how the mean function
+
+probability of independent events (known as Poisson trials) and all distributions, such as the Poisson-gamma (or negative binomial or NB) or Poisson-lognormal, are used as approximation to capture the over-dispersion observed in crash data.
+
+<sup>∗</sup> Corresponding author. Tel.: +1 979 458 3949; fax: +1 979 845 6481.
+
+*E-mail addresses:* [soldie@tamu.edu](mailto:soldie@tamu.edu) (B.-J. Park), [d-lord@tamu.edu](mailto:d-lord@tamu.edu) (D. Lord).
+
+<sup>1</sup> Tel.: +1 979 862 8492; fax: +1 979 845 6481.
+
+is modeled ([Mitra and Washington, 2007\).](#page-7-0) Models with a welldefined mean function may not have a structured variance. On the other hand, [Shankar et al. \(1998\)](#page-8-0) showed that when spatial and temporal effects are not explicitly included in the NB model, the random effect negative binomial model offered advantages.
+
+Despite the considerable efforts put in place to improve the performance of the NB models, several studies have documented important limitations associated with these models. As described above, research activities regarding the development of NB models using a fixed versus a varying dispersion parameter, and random parameter models with a fixed dispersion parameter are still on going. Varying dispersion parameter models may be preferred because one can determine sources influencing over-dispersion ([Hilbe, 2007\).](#page-7-0) However, finding appropriate covariates that influence the over-dispersion can be problematic if it is partly caused by unobserved variables or conditions. If the fixed dispersion model is preferred in terms of parameter parsimony, it may not tell us about the nature of the over-dispersion in the data. It only confirms that evidence of over-dispersion has been found and that this has been taken into account in the NB model [\(Land et al., 1996\).](#page-7-0) Furthermore, if the datasets are characterized by small sample sizes and low mean values, the performance of the NB models can be significantly affected in terms of parameter estimation ([Lord, 2006\)](#page-7-0) as well as goodness-of-fit ([Maher and Summersgill, 1996; Wood,](#page-7-0) [2002; Park and Lord, 2008\).](#page-7-0) On the other hand, some have reported that NB regression models have difficulties handling the heavily over-dispersed data with a very long-tail and relatively high mean value because a negligible probability is usually assigned to high counts [\(Guo and Trivedi, 2002\).](#page-7-0) Given these important limitations, [Lord et al. \(2008\)](#page-7-0) suggested using the Conway–Maxwell–Poisson (COM–Poisson) model for crash data analysis. They showed that the COM–Poisson model performed as good as the NB model in terms of goodness-of-fit statistics and predictive performance.
+
+As an alternative approach to address aforementioned problems, this paper documents an alternative formulation that could be used for capturing heterogeneity in crash count models through the use of finite mixture regression models (both for Poisson mixtures and NB mixtures). These models are compared to the results produced from the standard NB regression model. Modeling based on finite mixture distributions has a long history, and with the advancement of computing power and technology, it has continued to receive increasing attention in many areas, such as biometrics, genetics, medicine, and marketing ([Frühwirth-Schnatter, 2006\).](#page-7-0) The finite mixtures of Poisson regression models or NB regression models (abbreviated as FMP and FMNB, respectively, hereafter) are especially useful where count data were drawn from heterogeneous populations. There are many reasons to expect the existence of different sub-populations since the crash data are generally collected from various geographic, environmental and geometric design contexts over some fixed time periods. In such cases, it may be inappropriate to apply one aggregate NB regression model and the interpretation of the model could be misleading. Therefore, it would be reasonable to hypothesize that the individual crashes on highway entities (intersections, segments, etc.) are generated from a certain number (*K*) of hidden subgroups, or components that are unknown to the transportation safety analyst. The final outputs of FMNB regression models will be the number of components, component proportions, component-specific regression coefficients, and the degree of over-dispersion within each component.
+
+## **2. Background**
+
+This section describes the characteristics of NB models and the finite mixture models.
+
+## *2.1. Negative binomial distribution as a continuous mixture*
+
+Since the conventional Poisson model does not provide flexibility to accommodate frequently observed over-dispersion in crash data, several different mixed Poisson distributions have been applied by assuming a particular distribution in the Poisson mean rate (*mi*):
+
+$$m_i = \exp(\mathbf{x}_i \boldsymbol{\beta}) \cdot \varepsilon_i$$
+ (1)
+
+where *mi* is the Poisson mean rate for site *i*; **x***<sup>i</sup>* the covariate vector for site *i* where the first element is 1, (1 × *d*) vector; a vector of unknown coefficients, (*d* × 1) vector; ε a model error independent of all the covariates; and, *d* is the number of covariates including an intercept.
+
+Depending upon the parametric form imposed on *g*(ε*i*), various mixed Poisson regression models can be derived (e.g. Poissongamma, Poisson-lognormal, Poisson-inverse Gaussian, etc.). The negative binomial (NB) regression model arises if one assumes that *g*(ε*i*), or equivalently the distribution of *mi*, *f*(*mi*), follows a gamma distribution. The Poisson-gamma distribution is the most common distribution used for modeling crash data because its marginal distribution has a closed form and this mixture results in a conjugate model [\(Hauer, 1997\).](#page-7-0) The interpretation and derivation of the negative binomial as a Poisson-gamma mixture is well described in [Cameron and Trivedi \(1998\).](#page-7-0) Despite its reported limitations (see [Lord, 2006](#page-7-0) and references herein), the NB regression model is still very popular, especially since all statistical software programs have built-in functions that can handle such models ([Hilbe,](#page-7-0) [2007\).](#page-7-0)
+
+Unfortunately, the continuous parametric mixing distributions assumed in the Poisson mean rate may pose limitations in fitting the data, especially, with a small sample size or low sample mean value ([Lord, 2006; Park and Lord, 2008\).](#page-7-0) Furthermore, the choice of a particular distribution imposes a restrictive functional form between the mean and variance (e.g. quadratic relationship in the NB model), and is difficult to justify in practice because there is no *a priori* reason why the empirical frequency of crash data should be well approximated by that particular distribution, as discussed above.
+
+In addition, the continuous mixed Poisson models usually estimate a common parameter vector (-) and inverse dispersion parameter (-) for all the cross-sections ([Ramawamy et al., 1994\).](#page-7-0) In other words, they are estimated at the aggregate level with one standard probability distribution function, which can mask the possibility of heterogeneity in the coefficients of the covariates across the sites. As discussed above, since the crash data are generally collected from various geographic, environmental and geometric design contexts, there are many reasons to expect the different effects of each variable on the crash occurrence. To capture unobserved heterogeneity in these parameters, one can first classify the data based on some criteria, and then apply several Poisson or negative binomial regression models at a disaggregate level. However, there may be some arbitrariness involved in the criteria dividing the groups. [Gelman et al. \(2004, p. 467\)](#page-7-0) warned that this type of crude analysis completely ignores the uncertainty in the dividing indicators and thus can overestimate the differences between each model.
+
+# *2.2. Finite mixture model*
+
+The finite mixture model allows for extremely flexible modeling of heterogeneous data because it incorporates a combination of discrete and continuous representation of population heterogeneity. Accordingly, finite mixture models have been widely applied in many areas, such as biology, biometrics, genetics, medicine, and marketing. For a comprehensive list of the applications and numerical derivations of finite mixture models, readers are referred to [McLachlan and Peel \(2000\)](#page-7-0) and [Frühwirth-Schnatter \(2006\).](#page-7-0)
+
+The random vector **y** = (*y*1, *y*2, ..., *yN*) is said to arise from a finite mixture distribution, if the probability density function *p*(**y**) of this distribution has the following form:
+
+$$p(\mathbf{y}|\mathbf{\Theta}) = w_1 f_1(\mathbf{y}|\boldsymbol{\theta}_1) + w_2 f_2(\mathbf{y}|\boldsymbol{\theta}_2) + \dots + w_K f_K(\mathbf{y}|\boldsymbol{\theta}_K)$$
+ (2)
+
+where **-** = ((1, 2,..., <sup>K</sup> ) , **w**) denotes the vector of all parameters, and **w** = (w1, w2,...,wK ) is called a weight distribution whose elements are restricted to be positive and sum to unity wk > 0 and wk = 1 . A single density *f*(·|*k*) is referred to as the component distribution for component *k* (*k* = 1, 2, ..., *K*), and *K* is the number of components. In most applications, it is assumed that all component distributions arise from the same parametric distribution family, *f*(·|*k*). In our case, it is a Poisson or a NB distribution.
+
+The mean and the variance are given, respectively, by
+
+$$\mu = E(\mathbf{y}|\mathbf{\Theta}) = \sum_{k=1}^{K} \mu_k w_k \tag{3}$$
+
+$$\sigma^{2} = Var(\mathbf{y}|\mathbf{\Theta}) = \sum_{k=1}^{K} (\mu_{k}^{2} + \sigma_{k}^{2})w_{k} - \mu^{2}$$
+(4)
+
+provided that the component moments *<sup>k</sup>* = *E*(**y**|*k*) and <sup>2</sup> <sup>k</sup> = Var(**y**|k) exist [\(Frühwirth-Schnatter, 2006\).](#page-7-0)
+
+With this formulation, the heterogeneity in the data can be accounted for in two ways. First, it accounts for the population heterogeneity by choosing a finite number of unobserved latent components, each of which may be regarded as a sub-population. This is a discrete representation of heterogeneity in the data since the mean event rate is approximated by a finite number of support points. In this respect, the finite mixture model assumes that there is more than one component in the dataset. If such a distinct difference is not observed from modeling using Eq. (2), in other words, if the probability density function does not take the stated mixture density, the resulting parameter estimates would be very unstable and inaccurate. In such cases, it is possible to choose the traditional regression model (Poisson or NB regression model) that does not account for the heterogeneity due to the existence of different subpopulations. This can be done by setting *K* = 1 in Eq. (2). Second, depending on the choice of the component distribution, *f*(·|*k*), it can also accommodate heterogeneity within each component. For example, for FMP-*K* and FMNB-*K* regression models, the heterogeneity within each component is accounted for by including the explanatory variables in the mean event rate function. Using the NB distribution as a component distribution would explain additional over-dispersion within component not captured by those explanatory variables. Thus, the formulation is flexible enough to allow for both between-component and within-component variations. It should be noted that the finite mixture approach does not require any distributional assumptions for the mixing variable.
+
+The general setups for FMP-*K* and FMNB-*K* regression models can be extended from Eq. (2) and their means and variances are obtained from Eqs. (3) and (4).
+
+The FMP-*K* regression model assumes that the marginal distribution of *yi* follows a mixture of Poisson distributions:
+
+$$p(y_i|\mathbf{x}_i, \mathbf{\Theta}) = \sum_{k=1}^{K} w_k Pois(\mu_{k,i}) = \sum_{k=1}^{K} w_k \left( \frac{e^{-\mu_{k,i}} (\mu_{k,i})^{y_i}}{y_i!} \right)$$
+ (5)
+
+$$E(y_i|\mathbf{x}_i,\mathbf{\Theta}) = \sum_{k=1}^{K} \mu_{k,i} w_{k,i}$$
+(6)
+
+$$Var(y_i|\mathbf{x}_i, \boldsymbol{\Theta}) = E(y_i|\boldsymbol{\Theta}) + \left(\sum_{k=1}^K w_k \mu_{k,i}^2 - E(y_i|\boldsymbol{\Theta})^2\right)$$
+(7)
+
+where *k*,*<sup>i</sup>* = exp(**x***i<sup>k</sup>*) and **-**= {(-1, ..., -*<sup>K</sup>*), **w**}. It can be readily seen that unless all the component's means are the same (1,*<sup>i</sup>* = ··· = *K*,*i*), the variance is always greater than the mean.
+
+For the FMNB-*K* regression model, it is assumed that the marginal distribution of *yi* follows a mixture of negative binomial distributions:
+
+$$p(y_i|\mathbf{x}_i, \mathbf{\Theta}) = \sum_{k=1}^{K} w_k NB(\mu_{k,i}, \phi_k)$$
+
+$$= \sum_{k=1}^{K} w_k \left[ \frac{\Gamma(y_i + \phi_k)}{\Gamma(y_i + 1)\Gamma(\phi_k)} \left( \frac{\mu_{k,i}}{\mu_{k,i} + \phi_k} \right)^{y_i} \right]$$
+
+$$\times \left( \frac{\phi_k}{\mu_{k,i} + \phi_k} \right)^{\phi_k}$$
+(8)
+
+$$E(y_i|x_i, \mathbf{\Theta}) = \sum_{k=1}^{K} \mu_{k,i} w_{k,i}$$
+(9)
+
+$$Var(y_i|\mathbf{x}_i, \mathbf{\Theta}) = E(y_i|\mathbf{x}_i, \mathbf{\Theta}) + \left(\sum_{k=1}^K w_k \mu_{k,i}^2 \left(1 + \frac{1}{\phi_k}\right) - E(y_i|\mathbf{x}_i, \mathbf{\Theta})^2\right)$$
+(10)
+
+where *k*,*<sup>i</sup>* = exp(**x***i<sup>k</sup>*) and **-** = {(-<sup>1</sup>,..., -<sup>K</sup> ),(-1,...,-<sup>K</sup> ), **w**}. In this case, even if all the component's means are the same, the variance of *yi* is always greater than the mean. When *<sup>k</sup>* in each component goes to infinity, the FMNB-*K* model is reduced to the FMP-*K* model. Thus, the FMNB-*K* models allow for additional overdispersion within components not captured by the explanatory variables. If additional heterogeneity is present within components, the Poisson mixture model is misspecified. An implication of such additional heterogeneity is that the standard errors are underestimated ([Cameron and Trivedi, 1998\).](#page-7-0)
+
+It should be noted that the weight distribution (**w**) used in both FMP-*K* and FMNB-*K* was treated as a constant variable. The constant weight model can be extended to a more generalized model by parameterizing the weight distribution as a function of covariates [\(Wang et al., 1998; Scaccia and Green, 2003; Frühwirth-Schnatter](#page-8-0) [and Kaufmann, 2006\).](#page-8-0) This parameterization allows each observation to have a different weight that is dependent on the covariates, similar to the application of the varying dispersion parameter for the standard negative binomial model (see, e.g. [Miaou and Lord,](#page-7-0) [2003; Lord and Park, 2008\).](#page-7-0) Unfortunately, the use of varying weight factors was beyond the scope of this study, since the estimation process is very complex (i.e. there are various link functions that can be used to define the varying weights), and may not always provide the best modeling approach ([Frühwirth-Schnatter, 2006\).](#page-7-0)
+
+It is noteworthy that the finite mixture of regression models as defined in Eq. (5) or (8) embrace the zero-inflated Poisson (ZIP) or zero-inflated negative binomial (ZINB) regression models as a special case ([Cameron and Trivedi, 1998\);](#page-7-0) see [Lord et al. \(2005,](#page-7-0) [2007\)](#page-7-0) for a discussion about their use in highway safety. This can be obtained by setting *K* = 2 and 1,*<sup>i</sup>* = 0 for all *i.* However, the generalized two-component mixture model does not make this somewhat strict dual-state process assumption and allows mixing with respect to both zeroes and positives. The group separation is characterized by low mean with low variance and high mean with high variance. Recently, [Malyshkina et al. \(2009\)](#page-7-0) demonstrated a superior statistical fit of two-state Markov switching negative binomial models (with fixed weights) using time series crash data in Indiana interstate highway segments. Therefore, the FMP or FMNB models are expected to improve the goodness-of-fit relative to the conventional one-component NB model even when the sample mean is very low although this still needs to be verified in the future.
+
+#### **3. Model estimation method**
+
+Estimating a mixture models is not an easy task. Traditionally, the expectation-maximization (EM) algorithm has been most commonly applied based on the work of [Dempster et al. \(1977\), w](#page-7-0)ho realized that a finite mixture model may always be expressed in terms of an incomplete data problem by introducing the allocations as missing data. However, there are several drawbacks in using likelihood approach [\(McLachlan and Peel, 2000; Frühwirth-Schnatter,](#page-7-0) [2006\).](#page-7-0) The EM algorithm tends to lead to a local maximum and thus a grid of many different starting points is needed for finding the global maximum. It is also well known that the sample size has to be very large because the maximum likelihood method is based on the asymptotic theory. Furthermore, the calculation of standard errors is not straightforward when the likelihood function has unusual features. For the discussion of comparing various estimation methods for the finite mixture models, see [Frühwirth-Schnatter \(2006,](#page-7-0) [pp. 49–56\).](#page-7-0)
+
+In this paper, a Bayesian sampling approach was adopted which provides much richer inference than the maximum likelihood approach in that it can address the issue of parameter uncertainty via full posterior distribution. Following the work of [Diebolt and](#page-7-0) [Robert \(1994\)](#page-7-0) (data augmentation and Gibbs sampling), Bayesian mixture models can be applied routinely when the number of components is assumed to be known. According to [Richardson and](#page-8-0) [Green \(1997\), B](#page-8-0)ayesian method is the only sensible way if the number of mixture components is allowed to vary. However, this paper does not intend to address the mixture models with a varying number of components. Estimating the number of components (*K*) is a special kind of model choice problem, for which there is a number of possible solutions: using information-based criteria; Bayes factors; reversible jump MCMC; and birth-and-death process, etc. The first two methods were adopted in this paper because they are relatively easy to implement and therefore widely used. The last two methods assume that *K* is not fixed but variable, and is estimated within the modeling process, which was beyond the scope of this paper. Interested readers are referred to [Richardson and Green \(1997\)](#page-8-0) and [Stephens \(2000a\). I](#page-8-0)nstead, to determine the number of components in the mixture, a series of models with increasing numbers of components are fitted and then the most plausible model is selected by various model selection criteria which will be described later.
+
+## *3.1. Data augmentation and Gibbs sampling*
+
+The algorithm for the data augmentation and Gibbs sampling consists of three steps: first, the data are augmented with a latent random variable **z***<sup>i</sup>* = (*z*1,*i*, *z*2,*i*, ..., *zK*,*i*) which indicates the component membership of site *i*; second, conditional on **z***i*, the component parameters are drawn sequentially from the full conditional posterior distributions; third, conditional on knowing the component parameters, each component indicator vector **z***<sup>i</sup>* is drawn from a multinomial distribution, satisfying<sup>K</sup> <sup>k</sup>=1zk,i <sup>=</sup> 1. For details about the data augmentation and Gibbs sampling, see [Dempster et al.](#page-7-0) [\(1977\)](#page-7-0) and [Diebolt and Robert \(1994\).](#page-7-0)
+
+For the FMNB-*K*regression model, it is assumed that the parameters -1, ..., -*<sup>K</sup>*, -1, ..., -*<sup>K</sup>* and **w** are, *a priori*, mutually independent. As priors for the regression coefficient *<sup>k</sup>* and for the inverse dispersion parameter *<sup>k</sup>*, multivariate normal distribution and gamma distribution were specified, respectively. For the weight distribution **w**, the *Dirichlet* (*e*0, ..., *e*0) was used as a prior. With these prior settings, the posterior distribution of **w** follows the *Dirichlet* (*e*<sup>0</sup> <sup>+</sup> *<sup>n</sup>*1, ..., *<sup>e</sup>*<sup>0</sup> <sup>+</sup> *nK*), where nk <sup>=</sup> <sup>N</sup> <sup>i</sup>=1zk,i denotes the number of observations allocated to component *k.* However, the conditional distributions for *<sup>k</sup>* and *<sup>k</sup>* do not belong to any standard distribution family. This paper used the Random-Walk Metropolis algorithm with a normal distribution as a proposal density. The acceptance rates were tuned to lie between 25% and 45%. This range of acceptance rates is generally accepted for Metropolis algorithms [\(Roberts, 1996\).](#page-8-0) The Software R ([R Development Core Team, 2006\)](#page-7-0) was used for coding the algorithm.
+
+## *3.2. Model selection*
+
+To determine the appropriate model and the number of components, various model selection criteria were examined: information-based criteria (AIC, BIC, and DIC) and Bayes factor via marginal likelihoods. The Akaike information criterion, or AIC is defined as −2*LL* + 2*p*, where *p* is the number of parameters in the model. It penalizes the models by the number of parameters included. Smaller values represent better overall fit. The Bayesian information criterion, or BIC is uses a penalty term of *p* log(*n*), where *n* is the total number of observation (868 in this case study). The BIC is more conservative than the AIC by requiring a greater improvement in fit before it will accept a more complex model [\(Burnham and Anderson, 2002\).](#page-7-0) The Deviance information criterion, or DIC is defined as Dˆ + 2(D¯ − Dˆ), where D¯ is the average of the deviance (−2*LL*) over the posterior distribution, and Dˆ is the deviance calculated at the posterior mean parameters. As with AIC and BIC, DIC uses pD = D¯ − Dˆ (effective number of parameters) as a penalty term on the goodness-of-fit. Differences in DIC from 5 to 10 indicate that one model is clearly better [\(Spiegelhalter et al.,](#page-8-0) [2002\).](#page-8-0)
+
+Formal Bayesian model assessment is based on the Bayes factor, *B*12, for comparing model *M*<sup>1</sup> to model *M*<sup>2</sup> after observing the data [\(Lewis and Raftery, 1997\).](#page-7-0) The Bayes factor is the ratio of the marginal likelihoods of the two models being compared (*B*<sup>12</sup> = *p*(**y**|*M*1)/*p*(**y**|*M*2)). However, in practice, computing Bayes factors for a particular set of models can be demanding because it requires either complicated multidimensional integrals or some kind of stochastic sampling from the prior distribution. For calculating the marginal likelihood, we adopted the method developed by [Lewis and Raftery \(1997\),](#page-7-0) who suggested using the posterior simulation output for the computation of the marginal likelihoods (so-called Laplace-Metropolis estimator). The approximation of the marginal likelihood is carried out on the logarithmic scale such as
+
+$$\log(p(\mathbf{y}|M)) \approx \frac{d}{2} \log(2\pi) + \frac{1}{2} \log\{|\mathbf{H}^*|\} + \log\{f(\mathbf{y}|\boldsymbol{\Theta}^*)\}$$
+
+$$+ \log\{\pi(\boldsymbol{\Theta}^*)\}$$
+(11)
+
+where *d* is the number of parameters, log{*f*(**y**|**-**\*)} is the loglikelihood of data at **-**\*, and log{(**-**\*)} is the log-likelihood of prior distribution at **-**\*. One way of estimating **-**\* is to find the value of  at which log{*f*(*y*|**-**\*)} + log{(**-**\*)} achieves its maximum from the posterior simulation output. |**H**\*| is the determinant of the variance–covariance matrix estimated from the Hessian at the posterior mode, and it is asymptotically equal to the posterior variance–covariance matrix. This can be estimated from the sample variance–covariance matrix of the posterior simulation output. Assuming that the prior probabilities for the competing models are equal, *B*<sup>12</sup> is expressed as follows:
+
+$$\log(B_{12}) = \log\{p(\mathbf{y}|M_1)\} - \log\{p(\mathbf{y}|M_2)\}$$
+(12)
+
+<span id="page-4-0"></span>According to [Kass and Raftery \(1995\), t](#page-7-0)he values between 3 and 20 are positive and the values between 20 and 150 are strong in support of model 1.
+
+# **4. Data and mean functional form**
+
+To test the applicability of the proposed model, data collected in 1995 at urban 4-legged signalized intersections in Toronto, Canada were used. There are two main reasons for selecting this dataset for this study. First, the data have been used extensively for various study purposes and has been found to be of relatively good quality ([Miaou and Lord, 2003; Lord et al., 2008; Lord, 2000; Persaud](#page-7-0) [et al., 2002\).](#page-7-0) Second, more importantly, despite many factors that may have influenced crash occurrences around and within intersections, many transportation safety analysts have often favored using traffic flow-only models over models with covariates, even though the former models may be affected by the omitted variables bias ([Hauer, 1997; Persaud et al., 2001\).](#page-7-0) They are often preferred over models that include several covariates because they can be easily re-calibrated when they are developed in one jurisdiction and applied to another ([Persaud et al., 2002; Lord and Bonneson,](#page-7-0) [2005\).](#page-7-0)
+
+As initially discussed by [Miaou and Lord \(2003\)](#page-7-0) and later confirmed by [Mitra and Washington \(2007\), t](#page-7-0)he un-modeled heterogeneity across sites might be structured spatially in some way, especially when a limited number of covariates are used in the model. This study speculates that part of the heterogeneity could come from the existence of the several different sub-populations. There are many evidences to support this speculation in the data. For example, the data were collected across different business environment (e.g. shopping centers, schools, office compounds, etc.). The data contain a mix of fixed and actuated traffic signals with permissive, semi-protected, and protected left turns. It also includes divided and undivided approaches with different speed limits and different number of approaching lanes. Therefore, once the mixture model is estimated, one can go back to the data and see if there are common traits among the different observations that have separated the dataset (if they are known).
+
+The summary statistics are provided in Table 1. It contains 868 intersections, which have a total of 10,030 reported crashes. Individual intersections experienced crashes from 0 to 54 crashes. Entering traffic volumes vary widely from intersections to intersections: from about 5469 to 72,178 vehicles/day for major approaches and from 53 to about 42,644 vehicles/day for minor approaches. For a detailed description of the dataset, the readers are referred to [Lord](#page-7-0) [\(2000\).](#page-7-0)
+
+Similar to previous research that made use of this dataset, the mean functional form for each component was the following:
+
+$$\mu_{k,i} = \beta_{k,0} F_{1i}^{\beta_{k,1}} F_{2i}^{\beta_{k,2}} \tag{13}$$
+
+where *k*,*<sup>i</sup>* is the *k*th component's estimated number of crashes for intersection *i*; *F*1*<sup>i</sup>* the entering flows in veh/day from the major approaches at intersection *i*; *F*2*<sup>i</sup>* the entering flows in veh/day from the minor approaches at intersection *i*; and, *<sup>k</sup>* = (ˇ*k*,0, ˇ*k*,1, ˇ*k*,2) the estimated regression coefficients for component *k*.
+
+**Table 1** Summary statistics for application dataset.
+
+|                     | Min.  | Max.   | Average  | Standard deviation |
+|---------------------|-------|--------|----------|--------------------|
+| Crashes             | 0     | 54     | 11.56    | 10.02              |
+| Major-approach AADT | 5,469 | 72,178 | 28044.81 | 10660.39           |
+| Minor-approach AADT | 53    | 42,644 | 11010.18 | 8599.40            |
+
+**Table 2** Posterior means, standard deviations, 95% credible intervals for Poisson mixtures.
+
+| Poisson mixtures                                                 | w                                      | Ln(ˇ0)                                           | ˇ1                                           | ˇ2                                         |   |
+|------------------------------------------------------------------|----------------------------------------|--------------------------------------------------|----------------------------------------------|--------------------------------------------|---|
+| Single component<br>Estimate<br>(Std. Dev.)<br>(2.5%)<br>(97.5%) | 1.0                                    | −10.2294<br>(0.2837)<br>(−10.7860)<br>(−9.6770)  | 0.6023<br>(0.0287)<br>(0.5471)<br>(0.6589)   | 0.7038<br>(0.0139)<br>(0.6765)<br>(0.7311) | – |
+| FMP-2                                                            |                                        |                                                  |                                              |                                            |   |
+| Component 1<br>Estimate<br>(Std. Dev.)<br>(2.5%)<br>(97.5%)      | 0.492<br>(0.038)<br>(0.418)<br>(0.566) | −9.3231<br>(0.4564)<br>(−10.2046)<br>(−8.4356)   | 0.6107<br>(0.0452)<br>(0.5230)<br>(0.6993)   | 0.6286<br>(0.0219)<br>(0.5842)<br>(0.6713) | – |
+| Component 2<br>Estimate<br>(Std. Dev.)<br>(2.5%)<br>(97.5%)      | 0.508<br>(0.038)<br>(0.434)<br>(0.581) | −11.4200<br>(0.6679)<br>(−12.7555)<br>(−10.1314) | 0.6031<br>(0.0639)<br>(0.4780)<br>(0.7309)   | 0.7871<br>(0.0337)<br>(0.7213)<br>(0.8552) | – |
+| FMP-3                                                            |                                        |                                                  |                                              |                                            |   |
+| Component 1<br>Estimate<br>(Std. Dev.)<br>(2.5%)<br>(97.5%)      | 0.114<br>(0.028)<br>(0.066)<br>(0.178) | −5.3116*<br>(2.9760)<br>(−11.0566)<br>(0.4982)   | 0.1021*<br>(0.2516)<br>(−0.4148)<br>(0.5684) | 0.6238<br>(0.1930)<br>(0.1764)<br>(0.9731) | – |
+| Component 2<br>Estimate<br>(Std. Dev.)<br>(2.5%)<br>(97.5%)      | 0.329<br>(0.049)<br>(0.229)<br>(0.424) | −9.0683<br>(0.5763)<br>(−10.1907)<br>(−7.9200)   | 0.6190<br>(0.0559)<br>(0.5114)<br>(0.7300)   | 0.6018<br>(0.0279)<br>(0.5465)<br>(0.6556) | – |
+| Component 3<br>Estimate<br>(Std. Dev.)<br>(2.5%)<br>(97.5%)      | 0.557<br>(0.044)<br>(0.467)<br>(0.640) | −11.0158<br>(0.6530)<br>(−12.3389)<br>(−9.7712)  | 0.6139<br>(0.0625)<br>(0.4905)<br>(0.7372)   | 0.7558<br>(0.0352)<br>(0.6920)<br>(0.8308) | – |
+
+<sup>\*</sup> Not significant at 5% significance level.
+
+# **5. Model results**
+
+This section describes the results of the analysis. We first fitted the data with increasing number of components; for Poisson mixtures (FMP), models with *K* = 2, 3, 4 were estimated, and for NB mixtures (FMNB), models with *K* = 2, 3 were fitted. For each model, total 3 × 10<sup>5</sup> MCMC iterations were used, keeping every 10th samples, and half the iterations were discarded (burn-in period). From the remained 15,000 samples, the posterior means, standard deviations and 95% credible intervals were calculated.
+
+**Table 3** Posterior means, standard deviations, 95% credible intervals for NB mixtures.
+
+| NB mixtures   | w                | Ln(ˇ0)     | ˇ1       | ˇ2       |           |  |  |
+|---------------|------------------|------------|----------|----------|-----------|--|--|
+|               | Single component |            |          |          |           |  |  |
+| Estimate      |                  | −10.2300   | 0.6190   | 0.6854   | 7.0894    |  |  |
+| (Std. dev.)   |                  | (0.4659)   | (0.0459) | (0.0216) | (0.6156)  |  |  |
+| (2.5%)        | 1.0              | (−11.1707) | (0.5296) | (0.6428) | (5.9590)  |  |  |
+| (97.5%)       |                  | (−9.3241)  | (0.7118) | (0.7273) | (8.3760)  |  |  |
+| Two component |                  |            |          |          |           |  |  |
+| Component 1   |                  |            |          |          |           |  |  |
+| Estimate      | 0.430            | −10.9407   | 0.8588   | 0.5056   | 9.3692    |  |  |
+| (Std. dev.)   | (0.153)          | (1.3641)   | (0.1595) | (0.0812) | (1.6220)  |  |  |
+| (2.5%)        | (0.150)          | (−13.8766) | (0.5991) | (0.3199) | (6.8739)  |  |  |
+| (97.5%)       | (0.731)          | (−8.3865)  | (1.2297) | (0.6384) | (12.9768) |  |  |
+| Component 2   |                  |            |          |          |           |  |  |
+| Estimate      | 0.570            | −9.7842    | 0.3987   | 0.8703   | 8.2437    |  |  |
+| (Std. dev.)   | (0.153)          | (1.0447)   | (0.1289) | (0.0782) | (1.3502)  |  |  |
+| (2.5%)        | (0.268)          | (−11.8434) | (0.1116) | (0.7445) | (6.0746)  |  |  |
+| (97.5%)       | (0.849)          | (−7.6601)  | (0.6181) | (1.0497) | (11.2873) |  |  |
+
+**Table 4** Model selection criteria.
+
+| Models           | No. of parameters | LL      | AIC    | BIC    | DIC    | log p(y Mk) |
+|------------------|-------------------|---------|--------|--------|--------|-------------|
+| Standard Poisson | 3                 | −2791.5 | 5587.1 | 5596.6 | 5589.3 | −2811.494   |
+| FMP-2            | 7                 | −2560.4 | 5134.7 | 5168.1 | 5134.6 | −2598.153   |
+| FMP-3            | 11                | −2529.5 | 5081.1 | 5133.5 | 5080.0 | −2582.033   |
+| FMP-4            | 15                | −2519.7 | 5069.4 | 5141.3 | 5060.4 | −2583.296   |
+| Standard NB      | 4                 | −2534.6 | 5077.3 | 5096.3 | 5077.3 | −2559.311   |
+| FMNB-2           | 9                 | −2525.9 | 5069.8 | 5112.6 | 5068.7 | −2569.193   |
+| FMNB-3           | 14                | −2518.6 | 5065.2 | 5131.9 | 5060.6 | −2570.023   |
+
+Convergence was checked by monitoring the trace plots of the samples, marginal posterior distributions of model parameters and the autocorrelations.
+
+The results for the Poisson mixtures (for *K* = 2 and 3) are given in [Table 2](#page-4-0) along with the standard Poisson regression model (one single distribution). The results for the NB mixtures (*K* = 2) and the standard NB regression model are provided in [Table 3.](#page-4-0) The results for FMP-4 and FMNB-3 are not presented in [Tables 2 and 3,](#page-4-0) because it was noticed that the marginal densities for all parameters were very wide with multiple modes and non-vanishing high autocorrelations. This indicated difficulties with convergence and the nonidentifiability of the models. Nonidentifiability of finite mixtures of regression models is caused not only by the invariance of a finite mixture distribution to relabeling the components (known as "label switching problem"), but also by potential over-fitting. When the models are unidentified, it is meaningless to draw inference directly from MCMC output using ergodic averaging [\(Frühwirth-](#page-7-0)Schnatter, [2006\).](#page-7-0)
+
+Table 4 shows the computed values of log-likelihood, AIC, BIC and DIC for each model. Log-likelihood, indicated as LL, was obtained from the greatest values from the posterior simulation output for the individual models. With the exception of the AIC criteria, none of the evaluation measures showed that Poisson mixtures performed better than the standard negative binomial regression.
+
+Based on the DIC criteria, FMP-4 or FMNB-3 appears to be the best model. However, for these two models, the MCMC output displayed non-vanishing autocorrelation with extremely high lags and the marginal distributions of the model parameters showed multiple modes, indicating difficulties with convergence and label switching problems. Thus, the DIC values from such models are not reliable and should not be used to compare models ([Spiegelhalter](#page-8-0) [et al., 2002\).](#page-8-0) Instead, FMNB-2 model could be selected as the best model. As shown in the trace plots in[Fig. 1](#page-6-0) (left-hand side), although there was evidence of label switching at the initial part of the simulation, after some period the samples appear to be performing well. It can be seen that the component-wise regression parameters are well separated. The marginal posterior distributions of regression parameters (after a burn-in period) in [Fig. 1](#page-6-0) (right-hand side) are very close to a normal distribution. For the parameters, it was observed that the occasional very large samples skewed the marginal posterior distribution to the right with a relatively long tail.
+
+Using the BIC and Bayes factor (marginal likelihood) criteria, the standard NB regression model could be selected as the best model. This suggests that FMNB-2 model did not make a significant improvement in terms of fit by adding additional parameters. Actually, for this particular dataset, the standard NB regression model itself produced a very satisfactory goodness-of-fit. This is why these criteria favored the simpler model. When the predicted crash mean values for each intersection were compared between the two models, there were little differences by showing almost same predictive capabilities. However, when the variances were compared, they were significantly reduced in FMNB-2 model, especially for those intersections with high mean values.
+
+**Table 5** Summary statistics for each group.
+
+|                     | Min.  | Max.   | Average  | Standard deviation |
+|---------------------|-------|--------|----------|--------------------|
+| Component 1         |       |        |          |                    |
+| Crashes             | 1     | 48     | 11.61    | 7.85               |
+| Major-approach AADT | 5,967 | 56,623 | 28724.45 | 10464.09           |
+| Minor-approach AADT | 53    | 30,824 | 7147.26  | 6173.03            |
+| Component 2         |       |        |          |                    |
+| Crashes             | 0     | 54     | 11.54    | 10.53              |
+| Major-approach AADT | 5,469 | 72,178 | 27861.98 | 10712.77           |
+| Minor-approach AADT | 823   | 42,644 | 12049.33 | 8863.41            |
+
+In summary, the four model selection criteria used in this research produced conflicting conclusions, especially, between the FMNB-2 and the standard NB models (if we exclude FMNB-3). As described above, it was confirmed that the BIC and Bayes factor are more conservative than AIC and DIC criteria (see [Burnham and](#page-7-0) [Anderson, 2002\).](#page-7-0) Even though the FMNB-2 was not supported by all selection criteria, the parameter estimate results [\(Fig. 1](#page-6-0) and [Table 3\)](#page-4-0) suggest that there is something going on in the dataset that remains unexplained by the standard NB model. Thus, in this case, the FMNB-2 should be the recommended model, since it provides additional information about the data. This is described in the next paragraph.
+
+Based on the FMNB-2 model, to study the effects of the traffic flow, the data were classified into two groups by assigning each site to the component with the highest posterior probability. When the component proportions **w** is given, the posterior probability that the observation *yi* belongs to one component is given by the following equation by Bayes' rule [\(Frühwirth-Schnatter, 2006\):](#page-7-0)
+
+$$Pr(z_{k,i}|\boldsymbol{\Theta}, \mathbf{x}_i, y_i) = \frac{p(y_i|\mathbf{x}_i, \boldsymbol{\beta}_k, \phi_k) \cdot w_k}{\sum_{j=1}^K p(y_i|\mathbf{x}_i, \boldsymbol{\beta}_j, \phi_j) \cdot w_j} \propto p(y_i|\mathbf{x}_i, \boldsymbol{\beta}_k, \phi_k) \cdot w_k,$$
+
+$$k = 1, \dots, K$$
+(14)
+
+The calculated posterior probabilities of components 1 and 2 were 21.2% (184 obs.) and 78.8% (684 obs.), respectively, and they were grouped accordingly. Table 5 shows the summary statistics for each group. Compared with [Table 1,](#page-4-0) there is a striking difference in the average value of minor road approaching AADT. Most of low minor-approach AADT was assigned to component 1 resulting in a low average value, and many of high minorapproach AADT is associated with component 2. Therefore, it is obvious that the variability in minor-approach AADT is one source of over-dispersion along with other unobserved variables.2 The next step would be to examine each group more closely to see if they have common characteristics, such as traffic light phasing scheme, intersection geometry, lane configuration or unique geographical locations among others. Unfortunately, due to the
+
+<sup>2</sup> Another possible explanation is that if we had formulated the FMNB-2 model with a variable weight by including the minor-approach AADT as a covariate, it might have improved the model. We thank a reviewer for this suggestion.
+
+<span id="page-6-0"></span>![](_page_6_Figure_2.jpeg)
+
+**Fig. 1.** MCMC trace plots and marginal posterior distributions.
+
+age of the dataset, we are not able to get access to this kind of information.
+
+## **6. Summary and discussion**
+
+This paper has proposed an alternative formulation that could capture the heterogeneity in crash count models through the use of finite mixture regression models. Seven models were estimated using data collected at signalized 4-legged intersections in Toronto, Ont. Overall, the results show that the standard NB and the FMNB-2 regression models performed almost similarly; each one being favored over the other based on some of the four evaluation criteria. Despite this mixed assessment, the parameter estimates of the FMNB-2 clearly show that important characteristics observed in the data are not captured by the standard NB model.
+
+Given this modeling outcome, two points needs to be discussed. First, it seems that the dataset may have been generated from two distinct sub-populations, with each population having its own regression coefficients and degrees of over-dispersion. Although over-dispersion in crash data can be dealt with in a variety of ways, the mixture model provided the nature of the over-dispersion in the data. This result partly supports those found by [Miaou and Song](#page-7-0) [\(2005\)](#page-7-0) who used the same data. They showed that the inclusion of a spatial effect (induced by omitted variables) in the model could significantly improve the overall goodness-of-fit of the model. Second, from the post analysis of the mixture model, we could partly associate the source of over-dispersion with the minor-approach AADT. This is important because those intersections with different characteristics of minor approach flows may warrant different interventions to improve their safety. If additional descriptive data such as signal types, intersection geometry or number of approach<span id="page-7-0"></span>ing lanes are available, this finding also makes it possible for us to go back to the data and relate those variables with the separation of the dataset. This approach would be advantageous over the analysis with arbitrarily grouped intersections.
+
+Another point that should be noted is the fact that the standard NB regression model can be misleading when the data were actually generated by two or more component finite mixtures of Poisson regression models. In line with this study, we tested this point with simulated data generated by two-component finite mixture Poisson distribution (not included in this paper). The standard NB regression model accounted for almost all over-dispersion existing in the data and provided a satisfactory goodness-of-fit. However, because of model misspecification, it failed intrinsically to capture the existence of coefficient heterogeneity among components and it had a very poor prediction performance as compared to the FMP-2. This supports the idea that the analysts should consider the application of the finite mixture models when the data are suspected to comprise observations from several sub-populations.
+
+Despite many advantages of the finite mixture models, there are still several unresolved issues especially in terms of model parameter estimation through a Bayesian approach. First, because of the nonidentifiability caused by the invariance to relabeling the components, numerous authors suggested different approaches for relabeling the MCMC draws; imposing constraints on the parameters [\(Richardson and Green, 1997\),](#page-8-0) clustering methods ([Stephens,](#page-8-0) [2000b\),](#page-8-0) or random permutation (Frühwirth-Schnatter, 2006). Judging from the literature review in this area, it seems that there is no consensus on using a unique method yet. This paper, instead, excluded the unidentified models (i.e. FMP-4 or FMNB-3) and confined the analyses to the models whose label switching problem can be easily corrected by inspecting the MCMC trace plots. This is an obvious limitation of this paper and further work should be carried out.
+
+Future work also includes the examining the performance of the finite mixture models when the data have small sample mean values and small sample sizes. Since the estimate of the dispersion parameter is significantly biased under this situation, the results from the finite mixture models can provide an alternative estimates for the dispersion parameters for each component. Regarding the mean functional form (Eq. [\(13\)\),](#page-4-0) it is quite simplistic and clearly has many important missing variables that are known to influence crash frequencies. In such case, one would expect that a more flexible modeling form suggested in this paper would naturally fit the data better. In this respect, the analysis can be extended to test the model with more varied datasets to examine whether the suggested model would work better in a more fully specified model (both for fixed and varying weights). Finally, developing a COM–Poisson mixture model may prove to be useful for analyzing motor vehicle crashes.
+
+#### **Acknowledgements**
+
+The authors would like to thank Dr. Fred L. Mannering and Ms. Nataliya Malyshkina from Purdue University for their helpful comments and suggestions on an earlier version of the paper.
+
+# **References**
+
+- Burnham, K.P., Anderson, D.R., 2002. Model Selection and Multi-Model Inference: A Practical Information Theoretic Approach. Springer-Verlag, Now York.
+- Cameron, A.C., Trivedi, P.K., 1998. Regression Analysis of Count Data. Cambridge University Press, Cambridge, UK.
+- Dempster, A.P., Laird, N.M., Rubin, D.B., 1977. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B 39, 1–37.
+- Diebolt, J., Robert, C.P., 1994. Estimation of finite mixture distributions through Bayesian sampling. Journal of the Royal Statistical Society, Series B 56, 363–375.
+
+- El-Basyouny, K., Sayed, T., 2006. Comparison of two negative binomial regression techniques in developing accident prediction models. Transportation Research Record 1950, 9–16.
+- Frühwirth-Schnatter, S., 2006. Finite Mixture and Markov Switching Models. Springer Series in Statistics. Springer, New York.
+- Frühwirth-Schnatter, S., Kaufmann, S., 2006. Model-based Clustering of Multiple Time Series. Research Report IFAS, [http://www.ifas.jku.at](http://www.ifas.jku.at/).
+- Geedipally, S.R., Lord, D., 2008. Effects of the varying dispersion parameter of Poisson-models on the estimation of confidence interval of crash prediction models. Transportation Research Record 2061, 46–54.
+- Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B., 2004. Bayesian Data Analysis, 2nd ed. Chapman & Hall/CRC.
+- Guo, J.Q., Trivedi, P.K., 2002. Flexible parametric models for long-tailed patent count distributions. Oxford Bulletin of Economics & Statistics 64 (1), 63–82.
+- Hauer, E., 1997. Observational Before–After Studies in Road Safety. Pergamon Press/Elsevier Science Ltd., Oxford, UK.
+- Hauer, E., 2001. Overdispersion in modeling accidents on road sections and in empir-
+- ical Bayes estimation. Accident Analysis and Prevention 33 (6), 799–808. Heydecker, B.G., Wu, J., 2001. Identification of sites for road accident remedial work by Bayesian statistical methods: an example of uncertain inference. Advances in
+- Engineering Software 32, 859–869. Hilbe, J.M., 2007. Negative Binomial Regression. Cambridge University Press, UK.
+- Kass, R.E., Raftery, A.E., 1995. Bayes factors and model uncertainty. Journal of the American Statistical Association 90, 773–795.
+- Land, K.C., McCall, P.L., Nagi, D.S., 1996. A comparison of Poisson, negative binomial, and semiparametric mixed Poisson regression models. Sociological Methods & Research 24 (4), 387–442.
+- Lewis, S.M., Raftery, A.E., 1997. Estimating Bayes factors via posterior simulation with the Laplace-Metropolis estimator. Journal of the American Statistical Association 92, 648–655.
+- Lord, D., 2000. The Prediction of Accidents on Digital Networks: Characteristics and Issues Related to the Application of Accident Prediction Models. PhD Dissertation. Department of Civil Engineering, University of Toronto, Toronto.
+- Lord, D., 2006. Modeling motor vehicle crashes using Poisson-gamma models: examining the effects of low sample mean values and small sample size on the estimation of the fixed dispersion parameter. Accident Analysis and Prevention 38 (4), 751–766.
+- Lord, D., Bonneson, J.A., 2005. Calibration of predictive models for estimating the safety of ramp design configurations. Transportation Research Record 1908, 88–95.
+- Lord, D., Guikema, S.D., Geedipally, S., 2008. Application of the Conway–Maxwell–Poisson generalized linear model for analyzing motor vehicle crashes. Accident Analysis and Prevention 40 (3), 1123–1134.
+- Lord, D., Park, P.Y.-J., 2008. Investigating the effects of the fixed and varying dispersion parameters of Poisson-gamma models on empirical Bayes estimates. Accident Analysis and Prevention 40 (4), 1441–1457.
+- Lord, D., Washington, S.P., Ivan, J.N., 2005. Poisson Poisson-gamma and zero inflated regression models of motor vehicle crashes: balancing statistical fit and theory. Accident Analysis and Prevention 37 (1), 35–46.
+- Lord, D., Washington, S.P., Ivan, J.N., 2007. Further notes on the application of zero inflated models in highway safety. Accident Analysis and Prevention 39 (1), 53– 57.
+- Maher, M., Summersgill, I., 1996. A comprehensive methodology for the fitting of predictive accident models. Accident Analysis and Prevention 28 (6), 281–296.
+- Malyshkina, N.V., Mannering, F.L., Tarko, A.P., 2009. Markov switching negative binomial models: An application to vehicle accident frequencies. Accident Analysis and Prevention 41 (2), 217–226.
+- McLachlan, G., Peel, D., 2000. Finite Mixture Models. John Wiley & Sons Inc.
+- Miaou, S.-P., Lord, D., 2003. Modeling traffic crash-flow relationships for intersections: dispersion parameter, functional form, and Bayes versus empirical Bayes methods. Transportation Research Record 1840, 31–40.
+- Miaou, S.-P., Song, J.J., 2005. Bayesian ranking of sites for engineering safety improvements: decision parameter, treatability concept, statistical criterion, and spatial dependence. Accident Analysis and Prevention 37 (4), 699– 720.
+- Miranda-Moreno, L.F., Fu, F.F., Saccomanno, L., Labbe, A., 2005. Alternative risk models for ranking locations for safety improvement. Transportation Research Record 1908, 1–8.
+- Mitra, S., Washington, S., 2007. On the nature of over-dispersion in motor vehicle crash prediction models. Accident Analysis and Prevention 39 (3), 459– 468.
+- Park, B.-J., Lord, D., 2008. Adjustment for the maximum likelihood estimate of the negative binomial dispersion parameter. Transportation Research Record 2061, 9–19.
+- Persaud, B.N., Lord, D., Palminaso, J., 2002. Issues of calibration and transferability in developing accident prediction models for urban intersections. Transportation Research Record 1784, 57–64.
+- Persaud, B.N., Retting, R.A., Gårder, P.E., Lord, D., 2001. Safety effect of roundabout conversions in the United States: empirical Bayes observational before–after study. Transportation Research Record 1751, 1–8.
+- Ramawamy, V., Anderson, E.W., DeSarbo, W.S., 1994. A disaggregate negative binomial regression procedure for count data analysis. Management Science 40 (3), 405–417.
+- R Development Core Team, 2006. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3- 900051-07-0; [http://www.R-project.org](http://www.r-project.org/) (accessed January 2008).
+
+- <span id="page-8-0"></span>Richardson, S., Green, P.J., 1997. On Bayesian analysis of mixtures with an unknown number of components (with discussion). Journal of the Royal Statistical Society, Series B 59, 731–792.
+- Roberts, G.O., 1996. Markov chain concepts related to sampling algorithms. In: Gilks, W.R., Richardson, S., Spiegelhalter, D.J. (Eds.), Markov Chain Monte Carlo in Practice. Chapman & Hall, London, pp. 45–57.
+- Scaccia, L., Green, P.J., 2003. Bayesian growth curves using normal mixtures with nonparametric weights. Journal of Computational and Graphical Statistics 12, 308–331.
+- Shankar, V.N., Albin, R.B., Milton, J.C., Mannering, F.L., 1998. Evaluating median crossover likelihoods with clustered accident counts: an empirical inquiry using the random effects negative binomial model. Transportation Research Record 1635, 44–48.
+- Spiegelhalter, D.J., Best, N.G., Carlin, B.P., van der Linde, A., 2002. Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society, Series B 64, 583–639.
+- Stephens, M., 2000a. Bayesian analysis of mixture models with an unknown number of components—an alternative to reversible jump methods. The Annals of Statistics 28, 40–74.
+- Stephens, M., 2000b. Dealing with label switching in mixture models. Journal of the Royal Statistical Society, Series B 62, 795–809.
+- Wang, P.M., Cockburn, I.M., Puterman, M.L., 1998. Analysis of patent data—a mixed Poisson regression model. Journal of Business and Economic Statistics 16 (1), 27–41.
+- Wood, G.R., 2002. Generalized linear accident models and goodness of fit testing. Accident Analysis and Prevention 34 (1), 417–427.

Zotero/001_artiklid/Road Safety Indicators/pathivadainvestigatingmotorcyclecrashespre2024.md

@@ -0,0 +1,547 @@
+---
+category: literaturenote
+citekey: pathivadainvestigatingmotorcyclecrashespre2024
+title: Investigating motorcycle crashes pre- and post-COVID-19 pandemic along Kentucky’s urban roadway segments
+authors: "Pathivada, Bharat Kumar; Banerjee, Arunabha; Haleem, Kirolos"
+year: 2024
+date: 2024-00-00 2024
+doi: 10.1016/j.jsr.2024.09.002
+publication: Journal of Safety Research
+url: "https://doi.org/10.1016/j.jsr.2024.09.002"
+zotero_key: ES4ZCJ5T
+zotero_storage: MJBCGJAB
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Pathivada, Bharat Kumar"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+Contents lists available at [ScienceDirect](www.sciencedirect.com/science/journal/00224375)
+
+## Journal of Safety Research
+
+journal homepage: [www.elsevier.com/locate/jsr](https://www.elsevier.com/locate/jsr)
+
+![](_page_0_Picture_5.jpeg)
+
+# Investigating motorcycle crashes pre- and post-COVID-19 pandemic along Kentucky's urban roadway segments
+
+![](_page_0_Picture_7.jpeg)
+
+Bharat Kumar Pathivada , Arunabha Banerjee , Kirolos Haleem \*
+
+*Center for Transportation Research (CTR), School of Engineering & Applied Sciences, Western Kentucky University, 1906 College Heights Blvd, EBS 2122, Bowling Green, KY 42101, United States*
+
+#### ARTICLE INFO
+
+*Keywords:* Motorcycle Crashes COVID-19 Pandemic Urban Two-Lane Multilane Conway-Maxwell-Poisson Heterogeneous Conway-Maxwell-Poisson Empirical Bayes
+
+#### ABSTRACT
+
+*Introduction:* This study developed safety performance functions (SPFs) for motorcycle-related crashes along Kentucky's urban roadway segment facilities (separately along both two-lane undivided and multilane road segments) for each of the pre-COVID-19 pandemic (2015–2019) and post-pandemic (2020–2022) periods. A total of 2,963 motorcycle crashes over eight years (2015 through 2022) across 2,642 urban roadway segments and several road-specific features (e.g., shoulder width and annual average daily traffic "AADT") were used. *Method:*  The Conway-Maxwell-Poisson (CMP) model and heterogeneous Conway-Maxwell-Poisson (HTCMP) model "that accounts for unobserved heterogeneity" were applied and compared. *Results:* For both urban two-lane undivided and multilane segment facilities, the HTCMP models outperformed their CMP counterparts pre- and postpandemic, based on various goodness-of-fit measures (e.g., Akaike information criterion "AIC" and pseudo R<sup>2</sup> ) and prediction performance (e.g., mean absolute deviance "MAD" and mean square prediction error "MSPE"). From the fitted SPFs, higher AADT and presence of horizontal curve were significantly associated with increased motorcycle crash frequency, while the presence of roadside steel strong-post W-beam guardrail was significantly associated with reduced motorcycle crash frequency along all urban segments, both pre- and post-pandemic. Furthermore, posted speed limits (≥55 mph) were significantly associated with reduced motorcycle crashes along urban multilane segments, while collector and local road types were significantly associated with reduced motorcycle crashes along urban two-lane segments. *Practical Applications:* Countermeasures were then proposed to improve motorcyclists' safety along urban roadways (e.g., installing advance warning signs and/or chevrons along urban roadway segments with sharp horizontal curves).
+
+## **1. Introduction**
+
+According to the National Highway Traffic Safety Administration (NHTSA), motorcyclists are overrepresented in traffic fatalities in the United States, with 3.5% of registered motorcycles contributing to 14% of all traffic fatalities and 17% of all motor-vehicle occupant fatalities ([NHTSA, 2023a](#page-12-0)). Motorcyclists are highly susceptible to traffic casualties as they are vulnerable road users, primarily due to the inherent instability of motorcycles and the lack of in-vehicle protection for riders. For example, in the United States in 2021, the fatality rate for motorcyclists per 100 million vehicle miles traveled was 24 times and 40 times the occupant fatality rate of passenger cars and light trucks, respectively ([NHTSA, 2023a\)](#page-12-0). In Kentucky, less than 1% of registered motorcycles contributed to more than 7% of traffic fatalities [\(KTC, 2023\)](#page-11-0). Furthermore, the emergence of the severe acute respiratory syndrome (SARS-
+
+CoV-2), commonly referred to as the 2019 novel coronavirus (COVID-19), has led to drastic changes and disruptions in travel behavior and traffic safety. Several studies have highlighted reduced traffic flow while road fatalities and unsafe driving behaviors significantly increased during the pandemic (e.g., [Chand et al., 2021; Yasin et al., 2021; Shaik](#page-11-0) & [Ahmed, 2022](#page-11-0)).
+
+A preliminary investigation of motorcycle crash density in Kentucky during pre- and post-pandemic periods revealed that motorcycle crashes were highly concentrated in urban areas (refer to [Fig. 1](#page-1-0)a and b). Please note that the term "post-COVID-19" (or post-pandemic) in this study implies "during and post COVID-19." It is evident from [Fig. 1](#page-1-0) that, in both pre- and post-pandemic periods, motorcycle crashes were highly concentrated in urban metropolitan areas in Kentucky (e.g., Louisville, Lexington, and Covington). This is expected since urban areas have significant traffic exposure compared to rural areas. According to the
+
+*E-mail addresses:* [bharatkumar.pathivada@wku.edu](mailto:bharatkumar.pathivada@wku.edu) (B.K. Pathivada), [arunabha.banerjee@wku.edu](mailto:arunabha.banerjee@wku.edu) (A. Banerjee), [kirolos.haleem@wku.edu](mailto:kirolos.haleem@wku.edu) (K. Haleem).
+
+<sup>\*</sup> Corresponding author.
+
+<span id="page-1-0"></span>![](_page_1_Figure_2.jpeg)
+
+**Fig. 1.** Motorcycle Crash Density in the State of Kentucky: (*a*) Pre-COVID-19 Pandemic (2015–2019) and (*b*) Post-COVID-19 Pandemic (2020–2022).
+
+recent NHTSA report, motorcyclist fatalities in urban areas increased by 15% since 2020 [\(NHTSA, 2023b](#page-12-0)).
+
+Despite motorcycle crashes accounting for a large percentage of traffic fatalities in the United States, studies that developed motorcyclespecific safety performance functions (SPFs) are relatively limited. Considering the relevance of motorcycle crashes and the impact of the pandemic, this study comprehensively investigates motorcycle crash frequency along urban roadway segment facilities in the state of Kentucky in each of the pre- and post-pandemic periods. The study objectives are three-fold: (i) develop motorcycle-specific SPFs along urban two-lane undivided and multilane road segments, separately for each of the pre-pandemic (2015–2019) and post-pandemic (2020–2022) periods; (ii) rank motorcycle-specific high-crash locations along each of the urban two-lane undivided and urban multilane segments using the Empirical Bayes (EB) method; and (iii) propose targeted safety countermeasures to enhance motorcyclists' safety along Kentucky's urban roadways. To comprehensively investigate motorcycle crashes while considering the effect of COVID-19, this study analyzes motorcycle crashes during the pre-COVID-19 pandemic (2015–2019) and post-COVID-19 pandemic (2020–2022) periods separately.
+
+## **2. Literature review**
+
+## *2.1. Motorcycle-related safety studies*
+
+Scanning the safety literature, studies on motorcycle safety have mostly focused on analyzing crash injury severity. For example, [Santos](#page-12-0)  [et al. \(2023\)](#page-12-0) investigated factors associated with motorcyclist injury severity while comparing several machine-learning algorithms. Tenyear crash records in Portugal between 2010 and 2019 were used. The results indicated that factors including alcohol consumption, road type, road condition, location, motorcycle age, and rider's gender were significantly associated with motorcyclist injury severity. Recently, [Zubaidi et al. \(2023\)](#page-12-0) investigated factors contributing to motorcyclist crash injury severity using the random parameters logit model with heterogeneity in means and variance. A total of 10,067 motorcycle crashes in Florida involving senior motorcyclists (≥65 years old)
+
+between 2012 and 2018 were used. The results showed that driver, temporal, spatial, roadway, and driver features were significantly associated with motorcyclist injury severity. Other motorcycle-related crash injury severity studies can be found in [Cheng et al. \(2017\),](#page-11-0)  [Rezapour et al. \(2021\)](#page-11-0), and [Islam \(2021\).](#page-11-0)
+
+Within the safety literature, there are limited studies that have examined motorcycle crash frequencies in the United States. [Harnen](#page-11-0)  [et al. \(2006\)](#page-11-0) employed a negative binomial (NB) model to investigate motorcycle crash frequency at urban intersections in Malaysia. Motorcycle crashes at 104 signalized and unsignalized intersections between 1997 and 2000 were used. The results showed that the number of vehicles entering the intersection, approach speed, lane width, number of lanes, control type, shoulder width, and land use type were significant in predicting motorcycle crash frequency. [Haque et al. \(2010\)](#page-11-0) investigated motorcycle crash frequency at signalized intersections using Bayesian hierarchical models. A total of 2,348 motorcycle crashes at four-legged and T-signalized intersections in Singapore between 2003 and 2006 were used. The number of lanes, presence of wider medians, and uncontrolled left-turn lanes at major roads were significantly associated with increased motorcycle crashes at the four-legged intersections. On the other hand, the presence of exclusive right-turn lanes at both major and minor roads significantly increased motorcycle crashes at Tsignalized intersections in Singapore.
+
+[Manan et al. \(2013\)](#page-12-0) developed SPFs for fatal motorcycle crashes on Malaysian primary roads using the standard NB model. Fatal motorcycle crashes over 124 road sections between 2007 and 2009 were used. The results showed that motorcycle fatalities per kilometer were affected by the average daily number of motorcycles and number of access points per kilometer. [Flask et al. \(2014\)](#page-11-0) investigated multivehicle-motorcycle crashes using the Bayesian negative binomial model. A total of 3,804 multivehicle motorcycle crashes on segments between 2006 and 2011 in Ohio were used. The findings showed that smaller lane and shoulder widths, increased horizontal degree of curvature, and vertical grade were significantly associated with crash frequency. [Lyon et al. \(2017\)](#page-12-0) developed SPFs for motorcycle crashes while applying the standard NB model. A total of five-year motorcycle crash records from Florida, Pennsylvania, and Virginia were used. It was concluded that potential bias from missing motorcycle AADT was negligible and total AADT was adequate for developing motorcycle SPFs.
+
+Recently, [Ospina-Mateus et al. \(2021\)](#page-12-0) developed motorcycle SPF using the standard NB model. A total of 2,884 motorcycle crashes between 2016 and 2017 in Cartagena, Columbia were used. The results showed that the average volume of daily motorcyclists, commercial areas, number of access points per kilometer, and road type were significantly associated with motorcycle crash frequency.
+
+## *2.2. Safety investigations along urban facilities*
+
+[Sawalha and Sayed \(2001\)](#page-12-0) employed the NB model while estimating the safety performance of urban arterials in the Greater Vancouver Regional District, British Columbia, Canada using crash data on 58 arterials between 1994 and 1996. The results revealed that section length, traffic volume, intersection density, driveway density, pedestrian crosswalk density, number of traffic lanes, median type, and land use were significantly associated with crash frequency on urban arterials. [Zhang et al. \(2012\)](#page-12-0) employed the generalized additive model to predict crash frequency on different urban roadway segments (urban freeway mainlines, freeway system interchange areas, urban arterials, and urban frontage roads). A total of 46,813 "KAB" crash records between 1996 and 2000 in Houston, Texas were used. The results showed a linear relationship between crash frequency and traffic flow for urban freeway mainlines, freeway system interchange areas, and arterial segments. However, at other urban facilities, a non-linear relationship existed.
+
+[Cunto et al. \(2015\)](#page-11-0) evaluated the applicability and transferability of the Highway Safety Manual (HSM) SPFs for urban intersections in Fortaleza, Brazil. A total of 89 urban four-legged signalized and 92 urban
+
+unsignalized intersections were investigated. The results indicated that the baseline HSM SPFs can be cautiously applied to Brazilian roads. [Alkaaf and Abdel-Aty \(2015\)](#page-11-0) developed local SPFs using an NB model for urban four-lane divided road segments with angle parking in Riyadh, Saudi Arabia. The results showed that fatal and injury crash frequency on Riyadh's urban four-lane divided road segments were best described as a function of AADT, segment length, speed limit, and driveway density.
+
+[Al-Omari et al. \(2021\)](#page-11-0) developed simple and full SPFs based on areatype classification in Florida between 2015 and 2019 while applying the standard NB model. The results showed that crash frequency was influenced by AADT, presence of signalized intersections, speed limit, and shoulder width. Recently, [Alkaaf and Abdel-Aty \(2022\)](#page-11-0) investigated the transferability of SPFs for urban four-lane divided roadways in Muscat, using a total of 627 crashes during the 2010–2011 period. The study results indicated that HSM-calibrated SPFs provided the best fit for the crash data in the city of Muscat. Furthermore, crash frequency on urban four-lane divided roadways was best predicted as a function of AADT, segment length, and speed limit.
+
+## *2.3. Study contribution to the motorcycle safety state-of-the-art*
+
+A comprehensive review of motorcycle safety studies revealed that, to the best of the authors' knowledge, limited studies have developed motorcycle crash-specific SPFs in the United States, especially along urban roadway segments and for each of the pre-pandemic and postpandemic periods separately. This study takes the initiative and develops motorcycle-specific SPFs along each of the urban two-lane undivided and urban multilane roadway segments in Kentucky. Furthermore, considering the significant impact of COVID-19 on traffic patterns, this study separately investigates motorcycle SPFs during the pre-pandemic (2015–2019) and post-pandemic (2020–2022). The study findings will aid in more effective countermeasure selection to enhance motorcyclist safety along urban roads, particularly post the COVID-19 pandemic.
+
+## **3. Methodology**
+
+## *3.1. Conway-Maxwell-Poisson (CMP) model*
+
+Despite being widely used as an SPF tool, the NB model is limited to over-dispersed count data (i.e., when crash variance is greater than its mean). In this study, motorcycle crash variance was found to be less than its mean (i.e., under-dispersed data), so a more flexible Conway-Maxwell-Poisson (CMP) distribution was used as an SFP tool. The CMP (or COM-Poisson) distribution generalizes the well-known Poisson distribution and is suitable for fitting both over-dispersed and underdispersed count data [\(Shmueli et al., 2005](#page-12-0)). The probability function of a CMP distribution is given as follows ([Shmueli et al., 2005; Sellers](#page-12-0) & [Shmueli, 2010\)](#page-12-0):
+
+$$P(Y = y) = \frac{\lambda^{y}}{(y!)^{y}Z(\lambda, y)}, \ y = 0, 1, 2, \dots$$
+ (1)
+
+$$Z(\lambda, \nu) = \sum_{n=0}^{\infty} \frac{\lambda^n}{(n!)^{\nu}}$$
+ (2)
+
+where: *Y* is the discrete count, *λ* is a centering parameter that approximates the mean of the observations, *Z*(*λ, v*) is a normalizing constant, and *v* is the dispersion parameter, such that *v >* 1 represents under-dispersion and *v <* 1 represents over-dispersion. The CMP distribution does not have closed-form expressions for its moments in terms of the parameters *λ* and *v*. Using an asymptotic approximation for *Z*(*λ, v*), the expected value and variance can be approximated as follows ([Sellers](#page-12-0)  & [Shmueli, 2010; Sellers et al., 2012\)](#page-12-0):
+
+$$E[Y] = \frac{\partial log Z(\lambda, \nu)}{\partial log \lambda} \approx \lambda^{1/\nu} - \frac{\nu - 1}{2\nu}$$
+(3)
+
+$$Var[Y] = \frac{\partial E[Y]}{\partial log\lambda} \approx \frac{1}{\nu} \lambda^{1/\nu}$$
+ (4)
+
+## *3.2. Heterogeneous Conway-Maxwell-Poisson (HTCMP) model*
+
+An extension of the CMP model is the heterogeneous CMP (or HTCMP) model, which accounts for unobserved heterogeneity by allowing the dispersion parameter to vary as a function of site-specific features. The HTCMP uses the same probability density function as the CMP model, where the dispersion parameter is expressed as a function of site-specific attributes, as follows:
+
+$$ln(v) = \gamma \psi_i \tag{5}$$
+
+where: *ψi*is the vector of site-specific characteristics and *γ* is the vector of regression coefficients corresponding to *ψ<sup>i</sup>* . Note that if none of the variables are statistically significant in the dispersion part, the HTCMP model reduces to the CMP model, where the dispersion parameter is fixed across the sites (or segments).
+
+#### *3.3. Model comparison and selection*
+
+For each of the CMP and HTCMP models, goodness-of-fit (GOF) measures, including deviance statistic (*D*), Akaike information criterion (AIC), Bayesian information criterion (BIC), and McFadden pseudo R2 were used to evaluate the overall fit and compare the fitted models. The GOF measures are calculated as follows:
+
+$$D = 2*(LL_0 - LL_\beta) \tag{6}$$
+
+$$AIC = -2*LL_{\beta} + 2*P \tag{7}$$
+
+$$BIC = -2*LL_{\beta} + P*ln(n) \tag{8}$$
+
+$$\textit{McFadden pseudo } R^2 = 1 - \left(\frac{LL_{\beta}}{LL_0}\right)$$
+
+where: *LL*0 is log-likelihood of the null model (i.e., constant-only model), *LLβ* is log-likelihood at convergence, *P* is the number of model parameters, and *n* is the number of observations (or segments). Models with lower AIC and BIC values and higher pseudo R2 imply better-fit ones.
+
+Furthermore, the likelihood ratio test (LRT) was also applied to evaluate the relative performance of the fitted models. The LRT is based on the differences in the log-likelihoods of the fitted models (*LL<sup>β</sup>*<sup>1</sup> *,LL<sup>β</sup>*<sup>2</sup> ), which follows a chi-squared distribution with degrees of freedom equal to the difference between the number of estimated parameters in both models (*k*).
+
+$$LRT = 2*(LL_{\beta_2} - LL_{\beta_1}) \cong \chi^2_{(df.=k)}$$
+(10)
+
+To evaluate the predictive performance of the fitted SPFs using the validation dataset, mean absolute deviance (MAD) and mean square prediction error (MSPE) were used. The model with lower MAD and MSPE values provides better prediction performance. These measures are calculated as follows:
+
+$$MAD = \frac{\sum_{i=1}^{n} |\mu_i - y_i|}{n}$$
+ (11)
+
+$$MSPE = \frac{\sum_{i=1}^{n} (\mu_i - y_i)^2}{n}$$
+ (12)
+
+where: *yi* and *μi*are observed and predicted number of crashes at urban road segment *i*, respectively, and *n* is the number of sites (or segments) in the validation dataset.
+
+## *3.4. Empirical Bayes (EB) method*
+
+The Empirical Bayes (EB) method, widely employed in safety studies to rank high-crash locations, was used to rank motorcycle-specific highcrash urban road segments. The expected number of crashes is calculated as the weighted average of the observed and predicted number of crashes estimated by the fitted SPF model. The EB estimate (*NE*) is calculated as follows:
+
+$$EB = N_E = w^*N_P + (1 - w)^*N_O$$
+(13)
+
+$$w = \frac{1}{1 + \nu^* N_P} \tag{14}$$
+
+where: *NP* is the precited number of crashes estimated by the fitted SPF, *NO* is the number of observed crashes, *v* is the dispersion parameter from the SPF model, and *w* is the weighting factor. It is important to note that for the HTCMP model, the weighting factor (*w*) will vary by segment *i*, as a function of varying dispersion parameters (*vi*).
+
+High-crash segments were then ranked in descending order based on the excess expected crashes (EEC) (i.e., difference between the expected and predicted crashes at the sites), as follows:
+
+$$EEC = N_E - N_P (15)$$
+
+## **4. Data collection and processing**
+
+The used data were collected from two separate databases provided by the Kentucky Transportation Cabinet (KYTC). These include the crash database (consisting of eight-year "2015–2022" motorcycle-related crashes) and the roadway inventory database extracted from KYTC's Highway Information Extract View and Interface (HIEVi) ([KYTC, 2023](#page-11-0)). These two databases were then merged based on the roadway ID and milepost of each crash. Since this study focuses on investigating motorcycle crashes along urban roadway segments, crashes on rural roadways and other facilities, such as intersections and ramps, were excluded from the analysis.
+
+To investigate crash frequency along road segments, the road sections were divided into homogeneous segments based on the homogeneity of roadway geometric and traffic characteristics. In this regard, area type (rural or urban), roadway type (undivided or divided), number of traffic lanes, posted speed limit (PSL), and AADT were used to produce homogeneous segments. A homogeneous segment ended where at least one of these five features changed, and a new homogeneous segment subsequently started.
+
+Fig. 2 illustrates imaginary homogeneous segments produced based on changes in roadway type, PSL, and AADT. For example, homogeneous segment 1 concluded where AADT and PSL changed from 1,500 to 1,200 and 55 mph to 35 mph, respectively, while other roadway features (i.e., urban area, undivided roadway, and four traffic lanes) remained unchanged. Similarly, homogeneous segment 2 ended where PSL and roadway type changed from 35 mph to 25 mph and undivided to divided, respectively, while the other roadway characteristics remained consistent. The segment length for each homogeneous segment was computed as the difference between each segment's beginning and ending milepost. It is important to note that any segments shorter than 0.1 miles were excluded to avoid the issue of low exposure ([Hosseinpour](#page-11-0)  [et al., 2022\)](#page-11-0).
+
+As mentioned earlier, eight-year crash records were analyzed separately in two periods, i.e., pre-pandemic (2015–19) and post-pandemic (2020–2022), and the segmentation process was done separately for pre- and post-pandemic crash data. A total of 1,676 homogeneous segments were produced for the pre-pandemic period, with lengths varying from 0.101 to 4.65 miles and an average segment length of 0.79 miles. Of the 1,676 segments, 830 segments were urban two-lane undivided segments, and the remaining 846 segments were urban multilane ones. Similarly, a total of 966 homogeneous segments were produced for the post-pandemic period, with lengths varying from 0.1 to 4.65 miles and an average segment length of 0.80 miles. Of the 966 segments, 482 segments were urban two-lane undivided segments, and the remaining 484 segments were urban multilane ones. Note that approximately 20% of the total segments, i.e., 335 and 193 in the pre- and post-pandemic periods, respectively, were zero motorcycle-crash urban segments randomly collected across Kentucky to develop valid SPFs. Furthermore, for each roadway facility, homogeneous segments were randomly split into calibration (75%) and validation (25%) datasets, where the former was used to develop the motorcycle SPFs, and the latter was used to evaluate the prediction performance of the fitted SPFs.
+
+In total, 2,963 motorcycle crashes along urban roadway segments were analyzed. Of these, 1,272 crashes (841 pre-pandemic and 431 postpandemic) occurred on two-lane roadways, while 1,691 crashes (1,167 pre-pandemic and 524 post-pandemic) occurred on multilane roadways. Overall, the annual average crash frequencies decreased post-pandemic
+
+![](_page_3_Figure_16.jpeg)
+
+**Fig. 2.** Illustration Sketch Showing Homogeneous Urban Roadway Segments.
+
+across both urban two-lane and urban multilane segments. This reduction was expected, mainly due to reduced traffic volumes during and post-pandemic, which aligns with findings from other studies (e.g., Islam et al., 2022; Shaik & [Ahmed, 2022](#page-12-0)). Fig. 3 shows the histograms of motorcycle crash frequencies in the pre- and post-pandemic periods by the urban facility type (i.e., urban two-lane undivided and urban multilane roadway segments). As observed in Fig. 3a and b, both urban two-lane undivided and multilane segments had the highest representation of "one crash count" in each of the pre-pandemic and postpandemic periods. Thus, zero-inflated models were not the ideal modeling solution and were not used in the study. [Table 1](#page-5-0) shows descriptive statistics of the explanatory variables used in the SPF development.
+
+## **5. SPF results along urban two-lane undivided roadway segments**
+
+## *5.1. Motorcycle SPFs' interpretations*
+
+[Table 2](#page-6-0) shows the results of CMP and HTCMP SPF models for motorcycle crashes along urban two-lane undivided roadway segments for each of the pre- and post-pandemic periods. All the fitted models had significant deviance statistics, rejecting the hypothesis that they performed similarly to their constant-only models. The LRTs comparing the fitted HTCMP versus CMP models for each of the pre- and postpandemic periods indicated that the HTCMP models outperformed their respective CMP models at the 0.1% level of significance (P-value *<* 0.001).
+
+Overall, the GOF measures including Pseudo R2 and AIC suggested that HTCMP models provided better fit compared to their CMP counterparts. In addition, the HTCMP models had the lowest MAD and MSPE values based on the model validation performance, indicating better prediction capability. The superiority of the HTCMP models indicated that this model, with a varying dispersion parameter, could appropriately capture the under-dispersion in motorcycle crashes along urban two-lane undivided segments. For example, the HTCMP models for both pre- and post-pandemic periods showed significant roadway parameters (e.g., right shoulder width and presence of horizontal curve) in the dispersion part, which explained the under-dispersion.
+
+From [Table 2](#page-6-0), the model results showed that motorcycle crashes (pre- and post-pandemic) along urban two-lane undivided road segments were significantly associated with AADT, road functional class (collector and local roads), presence of horizontal curve, and presence of roadside guardrail. Pre-pandemic, posted speed limit (15–35 mph) was found to be a significant factor and was associated with a reduced motorcycle crash frequency. However, post-pandemic, posted speed
+
+![](_page_4_Figure_9.jpeg)
+
+**Fig. 3.** Motorcycle Crash Frequencies along: (*a*) Urban Two-Lane Undivided Roadway Segments and (*b*) Urban Multilane Roadway Segments.
+
+<span id="page-5-0"></span>**Table 1**  Descriptive Statistics of Explanatory Variables for Urban Segments Pre- and Post-Pandemic.
+
+| Variable Description                       |                          |       | Pre-Pandemic (2015–2019) |      |                 |      |       |                          |      |      | Post-Pandemic (2020–2022) |                 |      |       |       |       |  |
+|--------------------------------------------|--------------------------|-------|--------------------------|------|-----------------|------|-------|--------------------------|------|------|---------------------------|-----------------|------|-------|-------|-------|--|
+|                                            | Urban Two-Lane Undivided |       |                          |      | Urban Multilane |      |       | Urban Two-Lane Undivided |      |      |                           | Urban Multilane |      |       |       |       |  |
+|                                            | Min.                     | Max.  | Mean                     | SD   | Min.            | Max. | Mean  | SD                       | Min. | Max. | Mean                      | SD              | Min. | Max.  | Mean  | SD    |  |
+| Continuous variables (Calibration dataset) |                          |       |                          |      |                 |      |       |                          |      |      |                           |                 |      |       |       |       |  |
+| Natural logarithm of AADT                  | 4.04                     | 10.08 | 8.34                     | 0.96 | 7.04            | 12   | 9.92  | 0.8                      | 4.79 | 10.1 | 8.45                      | 0.87            | 7.64 | 12.04 | 9.91  | 0.79  |  |
+| Lane width (in feet)                       | 6                        | 21    | 10.91                    | 1.74 | 9               | 13   | 11.68 | 0.7                      | 8    | 20   | 10.95                     | 1.69            | 10   | 13    | 11.7  | 0.64  |  |
+| Right shoulder width (in feet)             | 0                        | 12    | 2.99                     | 2.64 | 0               | 16   | 5.63  | 4.4                      | 0    | 11   | 2.86                      | 2.62            | 0    | 13    | 5.3   | 4.34  |  |
+| Number of lanes                            | 2                        | 2     | 2                        | 0    | 3               | 12   | 4.41  | 1.2                      | 2    | 2    | 2                         | 0               | 3    | 11    | 4.41  | 1.09  |  |
+| Median width (in feet)*                    | na**                     | na    | na                       | na   | 0               | 99   | 16.71 | 21                       | na   | na   | na                        | na              | 0    | 99    | 15.48 | 18.34 |  |
+
+| Variable                                            | Variable Description                                                    |                             | Pre-Pandemic (2015–2019)        |                 |                                 |                 | Post-Pandemic (2020–2022)       |                    |                                 |
+|-----------------------------------------------------|-------------------------------------------------------------------------|-----------------------------|---------------------------------|-----------------|---------------------------------|-----------------|---------------------------------|--------------------|---------------------------------|
+|                                                     |                                                                         | Urban Two-Lane<br>Undivided |                                 |                 | Urban Multilane                 | Undivided       | Urban Two-Lane                  | Urban Multilane    |                                 |
+|                                                     |                                                                         | No. of<br>Sites             | No. of<br>Motorcycle<br>Crashes | No. of<br>Sites | No. of<br>Motorcycle<br>Crashes | No. of<br>Sites | No. of<br>Motorcycle<br>Crashes | No.<br>of<br>Sites | No. of<br>Motorcycle<br>Crashes |
+| Calibration dataset                                 | Number of sites and<br>motorcycle crashes in the<br>calibration dataset | 622                         | 628                             | 634             | 876                             | 361             | 319                             | 363                | 383                             |
+| Validation dataset                                  | Number of sites and<br>motorcycle crashes in the<br>validation dataset  | 208                         | 213                             | 212             | 291                             | 121             | 112                             | 121                | 141                             |
+|                                                     | Indicator dummy variables (Calibration dataset)                         |                             |                                 |                 |                                 |                 |                                 |                    |                                 |
+| Paved shoulder                                      | If true = 1, otherwise = 0                                              | 541                         | 568                             | 626             | 876                             | 225             | 212                             | 361                | 383                             |
+| Horizontal curve                                    | If present = 1, otherwise =<br>0                                        | 253                         | 296                             | 164             | 231                             | 148             | 153                             | 92                 | 96                              |
+| Vertical grade                                      | If present = 1, otherwise =<br>0                                        | 205                         | 206                             | 233             | 331                             | 179             | 167                             | 138                | 153                             |
+| Roadside guardrail<br>(steel strong-post<br>W-beam) | If present = 1, otherwise =<br>0                                        | 57                          | 42                              | 152             | 156                             | 29              | 12                              | 86                 | 68                              |
+
+| Variable                             | Variable                      |                 | Pre-Pandemic (2015–2019)     |                 |                              | Post-Pandemic (2020–2022) |                              |                    |                              |  |  |
+|--------------------------------------|-------------------------------|-----------------|------------------------------|-----------------|------------------------------|---------------------------|------------------------------|--------------------|------------------------------|--|--|
+|                                      | Description                   |                 | Urban Two-Lane Undivided     | Urban Multilane |                              |                           | Urban Two-Lane Undivided     |                    | Urban Multilane              |  |  |
+|                                      |                               | No. of<br>Sites | No. of Motorcycle<br>Crashes | No. of<br>Sites | No. of Motorcycle<br>Crashes | No. of<br>Sites           | No. of Motorcycle<br>Crashes | No.<br>of<br>Sites | No. of Motorcycle<br>Crashes |  |  |
+| Flat terrain type                    | If true = 1,<br>otherwise = 0 | 220             | 222                          | 272             | 389                          | 142                       | 128                          | 167                | 181                          |  |  |
+| Road Functional Class:               |                               |                 |                              |                 |                              |                           |                              |                    |                              |  |  |
+| Interstate/<br>principal<br>arterial | If true = 1,<br>otherwise = 0 | 20              | 31                           | 377             | 619                          | 8                         | 6                            | 194                | 239                          |  |  |
+| Minor arterial                       | If true = 1,<br>otherwise = 0 | 227             | 268                          | 215             | 277                          | 156                       | 158                          | 147                | 135                          |  |  |
+| Major collector                      | If true = 1,<br>otherwise = 0 | 269             | 260                          | 36              | 30                           | 148                       | 116                          | 20                 | 9                            |  |  |
+| Minor collector/<br>local            | If true = 1,<br>otherwise = 0 | 106             | 69                           | 6               | 0                            | 49                        | 39                           | 2                  | 0                            |  |  |
+| Undivided*                           | If true = 1,<br>otherwise = 0 | na              | na                           | 270             | 356                          | na                        | na                           | 151                | 163                          |  |  |
+| Posted Speed Limit:                  |                               |                 |                              |                 |                              |                           |                              |                    |                              |  |  |
+| ≤ 35 mph                             | If true = 1,<br>otherwise = 0 | 413             | 400                          | 162             | 190                          | 248                       | 211                          | 99                 | 99                           |  |  |
+| 40–50 mph                            | If true = 1,<br>otherwise = 0 | 111             | 125                          | 248             | 377                          | 72                        | 71                           | 151                | 153                          |  |  |
+| ≥ 55 mph                             | If true = 1,<br>otherwise = 0 | 98              | 103                          | 224             | 309                          | 41                        | 37                           | 113                | 131                          |  |  |
+| Type of Median Barriers:*            |                               |                 |                              |                 |                              |                           |                              |                    |                              |  |  |
+| Cable barriers                       | If true = 1,<br>otherwise = 0 | na              | na                           | 31              | 49                           | na                        | na                           | 15                 | 20                           |  |  |
+| Concrete barriers                    | If true = 1,<br>otherwise = 0 | na              | na                           | 63              | 109                          | na                        | na                           | 39                 | 53                           |  |  |
+| Other barriers                       | If true = 1,<br>otherwise = 0 | na              | na                           | 9               | 14                           | na                        | na                           | 7                  | 9                            |  |  |
+
+Note: Min. = minimum; Max. = maximum; SD=standard deviation; and AADT=annual average daily traffic.
+
+**Table 2**  Results of CMP and HTCMP Models for Motorcycle Crash SPFs along Urban Two-Lane Undivided Road Segments.
+
+| Variable                           |           | Pre-Pandemic (2015–2019) |            |             |               |            | Post-Pandemic (2020–2022) |               |            |             |               |            |  |
+|------------------------------------|-----------|--------------------------|------------|-------------|---------------|------------|---------------------------|---------------|------------|-------------|---------------|------------|--|
+|                                    | CMP Model |                          |            | HTCMP Model |               |            | CMP Model                 |               |            | HTCMP Model |               |            |  |
+|                                    | Coef.     | Std.<br>Error            | P<br>Value | Coef.       | Std.<br>Error | P<br>Value | Coef.                     | Std.<br>Error | P<br>Value | Coef.       | Std.<br>Error | P<br>Value |  |
+| Mean-Parameter Count Part          |           |                          |            |             |               |            |                           |               |            |             |               |            |  |
+| Constant                           | − 4.269   | 0.998                    | < 0.001    | − 3.432     | 0.896         | < 0.001    | − 4.974                   | 1.313         | < 0.001    | − 3.114     | 1.898         | 0.101      |  |
+| Ln(AADT)                           | 0.567     | 0.112                    | < 0.001    | 0.535       | 0.098         | < 0.001    | 0.713                     | 0.153         | < 0.001    | 0.572       | 0.210         | 0.007      |  |
+| Principal Arterial                 | 1.016     | 0.352                    | 0.004      | N/S*        | N/S           | N/S        | N/S                       | N/S           | N/S        | N/S         | N/S           | N/S        |  |
+| Major Collector                    | − 0.364   | 0.174                    | 0.037      | − 0.368     | 0.177         | 0.037      | − 0.717                   | 0.239         | 0.003      | − 1.736     | 0.400         | < 0.001    |  |
+| Minor Collector/Local              | − 0.900   | 0.274                    | 0.001      | − 0.946     | 0.271         | < 0.001    | N/S                       | N/S           | N/S        | − 1.785     | 0.557         | 0.001      |  |
+| Posted Speed Limit (15–35<br>mph)  | N/S       | N/S                      | N/S        | − 0.532     | 0.254         | 0.036      | N/S                       | N/S           | N/S        | N/S         | N/S           | N/S        |  |
+| Presence of Horizontal Curve       | 0.689     | 0.144                    | < 0.001    | 0.719       | 0.146         | < 0.001    | 1.077                     | 0.246         | < 0.001    | 1.600       | 0.351         | < 0.001    |  |
+| Presence of Roadside<br>Guardrails | − 0.951   | 0.255                    | < 0.001    | − 1.933     | 0.322         | < 0.001    | − 2.163                   | 0.452         | < 0.001    | − 2.530     | 0.494         | < 0.001    |  |
+| Dispersion Part                    |           |                          |            |             |               |            |                           |               |            |             |               |            |  |
+| Constant                           | 1.463     | 0.050                    | < 0.001    | 1.813       | 0.153         | < 0.001    | 1.801                     | 0.065         | <0.001     | 1.695       | 0.120         | < 0.001    |  |
+| Principal Arterial                 | na**      | na                       | na         | − 0.572     | 0.102         | < 0.001    | na                        | na            | na         | N/S         | N/S           | N/S        |  |
+| Major Collector                    | na        | na                       | na         | N/S         | N/S           | N/S        | na                        | na            | na         | − 0.340     | 0.124         | 0.006      |  |
+| Minor Collector/Local              | na        | na                       | na         | N/S         | N/S           | N/S        | na                        | na            | na         | − 0.633     | 0.187         | 0.001      |  |
+| Posted Speed Limit (15–35<br>mph)  | na        | na                       | na         | − 0.195     | 0.095         | 0.040      | na                        | na            | na         | 0.182       | 0.083         | 0.028      |  |
+| Right Shoulder Width               | na        | na                       | na         | 0.041       | 0.013         | 0.002      | na                        | na            | na         | N/S         | N/S           | N/S        |  |
+| Presence of Horizontal Curve       | na        | na                       | na         | N/S         | N/S           | N/S        | na                        | na            | na         | 0.270       | 0.112         | 0.016      |  |
+| Presence of Paved Shoulder         | na        | na                       | na         | − 0.244     | 0.123         | 0.047      | na                        | na            | na         | N/S         | N/S           | N/S        |  |
+| Presence of Roadside<br>Guardrails | na        | na                       | na         | − 0.863     | 0.206         | < 0.001    | na                        | na            | na         | 0.219       | 0.084         | 0.009      |  |
+| Presence of Gradient               | na        | na                       | na         | N/S         | N/S           | N/S        | na                        | na            | na         | N/S         | N/S           | N/S        |  |
+| Fixed Disp. Parameter              | 4.317     | 0.217                    | < 0.001    | na          | na            | na         | 6.057                     | 0.396         | < 0.001    | na          | na            | na         |  |
+
+|                                      | Pre-Pandemic (2015–2019)  |                            | Post-Pandemic (2020–2022) |                            |
+|--------------------------------------|---------------------------|----------------------------|---------------------------|----------------------------|
+|                                      | CMP Model                 | HTCMP Model                | CMP Model                 | HTCMP Model                |
+| Goodness-of-Fit (GOF) Statistics     |                           |                            |                           |                            |
+| Number of observations               | 622                       | 622                        | 361                       | 361                        |
+| Number of parameters                 | 8                         | 13                         | 6                         | 12                         |
+| Log-likelihood at convergence (LLβ)  | − 580.23                  | − 545.18                   | − 259.98                  | − 245.76                   |
+| Log-likelihood at zero (LL0)         | − 657.40                  | − 657.40                   | − 302.86                  | − 302.86                   |
+| Deviance (D) = (2*(LLβ − LL0))       | 154.34                    | 224.44                     | 85.76                     | 114.20                     |
+|                                      | (df = 6, P-Value < 0.001) | (df = 11, P-Value < 0.001) | (df = 4, P-Value < 0.001) | (df = 10, P-Value < 0.001) |
+| LRT (HTCMP vs CMP)                   | 70.10                     |                            | 28.44                     |                            |
+|                                      | (df = 5, P-Value < 0.001) |                            | (df = 6, P-Value < 0.001) |                            |
+| McFadden pseudo R2                   | 0.117                     | 0.171                      | 0.142                     | 0.189                      |
+| AIC (Akaike Information Criterion)   | 1176.46                   | 1116.36                    | 531.95                    | 515.52                     |
+| BIC (Bayesian Information Criterion) | 1211.93                   | 1173.98                    | 555.28                    | 562.19                     |
+| MAD (Mean Absolute Deviance)         | 0.508                     | 0.504                      | 0.372                     | 0.363                      |
+| MSPE (Mean Square Prediction Error)  | 0.497                     | 0.481                      | 0.315                     | 0.288                      |
+
+<sup>\*</sup> N/S=not statistically significant at 10% significance level.
+
+limit (15–35 mph) was not found significant along urban two-lane road segments.
+
+In the count part, higher AADT was associated with increased number of motorcycle crashes. This is expected since AADT reflects exposure and with higher exposure, motorcycle crash frequency increases. This finding is consistent with past studies (e.g., [Haque et al.,](#page-11-0)  [2010; Manan et al., 2013; Ospina-Mateus et al., 2021](#page-11-0)). Road functional class (major collector and minor collector/local roads) was associated with reduced motorcycle crash frequency. This could be since these roads have frequent stops and intersections, which compel motorcyclists to reduce speeds and proceed with caution. The presence of horizontal curve was associated with increased motorcycle crash frequency. This is plausible due to the heightened risk of losing motorcycle control at sharp curves/bends; thereby, increasing motorcycle crashes. This finding is consistent with [Flask et al. \(2014\).](#page-11-0) The presence of a roadside guardrail
+
+was associated with reduced motorcycle crash frequency. This is likely since motorcyclists perceive guardrails as a potential hazard [\(Michie](#page-12-0) & [Bronstad, 1994](#page-12-0)) and drive cautiously; thereby, reducing motorcycle crash frequency ([Elvik, 1995; Ben-Bassat](#page-11-0) & Shinar, 2011).
+
+Pre-pandemic, urban two-lane undivided road segments with posted speed limits of 15–35 mph were also associated with reduced motorcycle crash frequency. This could be attributed to lower motorcycle speeds, allowing more time to react to potential hazards and community awareness in urban areas. On the other hand, posted speed limits did not affect motorcycle crash frequency post-pandemic. This could be attributed to the reduced traffic volumes and increased vehicle speeds during and post-pandemic (Islam et al., 2022).
+
+<span id="page-6-0"></span><sup>\*</sup> only applicable to urban multilane segments.
+
+<sup>\*\*</sup> na = not applicable.
+
+<sup>\*\*</sup> na = not applicable.
+
+## *5.2. Identification of motorcycle-specific high-crash urban two-lane roadway segments*
+
+The results of the HTCMP models pre- and post-pandemic were incorporated into the EB method to compute EECs and rank the highcrash urban two-lane undivided road segments in Kentucky. Table 3 shows characteristics of the top 10 motorcycle-specific high-crash urban two-lane undivided road segments in each of the pre- and post-pandemic periods.
+
+From Table 3, pre-pandemic, the average AADT for the top 10 highcrash road segments listed was 10,076, which is greater than the overall average of all studied segments (*AADTallsegments* = 5,974). Similarly, postpandemic, the average AADT for listed road segments was 6,811, while the overall average of all studied segments (*AADTallsegments* = 6,183) was lower. Furthermore, an in-depth investigation of motorcycle crashes along these high-crash locations (HCLs) was performed to facilitate targeted safety countermeasures to enhance motorcyclists' safety along urban two-lane undivided road segments. [Fig. 4](#page-8-0) shows details of the most frequent crash characteristics within the top 10 high-crash urban two-lane undivided segments for both pre- and post-pandemic periods.
+
+It can be seen from [Fig. 4](#page-8-0) that single-vehicle collisions were the most frequent manner of collision pre- and post-pandemic, but they have significantly increased post-pandemic. Furthermore, the review of police crash narratives revealed that "lost motorcycle control," "unable to stop in time," and "failed to yield" were the most frequent causes of crashes along these top 10 high-crash segments. Specifically, postpandemic, "lost control of motorcycle" was responsible for 50% of crashes at these sites (almost three times higher than pre-pandemic period), which substantiates that unsafe driver behavior, such as speeding, significantly increased post-pandemic. This is consistent with the study by [Shaik and Ahmed \(2022\)](#page-12-0). Furthermore, it is evident from [Fig. 4](#page-8-0) that motorcyclists were the most frequent at-fault vehicle types and were responsible for 68% of motorcycle-related crashes (postpandemic) at the high-crash locations.
+
+#### **6. SPF results along urban multilane roadway segments**
+
+## *6.1. Motorcycle SPFs' interpretations*
+
+Similar to the SPFs fitted along urban two-lane undivided segments, CMP and HTCMP models were fitted along urban multilane segments for each of the pre- and post-pandemic periods. [Table 4](#page-9-0) shows the model results and GOF statistics of the fitted models for motorcycle SPFs along urban (fitted HTCMP versus CMP model indicated that the HTCMP models outperformed their respective CMP models at the 0.1% level of significance (P-value *<* 0.001). The HTCMP models had the highest Pseudo R2 and lowest AIC and BIC values; hence, indicating good model fit. Furthermore, from the model validation performance, the HTCMP models had the lowest MAD and MSPE values, indicating better
+
+prediction capability. Once more, the superiority of the HTCMP models suggested that the varying dispersion parameter could appropriately capture the motorcycle crash data under-dispersion at urban multilane segments. For example, the HTCMP models for both pre- and postpandemic showed significant roadway parameters in the dispersion part (e.g., presence of roadside guardrail and presence of cable median barrier).
+
+From [Table 4](#page-9-0), the model results showed that motorcycle crashes along urban multilane segments (pre- and post-pandemic) were significantly associated with AADT, right shoulder width, posted speed limit (≥55 mph), presence of horizontal curve, and presence of roadside guardrail. Pre-pandemic, the presence of concrete median barrier was also associated with reduced motorcycle crash frequency. However, post-pandemic, it did not affect motorcycle crash frequency.
+
+In the count part of the HTCMP models, higher AADT was associated with increased motorcycle crash frequency. This is expected as the number of vehicles on the road segment increases, the exposure to motorcycle crashes also increases [\(Lu, et al., 2013](#page-12-0)). The posted speed limit (≥55 mph) was associated with a reduced frequency of motorcycle crashes along urban multilane road segments. This might be because road segments with higher posted speed limits generally have better geometric design features (e.g., sight distance and wider lanes), which could contribute to the reduced motorcycle crash frequency. Similar findings were reported by [Hauer et al. \(2004\).](#page-11-0) To verify this finding, the average lane widths at urban multilane segments were contrasted by different posted speed limits, as shown in [Fig. 5](#page-10-0). From this figure, it is evident that segments with 55 mph or higher posted speed limits had wider lanes (i.e., better geometric design cross-section) compared to segments with lower posted speed limits.
+
+Wider right shoulders were associated with increased motorcycle crash frequency along urban multilane segments. This is likely since wider shoulders might give drivers a false sense of security, encouraging some to treat them as an additional lane for overtaking slower vehicles; hence, increasing the frequency of crashes. This is consistent with the findings of [Li et al. \(2013\)](#page-12-0). Also, to better investigate the impact of right shoulder widths on motorcycle crash frequency, [Fig. 6](#page-10-0) shows the plot of normalized motorcycle crash frequency (i.e., crash frequency per year) versus the right shoulder width (in feet). As seen from [Fig. 6](#page-10-0), motorcycle crashes significantly increase when the shoulder widths are less than 3 feet and 9 feet or greater.
+
+The presence of horizontal curve was associated with increased motorcycle crash frequency at the urban multilane segments. As discussed previously, this is likely since the risk of losing motorcycle control at horizontal curves increases ([Flask et al., 2014](#page-11-0)). The presence of a roadside guardrail was associated with reduced motorcycle crash frequency at urban multilane segments, possibly since motorcyclists drive cautiously nearby guardrails as they perceive these as potential hazard ([Elvik, 1995; Ben-Bassat](#page-11-0) & Shinar, 2011).
+
+During both pre- and post-pandemic, urban multilane segments with
+
+**Table 3**  Top 10 High Motorcycle-Crash Urban Two-Lane Undivided Segments.
+
+|      | Pre-Pandemic (2015–2019) |       |       |              | Post-Pandemic (2020–2022) |         |          |       |               |              |  |  |
+|------|--------------------------|-------|-------|--------------|---------------------------|---------|----------|-------|---------------|--------------|--|--|
+| Rank | Nobs.*                   | Np*   | EB*   | EEC* (EB-Np) | Segment AADT              | Nobs.** | **<br>Np | EB**  | EEC** (EB-Np) | Segment AADT |  |  |
+| 1    | 10                       | 2.730 | 8.975 | 6.245        | 15,403                    | 4       | 1.587    | 3.804 | 2.217         | 5,077        |  |  |
+| 2    | 5                        | 1.647 | 4.577 | 2.930        | 11,937                    | 3       | 1.081    | 2.804 | 1.723         | 15,403       |  |  |
+| 3    | 5                        | 1.770 | 4.696 | 2.925        | 5,221                     | 3       | 1.007    | 2.660 | 1.653         | 5,331        |  |  |
+| 4    | 5                        | 1.969 | 4.691 | 2.722        | 9,822                     | 3       | 0.913    | 2.499 | 1.586         | 1,612        |  |  |
+| 5    | 4                        | 1.445 | 3.741 | 2.296        | 13,013                    | 3       | 1.331    | 2.834 | 1.503         | 12,370       |  |  |
+| 6    | 4                        | 1.512 | 3.670 | 2.158        | 2,944                     | 3       | 1.239    | 2.734 | 1.496         | 3,280        |  |  |
+| 7    | 4                        | 1.085 | 3.148 | 2.063        | 15,403                    | 3       | 1.309    | 2.779 | 1.470         | 2,532        |  |  |
+| 8    | 4                        | 1.673 | 3.735 | 2.062        | 1,978                     | 2       | 0.696    | 1.772 | 1.076         | 12,825       |  |  |
+| 9    | 4                        | 1.825 | 3.594 | 1.769        | 9,104                     | 2       | 0.465    | 1.515 | 1.049         | 5,188        |  |  |
+| 10   | 3                        | 0.787 | 2.540 | 1.753        | 15,932                    | 2       | 0.293    | 1.277 | 0.984         | 4,487        |  |  |
+
+<sup>\*</sup> Unit of "crashes in five years" (note: Nobs. <sup>=</sup> Observed crashes; Np <sup>=</sup> Predicted crashes). \*\* Unit of "crashes in three years" (note: Nobs. <sup>=</sup> Observed crashes; Np <sup>=</sup> Predicted crashes).
+
+<span id="page-8-0"></span>![](_page_8_Figure_2.jpeg)
+
+**Fig. 4.** Most Frequent Crash Characteristics within Top Ten Motorcycle-Specific High-Crash Urban Two-Lane Undivided Segments.
+
+cable and concrete median barriers were significantly associated with reduced motorcycle crash frequency. This is likely since the presence of cable and concrete median barriers could lead to hazard strikes, prompting motorcyclists to drive cautiously; thereby, reducing the frequency of motorcycle crashes. This is in line with the findings of [Donnell](#page-11-0)  [and Mason \(2006\)](#page-11-0) and [Zou and Tarko \(2016\)](#page-12-0).
+
+## *6.2. Identification of motorcycle-specific high-crash urban multilane roadway segments*
+
+Once more, the results of the HTCMP model for pre- and postpandemic were incorporated into the EB method to compute EECs along urban multilane segments in Kentucky. [Table 5](#page-10-0) shows characteristics of the top 10 urban multilane segments with a high number of motorcycle crashes ranked with respect to the EEC values in each of the pre- and post-pandemic periods.
+
+Along the top 10 urban multilane segments, the average AADT prepandemic (*AADTtop*<sup>−</sup> *ten* = 31,415) and post-pandemic (*AADTtop*<sup>−</sup> *ten* = 44,289) were greater than the corresponding overall average across all studied roadway segments (*AADTpre*<sup>−</sup> *pandemic,allsegments* = 30,167, *AADTpost*<sup>−</sup> *pandemic,allsegments* = 29,242). Furthermore, an in-depth investigation of motorcycle crashes at these high-crash locations (HCLs) was performed to facilitate targeted safety countermeasures to enhance motorcyclists' safety along urban multilane segments. [Fig. 7](#page-11-0) shows details of the most frequent crash characteristics within the top 10 highcrash urban multilane segments for both pre- and post-pandemic periods.
+
+It can be seen from [Fig. 7](#page-11-0) that single-vehicle, rear-end, and angle collisions collectively accounted for over 80% of motorcycle crashes in both pre- and post-pandemic periods, with increased rear-end and angle collisions post-pandemic. Regarding the at-fault vehicles, motorcyclists were responsible for 73% of crashes post-pandemic at the high-crash segments, an increase of around 28% compared to the pre-pandemic period. Furthermore, the review of the police crash narratives revealed that the increase in rear-end and angle collisions postpandemic correlates with the following motorcyclists' errors: "unable to stop in time," "improper lane change," and "lost motorcycle control" along the top 10 high-crash segments. These findings show that unsafe driver behaviors, such as speeding and aggressive driving, experienced a
+
+significant increase during and post-pandemic. These findings highlight the need for advance assistance systems in motorcycles (e.g., lane change assistance, brake assistance systems) to account for motorcyclists' errors.
+
+## **7. Proposed motorcycle safety countermeasures**
+
+Based on the results of best-fit models (i.e., the HTCMP models in [Tables 2 and 4\)](#page-6-0) for pre- and post-pandemic and crash-specific characteristics at high motorcycle-crash segments (in Figs. 4 and 5), several motorcycle safety countermeasures are proposed to improve motorcyclists' safety along urban segments in Kentucky, as follows:
+
+Common countermeasures along urban two-lane and multilane roadway segments:
+
+- Installing advance warning signs and/or chevrons along sharp horizontal curves (based on the SPF model results).
+- Installing roadside guardrails (based on the SPF model results).
+- Promoting the adoption of latest safety technologies in motorcycles (e.g., lane change assist, collision warning, and brake assistance) to reduce motorcycle crashes related to driver error (e.g., inability to stop in time and improper lane changes) (based on in-depth investigation of HCLs).
+
+#### Countermeasures specific to urban two-lane undivided segments:
+
+• Installing edge line rumble strips to alert inattentive or speeding drivers drifting from their driving lane (since single-vehicle collisions were the most-frequent manner of collision from the in-depth investigation of HCLs).
+
+## Countermeasures specific to urban multilane segments:
+
+• Providing right shoulders between 3 and 9 feet along urban multilane segments (since there is a significant increase in motorcycle crashes on road segments with right shoulder widths less than 3 ft and greater than 9 ft) (as previously highlighted in [Fig. 6](#page-10-0)).
+
+<span id="page-9-0"></span>**Table 4**  Results of CMP and HTCMP Models for Motorcycle Crash SPFs along Urban Multilane Road Segments.
+
+| Variable                           |           | Pre-Pandemic (2015–2019) |            |             |               |            |           | Post-Pandemic (2020–2022) |            |             |               |            |  |
+|------------------------------------|-----------|--------------------------|------------|-------------|---------------|------------|-----------|---------------------------|------------|-------------|---------------|------------|--|
+|                                    | CMP Model |                          |            | HTCMP Model |               |            | CMP Model |                           |            | HTCMP Model |               |            |  |
+|                                    | Coef.     | Std.<br>Error            | P<br>Value | Coef.       | Std.<br>Error | P<br>Value | Coef.     | Std.<br>Error             | P<br>Value | Coef.       | Std.<br>Error | P<br>Value |  |
+| Mean-Parameter Count Part          |           |                          |            |             |               |            |           |                           |            |             |               |            |  |
+| Constant                           | − 8.275   | 1.024                    | <0.001     | − 6.785     | 0.932         | <0.001     | − 9.722   | 1.424                     | <0.001     | − 3.542     | 2.529         | 0.161      |  |
+| Ln(AADT)                           | 0.843     | 0.108                    | <0.001     | 0.689       | 0.096         | <0.001     | 1.115     | 0.151                     | <0.001     | 0.441       | 0.264         | 0.095      |  |
+| Interstate/Principal Arterial      | 0.402     | 0.115                    | 0.001      | N/S*        | N/S           | N/S        | N/S       | N/S                       | N/S        | N/S         | N/S           | N/S        |  |
+| Posted Speed Limit (≥55<br>mph)    | − 0.816   | 0.138                    | <0.001     | − 0.730     | 0.144         | <0.001     | − 0.632   | 0.263                     | 0.016      | − 0.596     | 0.289         | 0.040      |  |
+| Right Shoulder Width               | 0.030     | 0.014                    | 0.036      | 0.137       | 0.025         | <0.001     | 0.046     | 0.026                     | 0.081      | 0.092       | 0.029         | 0.001      |  |
+| Presence of Horizontal Curve       | 0.186     | 0.104                    | 0.072      | 0.233       | 0.104         | 0.025      | N/S       | N/S                       | N/S        | 1.256       | 0.355         | <0.001     |  |
+| Presence of Roadside<br>Guardrails | − 0.602   | 0.123                    | <0.001     | − 1.780     | 0.212         | <0.001     | − 0.964   | 0.227                     | <0.001     | − 1.746     | 0.315         | <0.001     |  |
+| Median Barrier: Cable              | − 1.368   | 0.253                    | <0.001     | N/S         | N/S           | N/S        | − 1.201   | 0.473                     | 0.011      | N/S         | N/S           | N/S        |  |
+| Median Barrier: Concrete           | − 0.687   | 0.233                    | 0.003      | − 0.664     | 0.217         | 0.002      | − 0.972   | 0.365                     | 0.008      | N/S         | N/S           | N/S        |  |
+| Dispersion Part                    |           |                          |            |             |               |            |           |                           |            |             |               |            |  |
+| Constant                           | 0.798     | 0.050                    | <0.001     | 0.937       | 0.086         | <0.001     | 1.423     | 0.064                     | <0.001     | 4.843       | 1.381         | <0.001     |  |
+| Principal Arterial                 | na**      | na                       | na         | − 0.312     | 0.058         | <0.001     | na        | na                        | na         | N/S         | N/S           | N/S        |  |
+| Posted Speed Limit (≤35<br>mph)    | na        | na                       | na         | N/S         | N/S           | N/S        | na        | na                        | na         | − 0.455     | 0.110         | <0.001     |  |
+| Right Shoulder Width               | na        | na                       | na         | 0.053       | 0.010         | <0.001     | na        | na                        | na         | N/S         | N/S           | N/S        |  |
+| Presence of Horizontal Curve       | na        | na                       | na         | N/S         | N/S           | N/S        | na        | na                        | na         | 0.450       | 0.127         | <0.001     |  |
+| Terrain Type: Flat                 | na        | na                       | na         | − 0.151     | 0.044         | 0.001      | na        | na                        | na         | − 0.234     | 0.077         | 0.002      |  |
+| Presence of Roadside<br>Guardrails | na        | na                       | na         | − 0.766     | 0.128         | <0.001     | na        | na                        | na         | − 0.587     | 0.169         | 0.001      |  |
+| Median Barrier: Cable              | na        | na                       | na         | 0.524       | 0.091         | <0.001     | na        | na                        | na         | 0.464       | 0.180         | 0.010      |  |
+| Median Barrier: Concrete           | na        | na                       | na         | N/S         | N/S           | N/S        | na        | na                        | na         | 0.386       | 0.178         | 0.030      |  |
+| Ln(AADT)                           | na        | na                       | na         | N/S         | N/S           | N/S        | na        | na                        | na         | − 0.325     | 0.138         | 0.019      |  |
+| Fixed Disp. Parameter              | 2.221     | 0.111                    | <0.001     | na          | na            | na         | 4.150     | 0.267                     | <0.001     | na          | na            | na         |  |
+
+|                                      | Pre-Pandemic (2015–2019)  |                            | Post-Pandemic (2020–2022) |                            |
+|--------------------------------------|---------------------------|----------------------------|---------------------------|----------------------------|
+|                                      | CMP Model                 | HTCMP Model                | CMP Model                 | HTCMP Model                |
+| Goodness-of-Fit (GOF) Statistics     |                           |                            |                           |                            |
+| Number of observations               | 634                       | 634                        | 363                       | 363                        |
+| Number of parameters                 | 10                        | 13                         | 8                         | 14                         |
+| Log-likelihood at convergence (LLβ)  | − 838.30                  | − 806.27                   | − 357.07                  | − 334.29                   |
+| Log-likelihood at zero (LL0)         | − 954.93                  | − 954.93                   | − 396.64                  | − 396.64                   |
+| Deviance (D) = (2*(LLβ − LL0))       | 233.26                    | 297.32                     | 79.14                     | 124.70                     |
+|                                      | (df = 8, P-Value < 0.001) | (df = 11, P-Value < 0.001) | (df = 6, P-Value < 0.001) | (df = 12, P-Value < 0.001) |
+| LRT (HTCMP vs CMP)                   | 64.06                     |                            | 45.56                     |                            |
+|                                      | (df = 3, P-Value < 0.001) |                            | (df = 6, P-Value < 0.001) |                            |
+| McFadden pseudo R2                   | 0.122                     | 0.156                      | 0.100                     | 0.157                      |
+| AIC (Akaike Information Criterion)   | 1696.59                   | 1638.53                    | 730.14                    | 696.59                     |
+| BIC (Bayesian Information Criterion) | 1741.11                   | 1696.41                    | 761.30                    | 751.11                     |
+| MAD (Mean Absolute Deviance)         | 0.842                     | 0.832                      | 0.610                     | 0.562                      |
+| MSPE (Mean Square Prediction Error)  | 1.306                     | 1.278                      | 0.856                     | 0.775                      |
+
+<sup>\*</sup> N/S = not statistically significant at 10% significance level.
+
+## **8. Conclusions**
+
+Limited studies have developed motorcycle crash-specific SPFs in the United States, especially along urban roadway segments and for each of the pre-pandemic and post-pandemic periods separately. This study took the initiative and developed motorcycle SPFs along urban two-lane undivided and multilane segments in Kentucky separately in each of the pre-pandemic (2015–2019) and post-pandemic (2020–2022) periods. For both pre- and post-pandemic periods, CMP and HTCMP models were developed and compared for each urban segment facility while accounting for motorcycle crash under-dispersion. For both urban two-lane undivided and multilane segment facilities, the HTCMP models outperformed their CMP counterparts pre- and post-pandemic, based on GOF statistics and prediction performance. The superiority of the HTCMP model suggests that the varying dispersion parameter could efficiently handle the motorcycle crash under-dispersion by incorporating site-specific variables in the dispersion part of the model.
+
+From the developed SPFs, higher AADT and presence of horizontal
+
+curve were associated with increased motorcycle crash frequency, while the presence of roadside steel strong-post W-beam guardrail significantly reduced motorcycle crash frequency along all urban segments, both pre- and post-pandemic. Furthermore, posted speed limits (≥55 mph) were significantly associated with reduced motorcycle crash frequency along urban multilane segments, while collector and local road types were associated with reduced motorcycle crash frequency along urban two-lane segments.
+
+The results of the HTCMP models (for both urban two-lane and multilane) were then used to rank the top 10 motorcycle-specific highcrash segments in each of the pre- and post-pandemic periods using the EB method. An in-depth investigation of motorcycle crashes at the identified high-crash locations was also performed. Based on the study findings, several countermeasures were proposed to enhance motorcyclists' safety along both urban two-lane and multilane segments (e.g., installing chevrons and roadside guardrails at curved sections). Along urban multilane segments, it is recommended to provide right shoulders between 3 and 9 feet (since a significant increase in motorcycle crashes
+
+<sup>\*\*</sup> na = not applicable.
+
+<span id="page-10-0"></span>![](_page_10_Figure_2.jpeg)
+
+**Fig. 5.** Average Lane Widths at Urban Multilane Segments vs. Posted Speed Limits.
+
+![](_page_10_Figure_4.jpeg)
+
+**Fig. 6.** Normalized Motorcycle Crash Frequency at Urban Multilane Segments vs. Right Shoulder Widths.
+
+**Table 5**  Top 10 High Motorcycle-Crash Urban Multilane Segments.
+
+|      | Pre-Pandemic (2015–2019) |       |       |                 |              | Post-Pandemic (2020–2022) |          |       |                  |              |  |  |
+|------|--------------------------|-------|-------|-----------------|--------------|---------------------------|----------|-------|------------------|--------------|--|--|
+| Rank | Nobs.*                   | Np*   | EB*   | EEC*<br>(EB-Np) | Segment AADT | Nobs.**                   | **<br>Np | EB**  | EEC**<br>(EB-Np) | Segment AADT |  |  |
+| 1    | 10                       | 3.611 | 9.061 | 5.450           | 16,311       | 6                         | 1.704    | 5.306 | 3.602            | 23,881       |  |  |
+| 2    | 9                        | 2.254 | 7.438 | 5.185           | 81,002       | 6                         | 2.080    | 5.488 | 3.408            | 40,523       |  |  |
+| 3    | 8                        | 2.435 | 6.997 | 4.563           | 32,638       | 5                         | 2.185    | 4.702 | 2.517            | 153,657      |  |  |
+| 4    | 8                        | 3.220 | 7.463 | 4.242           | 13,144       | 4                         | 1.283    | 3.498 | 2.215            | 16,486       |  |  |
+| 5    | 6                        | 2.012 | 5.325 | 3.313           | 15,684       | 5                         | 2.467    | 4.653 | 2.185            | 20,130       |  |  |
+| 6    | 8                        | 4.622 | 7.763 | 3.141           | 29,531       | 4                         | 1.508    | 3.510 | 2.001            | 95,740       |  |  |
+| 7    | 6                        | 2.243 | 5.127 | 2.884           | 38,814       | 4                         | 1.797    | 3.768 | 1.971            | 51,078       |  |  |
+| 8    | 5                        | 1.201 | 3.955 | 2.754           | 14,586       | 4                         | 1.861    | 3.762 | 1.900            | 16,311       |  |  |
+| 9    | 5                        | 1.107 | 3.846 | 2.739           | 15,447       | 3                         | 1.029    | 2.801 | 1.771            | 15,708       |  |  |
+| 10   | 7                        | 3.887 | 6.590 | 2.703           | 56,997       | 3                         | 0.813    | 2.578 | 1.766            | 9,375        |  |  |
+
+<sup>\*</sup> Unit of "crashes in five years" (Note: Nobs. <sup>=</sup> Observed crashes; Np <sup>=</sup> Predicted crashes). \*\* Unit of "crashes in three years" (Note: Nobs. <sup>=</sup> Observed crashes; Np <sup>=</sup> Predicted crashes).
+
+<span id="page-11-0"></span>![](_page_11_Figure_2.jpeg)
+
+**Fig. 7.** Most Frequent Crash Characteristics within Top Ten Motorcycle-Specific High-Crash Urban Multilane Segments.
+
+was observed on road segments with right shoulder widths less than 3 ft and greater than 9 ft).
+
+It is important to note that the findings from the motorcycle-specific SPFs along urban segments in this study are specifically based on motorcycle crashes in Kentucky. The study results can be transferable to other states with similar motorcycle crash and traffic characteristics/ demographics as those in Kentucky. To build upon this study, future studies can compare the SPF model results along urban segment facilities in other states, which would help assess how these study findings can be transferred or generalized. Other potential research avenues include development of motorcycle crash-specific SPFs along other facilities, such as intersections and ramps, as well as along urban segments with raised medians and flushed medians separately (given that enough site and crash sample sizes exist).
+
+#### **CRediT authorship contribution statement**
+
+**Bharat Kumar Pathivada:** Data curation, Formal analysis, Investigation, Methodology, Software, Writing – original draft, Writing – review & editing. **Arunabha Banerjee:** Data curation, Writing – review & editing. **Kirolos Haleem:** Conceptualization, Funding acquisition, Validation, Writing – original draft, Writing – review & editing.
+
+## **Declaration of competing interest**
+
+The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
+
+## **Acknowledgment**
+
+The authors would like to acknowledge the Kentucky Transportation Cabinet (KYTC) for the grant provided to conduct this research and for providing the necessary crash and roadway data used in this research. The opinions, findings, and conclusions in this paper are those of the authors and not necessarily those of KYTC.
+
+#### **References**
+
+[Alkaaf, K. A., & Abdel-Aty, M. \(2015\). Transferability and calibration of highway safety](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0005)  [manual performance functions and development of new models for urban four-lane](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0005)  divided roads in Riyadh, Saudi Arabia. *[Transportation Research Record: Journal of the](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0005)  [Transportation Research Board, 2515](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0005)*(1), 70–77.
+
+[Alkaaf, K. A., & Abdel-Aty, M. \(2022\). Transferability of safety performance functions:](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0010) [The case of urban four-lane divided roadways in Muscat.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0010) *Vehicles, 4*(4), 1096–1108. [Al-Omari, M. M., Abdel-Aty, M., & Cai, Q. \(2021\). Crash analysis and development of](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0015) [safety performance functions for Florida roads in the framework of the context](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0015) classification system. *[Journal of Safety Research, 79](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0015)*, 1–13.
+
+[Ben-Bassat, T., & Shinar, D. \(2011\). Effect of shoulder width, guardrail and roadway](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0020) [geometry on driver perception and behavior.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0020) *Accident Analysis & Prevention, 43*(6), [2142](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0020)–2152.
+
+[Chand, S., Yee, E., Alsultan, A., & Dixit, V. V. \(2021\). A descriptive analysis on the impact](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0025)  [of COVID-19 lockdowns on road traffic incidents in Sydney, Australia.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0025) *International [Journal of Environmental Research and Public Health, 18](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0025)*(21), 11701.
+
+[Cheng, W., Gill, G. S., Sakrani, T., Dasu, M., & Zhou, J. \(2017\). Predicting motorcycle](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0030) [crash injury severity using weather data and alternative Bayesian multivariate crash](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0030)  [frequency models.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0030) *Accident Analysis & Prevention, 108*, 172–180.
+
+[Cunto, F., Sobreira, L., & Ferreira, S. \(2015\). Assessing the transferability of the highway](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0035)  [safety manual predictive method for urban roads in Fortaleza City Brazil.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0035) *Journal of [Transportation Engineering, 141](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0035)*(1), 04014072.
+
+[Donnell, E. T., & Mason, J. M., Jr \(2006\). Predicting the frequency of median barrier](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0040)  [crashes on pennsylvania interstate highways.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0040) *Accident Analysis & Prevention, 38*(3),
+
+[Elvik, R. \(1995\). The safety value of guardrails and crash cushions: A meta-analysis of](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0045) [evidence from evaluation studies.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0045) *Accident Analysis & Prevention, 27*(4), 523–549.
+
+[Flask, T., Schneider, W. H., IV, & Lord, D. \(2014\). A segment level analysis of multi](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0050)[vehicle motorcycle crashes in ohio using bayesian multi-level mixed effects models.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0050)  *[Safety Science, 66](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0050)*, 47–53.
+
+[Haque, M. M., Chin, H. C., & Huang, H. \(2010\). Applying Bayesian hierarchical models to](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0055)  [examine motorcycle crashes at signalized intersections.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0055) *Accident Analysis & [Prevention, 42](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0055)*(1), 203–212.
+
+[Harnen, S., Umar, R. R., Wong, S. V., & Hashim, W. W. \(2006\). Motorcycle accident](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0060) [prediction model for junctions on urban roads in Malaysia.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0060) *Advances in [Transportation Studies, 8](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0060)*, 31–40.
+
+[Hauer, E., Council, F. M., & Mohammedshah, Y. \(2004\). Safety models for urban four](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0065)lane undivided road segments. *[Transportation Research Record: Journal of the](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0065)  [Transportation Research Board, 1897](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0065)*(1), 96–105.
+
+[Hosseinpour, M., Love, R., Williams, B., & Haleem, K. \(2022\). Comprehensive](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0070)  [investigation of commercial motor vehicle crashes along roadway segments in](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0070)  Kentucky. *[Transportation Research Record: Journal of the Transportation Research](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0070) [Board, 2676](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0070)*(7), 655–671.
+
+[Islam, M. \(2021\). The effect of motorcyclists](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0075)' age on injury severities in single[motorcycle crashes with unobserved heterogeneity.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0075) *Journal of Safety Research, 77*, 125–[138](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0075).
+
+Kentucky Transportation Center (KTC). (2023). Kentucky traffic collision facts: 2021 Report. CollisionFacts2021.pdf (ky.gov). Accessed November 15, 2023.
+
+Kentucky Transportation Cabinet (KYTC). (2023). Highway information view and extract interface (HIVEi) KYTC – Query. HIVEiKYTC - Query. Accessed October 10, 2023.
+
+- <span id="page-12-0"></span>[Li, Z., Kepaptsoglou, K., Lee, Y., Patel, H., Liu, Y., & Kim, H. G. \(2013\). Safety effects of](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0090)  [shoulder paving for rural and urban interstate, multilane, and two-lane highways.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0090)  *[Journal of Transportation Engineering, 139](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0090)*(10), 1010–1019.
+- [Lu, J., Haleem, K., Alluri, P., & Gan, A. \(2013\). Full versus simple safety performance](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0095)  [functions: Comparison based on urban four-lane freeway interchange influence areas](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0095)  in Florida. *[Transportation Research Record: Journal of the Transportation Research](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0095) [Board, 2398](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0095)*(1), 83–92.
+- [Lyon, C., Persaud, B., & Himes, S. \(2017\). Investigating total annual average daily traffic](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0100)  [as a surrogate for motorcycle volumes in estimating safety performance functions for](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0100)  motorcycle crashes. *[Transportation Research Record: Journal of the Transportation](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0100)  [Research Board, 2637](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0100)*(1), 9–16.
+- Manan, M. M. A., Jonsson, T., & V´ [arhelyi, A. \(2013\). Development of a Safety](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0105) [Performance Function for Motorcycle Accident Fatalities on Malaysian Primary](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0105) Roads. *[Safety Science, 60](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0105)*, 13–20.
+- [Michie, J. D., & Bronstad, M. E. \(1994\). Highway guardrails: Safety feature or roadside](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0110)  hazard? *[Transportation Research Record: Journal of the Transportation Research Board,](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0110)  [1468](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0110)*, 1–9.
+- National Highway Traffic Safety Administration (NHTSA). (2023a). Motorcycle: Traffic safety facts 2021 data. 2021 data: Motorcycles (dot.gov). Accessed November 15,
+- 2023. National Highway Traffic Safety Administration (NHTSA). (2023b). Overview of motor
+- vehicle traffic crashes in 2021*.* Report No. DOT HS 813 435. Ospina-Mateus, H., Jim´[enez, L. A. Q., Lopez-Valdes, F. J., & Sana, S. S. \(2021\). Prediction](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0125)  [of motorcyclist traffic crashes in Cartagena \(Colombia\): Development of a safety](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0125)  performance function. *[RAIRO-Operations Research, 55](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0125)*(3), 1257–1278.
+- [Rezapour, M., Farid, A., Nazneen, S., & Ksaibati, K. \(2021\). Using machine leaning](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0130) [techniques for evaluation of motorcycle injury severity.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0130) *IATSS Research, 45*(3), 277–[285](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0130).
+- [Santos, K., Firme, B., Dias, J. P., & Amado, C. \(2023\). Analysis of motorcycle accident](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0135)  [injury severity and performance comparison of machine learning algorithms.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0135) *[Transportation Research Record: Journal of the Transportation Research Board](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0135)*. pp. [03611981231172507.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0135)
+- [Sawalha, Z., & Sayed, T. \(2001\). Evaluating safety of urban arterial roadways.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0140) *Journal of [Transportation Engineering, 127](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0140)*(2), 151–158.
+- [Sellers, K. F., & Shmueli, G. \(2010\). A flexible regression model for count data.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0145) *The [Annals of Applied Statistics, 4](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0145)*(2), 943–961.
+- [Sellers, K. F., Borle, S., & Shmueli, G. \(2012\). The COM-Poisson model for count data: A](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0150)  survey of methods and applications. *[Applied Stochastic Models in Business and](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0150) [Industry, 28](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0150)*(2), 104–116.
+- [Shaik, M. E., & Ahmed, S. \(2022\). An overview of the impact of COVID-19 on road traffic](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0155)  safety and travel behavior. *[Transportation Engineering, 9](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0155)*, Article 100119.
+- [Shmueli, G., Minka, T. P., Kadane, J. B., Borle, S., & Boatwright, P. \(2005\). A useful](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0160) [distribution for fitting discrete data: Revival of the Conway](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0160)–Maxwell–Poisson
+
+- distribution. *[Journal of the Royal Statistical Society Series C: Applied Statistics, 54](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0160)*(1), 27–[142.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0160)
+- [Yasin, Y. J., Grivna, M., & Abu-Zidan, F. M. \(2021\). Global impact of COVID-19](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0165)
+- pandemic on road traffic collisions. *[World Journal of Emergency Surgery, 16](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0165)*(1), 1–14. [Zhang, Y., Xie, Y., & Li, L. \(2012\). Crash frequency analysis of different types of urban](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0170) [roadway segments using generalized additive model.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0170) *Journal of Safety Research, 43*  [\(2\), 107](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0170)–114.
+- [Zou, Y., & Tarko, A. P. \(2016\). An insight into the performance of road](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0175)  barriers− [redistribution of barrier-relevant crashes.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0175) *Accident Analysis & Prevention, 96*[, 152](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0175)–161.
+- [Zubaidi, H., Tamakloe, R., Al-Bdairi, N. S. S., Alnedawi, A., & Obaid, I. \(2023\). Exploring](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0180)  [senior motorcyclist injury severity crashes: Random parameter model with](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0180)  [heterogeneity in mean and variance.](http://refhub.elsevier.com/S0022-4375(24)00121-X/h0180) *IATSS Research, 47*(1), 1–13.
+- **Dr. Bharat Kumar Pathivada** is a Post-Doctoral Research Associate at the Center for Transportation Research (CTR) in the School of Engineering and Applied Sciences at Western Kentucky University (WKU). He has over two years of research experience in the field of transportation engineering with a focus on traffic safety and human factors. His research interests include Traffic Safety Analysis, Crash Modeling, Road User Behavior, and Intelligent Transportation Systems (ITS). He has a Doctorate Degree in transportation engineering and has participated in accomplishing several externally funded projects.
+- **Dr. Arunabha Banerjee** is a Postdoctoral Research Associate at the Center for Transportation Research (CTR) in the School of Engineering and Applied Sciences at Western Kentucky University. His main interests are traffic safety analysis, crash modeling, and pedestrian walking behavior modeling. He also holds a Doctorate Degree in transportation engineering
+- **Dr. Kirolos Haleem** is an associate professor of civil (transportation) engineering at Western Kentucky University (WKU). Dr. Haleem is the founder & director of the Center for Transportation Research (CTR) at WKU. Kirolos received his master's and Ph.D. degrees in civil (transportation) engineering from the University of Central Florida (UCF) in 2007 and 2009, respectively. He has more than 15-year experience in transportation engineering. His areas of expertise include highway traffic safety analysis, application of statistical models and data mining techniques in transportation data analysis, driving simulator studies, and intelligent transportation systems (ITS). Kirolos led and conducted numerous transportation research projects sponsored by the Kentucky Department of Transportation (Kentucky DOT), Alabama Department of Transportation (ALDOT), Florida DOT, and the U.S. DOT.

Zotero/001_artiklid/Road Safety Indicators/pathivadamodelingmotorcyclecrashfrequency2025.md

@@ -0,0 +1,302 @@
+---
+category: literaturenote
+citekey: pathivadamodelingmotorcyclecrashfrequency2025
+title: Modeling motorcycle crash frequency on rural multilane segments in Kentucky
+authors: "Pathivada, Bharat Kumar; Banerjee, Arunabha; Haleem, Kirolos; Khan, Tathagatha; Justice, Dylan"
+year: 2025
+date: 2025-00-00 2025
+doi: 10.1016/j.aap.2025.108085
+publication: "Accident Analysis & Prevention"
+url: "https://doi.org/10.1016/j.aap.2025.108085"
+zotero_key: ESWMX2TN
+zotero_storage: DGSN6JP5
+collections: transpordi_planeerimine
+folder: 001_artiklid/Road Safety Indicators
+firstAuthor: "Pathivada, Bharat Kumar"
+status: converted
+---
+![](_page_0_Picture_1.jpeg)
+
+Contents lists available at [ScienceDirect](www.sciencedirect.com/science/journal/00014575)
+
+## Accident Analysis and Prevention
+
+journal homepage: [www.elsevier.com/locate/aap](https://www.elsevier.com/locate/aap)
+
+![](_page_0_Picture_5.jpeg)
+
+![](_page_0_Picture_6.jpeg)
+
+# Modeling motorcycle crash frequency on rural multilane segments in Kentucky☆
+
+Bharat Kumar Pathivada , Arunabha Banerjee [,](https://orcid.org/0000-0002-6314-295X) Kirolos Haleem \* , Tathagatha Khan [,](https://orcid.org/0000-0002-0017-2342) Dylan Justice
+
+*Center for Transportation Research (CTR), School of Engineering & Applied Sciences, Western Kentucky University, 1906 College Heights Blvd, EBS 2122, Bowling Green, KY 42101, United States*
+
+ARTICLE INFO
+
+*Keywords:* Motorcycle Crashes Rural Multilane Segments COVID-19 Pandemic Heterogeneous Conway-Maxwell-Poisson
+
+#### ABSTRACT
+
+Despite motorcycle crashes accounting for a large percentage of traffic fatalities in the U.S., studies investigating motorcycle crash frequency are relatively limited. This study took the initiative and developed motorcycle crashspecific safety performance functions (SPFs) along rural multilane road segments in Kentucky, separately for the pre-COVID-19 pandemic (2015–2019) and post-COVID-19 pandemic (2020–2022) periods. Eight years of motorcycle crash records (2015 through 2022) and site-specific characteristics (e.g., shoulder width and annual average daily traffic "AADT") were collected and used. Conway-Maxwell-Poisson (CMP) and heterogeneous Conway-Maxwell-Poisson (HTCMP) models were fitted and compared while accounting for motorcycle crash under-dispersion (i.e., when crash variance is less than its mean). The study results showed that, for both pre- and post-pandemic periods, the HTCMP models outperformed their CMP counterparts based on various goodness-offit measures (e.g., likelihood ratio test "LRT", Akaike information criterion "AIC", and McFadden pseudo Rsquared) and prediction performance measures (i.e., mean absolute deviance "MAD" and mean square prediction error "MSPE"). From the developed SPFs, for the pre-pandemic period, the presence of horizontal curves and undivided roadways were significantly associated with increased motorcycle crash frequency along rural multilane segments, while in the post-pandemic period, wider right shoulders and higher AADT were significantly associated with increased motorcycle crash frequency. The predicted crash frequencies while applying the best-fit models (i.e., the HTCMP models) were then used to identify and rank high-crash rural multilane segments in Kentucky. Based on the study findings, several countermeasures were proposed to improve motorcyclists' safety along Kentucky's rural multilane segments, e.g., adding centerline grooved rumble strips along undivided rural multilane roadways.
+
+## **1. Introduction**
+
+Motorcyclists are vulnerable road users with a heightened risk of traffic casualties due to the inherent instability of the motorcycles and the absence of in-vehicle protection. According to the National Highway Traffic Safety Administration (NHTSA), motorcycles contribute to 14 % of all traffic fatalities and 17 % of all motor vehicle occupant fatalities in the U.S., despite comprising only 3.5 % of registered vehicles [\(NHTSA,](#page-5-0)  [2023\)](#page-5-0). Furthermore, motorcycles accounted for over 7 % of traffic fatalities in Kentucky, while accounting for less than 1 % of registered vehicles [\(KTC, 2023\)](#page-5-0). Moreover, the emergence of the 2019 novel coronavirus (COVID-19), resulted in significant changes in travel behavior, travel patterns, and traffic safety ([Chand et al., 2021; Yasin](#page-5-0)  [et al., 2021\)](#page-5-0). For example, [Islam et al. \(2022\)](#page-5-0) examined risk compensation trends in road safety during COVID-19 along Interstate 4 in Orlando, Florida, and revealed that traffic volumes decreased by 13.6 % amid the pandemic, leading to reduced crash frequency, while average speeds increased by 11.3 %, resulting in increased crash severities.
+
+To address safety concerns, several studies have investigated motorcycle safety, but the majority of the studies were focused on motorcyclist crash injury severity (e.g., [Savolainen and Mannering,](#page-5-0)  [2007; Cheng et al., 2017; Farid and Ksaibati, 2021\)](#page-5-0). For example, studies have highlighted that factors such as motorcyclists' characteristics (age and ethnicity), crash-specific factors (collision type and speeding),
+
+*E-mail addresses:* [arunabha.banerjee@wku.edu](mailto:arunabha.banerjee@wku.edu) (A. Banerjee), [kirolos.haleem@wku.edu](mailto:kirolos.haleem@wku.edu) (K. Haleem).
+
+<sup>☆</sup> This article is part of a special issue entitled: 'RSS 2024' published in Accident Analysis and Prevention.
+
+<sup>\*</sup> Corresponding author.
+
+roadway characteristics (rural roads and road surface conditions), helmet usage, and impaired driving (under the influence of drug, alcohol, or medication) significantly contributed to motorcycle crash injury severity (e.g., [Savolainen and Mannering, 2007; Shaheed and Gkritza,](#page-5-0)  [2014; Alnawmasi and Mannering, 2019\)](#page-5-0). Motorcycle-specific safety studies, particularly focusing on motorcycle crash frequency, are rather limited, especially in the U.S. For example, [Flask et al. \(2014\)](#page-5-0) employed Bayesian multi-level mixed effects models to predict multi-vehicle motorcycle crash frequency along road segments in Ohio. Motorcycle crash data along 32,289 state-maintained road segments in Ohio were used. The results showed that motorcycle crash frequency was associated with narrower lane and shoulder widths and increased horizontal curvature and vertical grade. [Manan et al. \(2013\)](#page-5-0) developed safety performance functions (SPFs) for motorcycle crash casualties on Malaysian primary roads utilizing the negative binomial (NB) model. Three-year crash data between 2007 and 2009 from three different states in Malaysia were used. Their results showed that the number of access points per kilometer and the average daily number of motorcycles were significantly associated with motorcycle casualties.
+
+Rural multilane roadways pose unique safety challenges due to their high operating speeds and limited lighting, which increases the risk of traffic crashes. [Banihashemi \(2015\)](#page-5-0) investigated the effect of horizontal curvature on traffic crashes along rural multilane highways. Five-year crash data were used between 2007 and 2011 on rural multilane highway segments in Washington state. Their results showed that using a crash modification factor (CMF) for horizontal curvature significantly improved the crash prediction for tangent and curve segments along rural multilane highways. [Asal and Said \(2019\)](#page-5-0) developed local SPFs for rural multilane divided highways in Egypt while applying the NB models. A total of 2,153 crashes on 17 rural multilane roadways between 2007 and 2014 were used. Their results showed that curve density (i.e., number of horizontal curves per mile), lane width, and shoulder width were significant geometric features associated with crashes on rural multilane divided highways. Overall, the review of literature has shown that there exist limited studies that specifically investigated motorcycle crashes along rural multilane roadway segments.
+
+The NB (or Poisson-gamma) model is the most frequently used crash frequency modeling approach in transportation safety analysis [\(Lord](#page-5-0)  [and Mannering, 2010; Khaddar et al., 2023\)](#page-5-0). However, the NB model is limited to over-dispersed data and cannot handle under-dispersed count data [\(Lord and Mannering, 2010\)](#page-5-0). To address this limitation, studies have employed the Conway-Maxwell-Poisson (CMP) distribution, a flexible distribution that accounts for both over-dispersed and underdispersed data. For example, [Lord et al. \(2008\)](#page-5-0) developed and compared the motor-vehicle crash SPFs while applying the standard NB and CMP models. Crashes at rural four-lane highways in Texas and crashes at signalized four-legged intersections in Toronto were used. Their results suggested that CMP models performed similarly to the NB models in terms of goodness-of-fit (GOF) statistics and predictive performance. [Hosseinpour et al. \(2022\)](#page-5-0) developed SPFs for commercial motor vehicles (CMVs) along roadway segment facilities while employing CMP-based models. The study used fatal and injury CMV crash data between 2015 and 2019 in Kentucky. The authors compared the standard NB and CMP-based models and concluded that CMP-based models performed better than the NB model.
+
+A thorough review of the literature revealed that despite motorcycle crashes accounting for a significant portion of traffic fatalities in the U. S., there is a lack of motorcycle-specific safety studies, particularly those focusing on motorcycle crash frequency. Furthermore, the majority of motorcycle-related safety studies focused on crash injury severity, and limited studies have developed motorcycle crash-specific SPFs along rural multilane segments, especially while accounting for the pre-COVID-19 and post-COVID-19 periods separately. Please note that the term "post-COVID-19" (or post-pandemic) in this study implies "during and post COVID-19". This study takes the initiative to address this gap by comprehensively investigating motorcycle crash frequency along rural multilane roadway segments, considering each of the prepandemic (2015–2019) and post-pandemic (2020–2022) periods in Kentucky.
+
+## **2. Methodology**
+
+#### *2.1. SPF model development*
+
+The negative binomial (NB) model is commonly employed as an SPF tool; however, it is limited to handling over-dispersed count data and cannot accommodate under-dispersed count data. The CMP (or COM-Poisson) distribution generalizes the well-known Poisson distribution and accounts for both over-dispersed and under-dispersed count data ([Shmueli et al., 2005](#page-5-0)). The probability mass function of a CMP distribution is shown in Equations (1) and (2).
+
+$$P(Y = y) = \frac{\lambda^{y}}{(y!)^{y}Z(\lambda, y)} \tag{1}$$
+
+$$Z(\lambda, \nu) = \sum_{n=0}^{\infty} \frac{\lambda^n}{(n!)^{\nu}}$$
+ (2)
+
+where: *Y* is the discrete count, *λ* is a centering parameter that approximates the mean of the observations, and *v* is the dispersion parameter of the CMP distribution. The condition *v >* 1 corresponds to underdispersion and *v <* 1 to over-dispersion.
+
+A promising generalization of the CMP model is the heterogeneous CMP (or HTCMP) model, which allows the dispersion parameter to vary across the observations as a function of site-specific characteristics. The dispersion parameter of the HTCMP model is expressed as a function of site-specific attributes, as follows:
+
+$$\ln(\nu) = \gamma \psi_i \tag{3}$$
+
+where: *ψi*is the vector of site-specific characteristics, and *γ* is the vector of regression coefficients corresponding to *ψ<sup>i</sup>* . If none of the variables are significant in the dispersion part, the HTCMP model simplifies to the CMP model form, with a fixed dispersion parameter across all sites.
+
+## *2.2. Goodness-of-Fit (GOF) metrics*
+
+For each of the CMP and HTCMP models in pre- and post-pandemic periods, the overall GOF measure was determined by calculating the deviance statistic (*D*), as shown in Equation (4).
+
+$$D = 2*(LL_{\beta} - LL_0) \cong \chi^2_{(df, =p)}$$
+
+$$\tag{4}$$
+
+where: *LLβ*is the log-likelihood at convergence, *LL*0 is the log-likelihood of the null model (i.e., constant-only or intercept-only model), and *p* is the number of predictors in the final model. Other GOF measures used to assess the fitted models include the Akaike information criterion (AIC), Bayesian information criterion (BIC), and McFadden pseudo R<sup>2</sup> . The best-fit models have lower AIC and BIC values, while a higher McFadden pseudo R2 suggests a superior fit. These measures are computed as shown in Equations (5) through (7).
+
+$$AIC = -2*LL_{\beta} + 2*P \tag{5}$$
+
+$$BIC = -2*LL_{\beta} + P*ln(n) \tag{6}$$
+
+$$McFaddenpseudoR^2 = 1 - (LL_{\beta}/LL_0)$$
+ (7)
+
+where: *P* is the number of model parameters, and *n* is the number of observations. The likelihood ratio test (LRT) was also used to evaluate the relative performance of the fitted CMP and HTCMP models ([Pathivada et al., 2024\)](#page-5-0). The LRT is based on the differences in the loglikelihoods of the fitted models (*LLCMP, LLHTCMP*), which follows a Chisquared distribution with degrees of freedom equal to the difference
+
+between the number of estimated parameters in both models (*k*), see Equation (8).
+
+$$LRT = 2*(LL_{HTCMP} - LL_{CMP}) \cong \chi^{2}_{(d.f.=k)}$$
+(8)
+
+The predictive performance of the fitted SPFs with the validation dataset was also evaluated using the absolute deviance (MAD) and mean square prediction error (MSPE), as shown in Equations (9) and (10).
+
+$$MAD = \frac{\sum_{i=1}^{n} |\mu_{i} - y_{i}|}{n}$$
+ (9)
+
+$$MSPE = \frac{\sum_{i=1}^{n} (\mu_i - y_i)^2}{n}$$
+ (10)
+
+where: *yi* and *μi*are observed and predicted number of crashes at road segment *i*, respectively, and *n* is the number of sites in the validation dataset. The lower the MAD and MSPE values, the superior prediction performance is.
+
+#### **3. Data collection and processing**
+
+Crashes involving at least one motorcycle were classified as motorcycle-related crashes, and were identified based on the vehicle type(s) reported in the crash database. Eight years of motorcycleinvolved crash data from 2015 to 2022 were collected from the Kentucky Transportation Cabinet (KYTC) crash database. The eight-year crash records were analyzed separately in two periods, i.e., prepandemic (2015–19) and post-pandemic (2020–2022). The roadwayspecific inventory (e.g., posted speed limit, number of lanes, and shoulder width) and annual average daily traffic (AADT) were extracted from the KYTC's Highway Information Extract View and Interface (HIEVi) database [\(KYTC, 2024](#page-5-0)). The crash database was integrated with the HIEVi database based on the route ID and milepost of each crash. Since the present study investigates motorcycle crashes on rural multilane roadway segments, crashes on all other facilities were excluded from the analysis. After removing crash records with missing information, there were a total of 323 and 177 motorcycle crashes along rural multilane roadway segments during the pre- and post-pandemic periods, respectively.
+
+To investigate motorcycle crash frequency along roadway segments, the road sections were split into homogeneous segments based on the homogeneity in roadway geometric and traffic characteristics. Crash frequency was then obtained by aggregating motorcycle crashes over the homogeneous segments. In this regard, the features used to produce homogeneous segments were area type (rural or urban), roadway type (undivided or divided), number of traffic lanes, posted speed limit, and AADT. A homogeneous segment ended where at least one of these features changed, and a new homogeneous segment subsequently started.
+
+The segmentation process was done separately for the pre- and postpandemic crash data. A total of 277 and 173 homogeneous segments were produced for the pre- and post-pandemic periods, respectively. Note that approximately 20 percent of the total rural multilane segments had zero motorcycle-related crashes, randomly collected across Kentucky to develop valid SPFs. Fig. 1 shows motorcycle crash counts along rural multilane road segments for the pre- and post-pandemic periods. Motorcycle crash data in pre- and post-pandemic periods were underdispersed (i.e., the crash variance is less than the mean), making CMPbased models appropriate for fitting the SPFs. Furthermore, for each period, the homogeneous segments were then randomly split into calibration (75 %) and validation (25 %) datasets, where calibration datasets were used to develop crash prediction models (i.e., SPFs) and validation datasets were used to evaluate the prediction performance of the fitted models. [Table 1](#page-3-0) shows descriptive statistics of the explanatory variables used in the motorcycle SPF development along rural multilane segments.
+
+#### **4. Motorcycle SPF modeling results**
+
+It should be noted that before running separate models (pre- and post-pandemic), the authors have plotted the monthly crash trend in each of the pre-pandemic (2015–2019) and post-pandemic (2020–2022) periods and different trendlines in both periods were detected, which suggested and justified the need to fit separate models. Furthermore, the LRT was used to compare the two CMP models pre- and post-pandemic, as well as the two HTCMP models pre- and post-pandemic, and significant differences in the log-likelihoods pre- and post-pandemic were observed, which again justified the need to run separate models.
+
+The significant model estimates of fitted CMP and HTCMP models for pre- and post-pandemic periods are presented in [Table 2](#page-4-0). The deviance statistic (*D*) for all the fitted models was significant at the 0.1 % significance level (i.e., P-value *<* 0.001), indicating that the fitted models performed significantly better than their constant-only models.
+
+The GOF measures in [Table 2](#page-4-0) indicate that the HTCMP models were superior to their CMP counterparts. Specifically, the LRT results revealed that the HTCMP model outperformed its respective CMP model for both pre- and post-pandemic periods at the 5 % significance level. Additionally, AIC and McFadden pseudo R2 values indicate that the HTCMP models provided a better fit than their CMP counterparts. Furthermore, in terms of model validation performance, the HTCMP models exhibited the lowest MAD and MSPE values, indicating better prediction capability. The superiority of the HTCMP models implies that the varying dispersion parameter effectively captured under-dispersion at rural multilane segments. For instance, in both pre- and post-
+
+![](_page_2_Figure_17.jpeg)
+
+**Fig. 1.** Motorcycle Crash Counts along Rural Multilane Road Segments.
+
+<span id="page-3-0"></span>**Table 1**  Descriptive Statistics of Explanatory Variables on Rural Multilane Roadway Segments.
+
+| Variable                                        | Variable Description                            | Pre-Pandemic (2015–2019)                     |       |       |              | Post-Pandemic (2020–2022) |                              |       |       |  |
+|-------------------------------------------------|-------------------------------------------------|----------------------------------------------|-------|-------|--------------|---------------------------|------------------------------|-------|-------|--|
+|                                                 |                                                 | No. of Sites<br>No. of Motorcycle<br>Crashes |       |       | No. of Sites |                           | No. of Motorcycle<br>Crashes |       |       |  |
+| Calibration Dataset                             | Site and crash summary statistics               | 208                                          |       | 242   |              | 130                       |                              | 134   |       |  |
+| Validation Dataset                              | Site and crash summary statistics               | 69                                           |       | 81    |              | 43                        |                              | 43    |       |  |
+| Indicator Dummy Variables (Calibration dataset) |                                                 |                                              |       |       |              |                           |                              |       |       |  |
+| Road Functional Class                           |                                                 |                                              |       |       |              |                           |                              |       |       |  |
+| Interstate/Principal arterial                   | If true = 1, otherwise = 0                      | 181                                          |       | 218   |              | 113                       |                              | 120   |       |  |
+| Minor arterial                                  | If true = 1, otherwise = 0                      | 17                                           |       | 17    |              | 14                        |                              | 12    |       |  |
+| Major/Minor collector                           | If true = 1, otherwise = 0                      | 10                                           |       | 7     |              | 3                         |                              | 2     |       |  |
+| Posted Speed Limit                              |                                                 |                                              |       |       |              |                           |                              |       |       |  |
+| < 55 mph                                        | If true = 1, otherwise = 0                      | 24                                           |       | 20    |              | 10                        |                              | 10    |       |  |
+| ≥ 55 mph                                        | If true = 1, otherwise = 0                      | 184                                          |       | 222   |              | 120                       |                              | 124   |       |  |
+| Terrain Type                                    | If flat terrain = 1, otherwise = 0              | 67                                           |       | 72    |              | 46                        |                              | 48    |       |  |
+| Horizontal Curve                                | If present = 1, otherwise = 0                   | 53                                           |       | 71    |              |                           | 32                           |       | 36    |  |
+| Vertical Grade                                  | If present = 1, otherwise = 0                   | 93                                           |       | 115   |              | 64                        |                              | 69    |       |  |
+| Roadside Guardrail                              | If present = 1, otherwise = 0                   | 81                                           |       | 90    |              | 50                        |                              | 54    |       |  |
+| Undivided                                       | If road segment is undivided = 1, otherwise = 0 | 28                                           |       | 26    |              | 20                        |                              | 20    |       |  |
+| Median Barrier Type                             |                                                 |                                              |       |       |              |                           |                              |       |       |  |
+| Cable                                           | If true = 1, otherwise = 0                      | 23                                           |       | 34    |              | 21                        |                              | 24    |       |  |
+| Concrete                                        | If true = 1, otherwise = 0                      | 41                                           |       | 59    |              | 27                        |                              | 31    |       |  |
+| Other types (e.g., guardrails)                  | If true = 1, otherwise = 0                      | 10                                           |       | 14    |              | 2                         |                              | 2     |       |  |
+| None                                            | If true = 1, otherwise = 0                      | 131                                          |       | 131   |              |                           | 80                           |       | 77    |  |
+| Variable                                        | Variable Description                            | Pre-Pandemic (2015–2019)                     |       |       |              |                           | Post-Pandemic (2020–2022)    |       |       |  |
+|                                                 |                                                 | Min.*                                        | Max.* | Mean  | SD*          | Min.*                     | Max.*                        | Mean  | SD*   |  |
+| Continuous Variables (Calibration dataset)      |                                                 |                                              |       |       |              |                           |                              |       |       |  |
+| Ln(AADT)                                        | Natural log. of annual average daily traffic    | 4.28                                         | 11.30 | 9.67  | 0.90         | 7.98                      | 11.30                        | 9.70  | 0.81  |  |
+| Lane Width (ft)                                 | Lane width (in feet)                            | 10                                           | 13    | 11.94 | 0.28         | 11                        | 13                           | 11.95 | 0.28  |  |
+| RSW (ft)                                        | Right shoulder width (in feet)                  | 0                                            | 14    | 9.22  | 2.70         | 0                         | 14                           | 9.14  | 2.73  |  |
+| Number of Lanes                                 | Total number of lanes                           | 3                                            | 7     | 4.37  | 0.90         | 3                         | 6                            | 4.31  | 0.82  |  |
+| Median Width (ft)                               | Median Width (in feet)                          | 0                                            | 99    | 34.22 | 25.08        | 0                         | 99                           | 33.25 | 25.44 |  |
+
+<sup>\*</sup> Min. = minimum; Max. = maximum; and SD = standard deviation.
+
+pandemic periods, the HTCMP models showed significant roadway features (e.g., right shoulder width, presence of cable median barrier, and flat terrain) in the dispersion part of the model.
+
+For the pre-pandemic period, the model results showed that the presence of horizontal curves, higher AADT, and undivided roadways were positively associated with motorcycle crash frequency. In contrast, for the post-pandemic period, the model results showed that wider right shoulders, higher AADT, and undivided roadways were positively associated with motorcycle crashes along rural multilane road segments. For both pre- and post-pandemic periods, the posted speed limit (≥55 mph) was negatively associated with motorcycle crash frequency along rural multilane road segments. Additionally, the presence of median cable barriers was associated with reduced motorcycle crash frequency post-pandemic.
+
+The model results from [Table 2](#page-4-0) were intuitive. For example, higher AADT was found to be associated with increased number of motorcycle crashes along rural multilane segments. This is expected, as AADT reflects exposure, and higher exposure results in increased motorcycle crashes. This finding is consistent with other safety studies (e.g., [Manan](#page-5-0)  [et al., 2013; Haque et al., 2010; Ospina-Mateus et al., 2021](#page-5-0)). The posted speed limit (≥55 mph) was associated with reduced motorcycle crashes on rural multilane road segments. This might be since road segments with higher posted speed limits generally have better geometric design features (e.g., sight distance), which could contribute to the reduced motorcycle crash frequency. Similar findings were reported by [Hos](#page-5-0)[seinpour et al. \(2022\)](#page-5-0) and [Hauer et al. \(2004\)](#page-5-0). Undivided rural multilane roadway segments were found to be associated with increased motorcycle crashes, possibly due to the absence of a physical separator median, which increases the risk of head-on collisions. This is consistent with the findings of [Saeed et al. \(2019\)](#page-5-0).
+
+The presence of horizontal curve was found to be associated with increased motorcycle crash frequency. This could be because the risk of losing motorcycle control increases at or nearby horizontal curves. This is in line with the findings of [Flask et al. \(2014\)](#page-5-0). Rural multilane
+
+segments with cable median barriers were associated with reduced motorcycle crash frequency, likely because motorcyclists perceive the presence of median barriers as a potential hazard, prompting motorcyclists to drive cautiously; thereby, reducing the motorcycle crash frequency.
+
+As previously indicated, post-pandemic, wider right shoulders were found to be associated with increased motorcycle crash frequency. This could be since wider shoulders may give drivers a false sense of security, encouraging them to drive faster than what the roadway conditions permit. This aligns with the findings of [Shaik and Ahmed \(2022\),](#page-5-0) who reported that unsafe driver behavior significantly increased during and post-pandemic. To better understand the impact of right shoulder width on motorcycle crash frequency, [Fig. 2](#page-4-0) shows the plot of "normalized motorcycle crashes per segment" vs. "right shoulder width (in feet)". For better analysis accuracy and to adjust for exposure, motorcycle crashes were normalized based on the number of segments within a specific shoulder width range. As seen in [Fig. 2,](#page-4-0) there was a noticeable increase in motorcycle crash frequency along rural multilane segments with right shoulder widths 6 feet or wider.
+
+## **5. High-crash location identification**
+
+The predicted crash frequencies from the best-fit HTCMP models preand post-pandemic periods were used to rank the high-crash rural multilane segments using the empirical Bayes (EB) method. The EB method accounts for the regression-to-the-mean (RTM) bias and estimates the expected average crash frequency by combining the observed and predicted crashes using a weight factor, which is a function of the dispersion parameter from the fitted SPF. The high-crash locations (HCLs) were then identified and ranked based on the higher-thanexpected crashes. For more details on the EB method, readers are referred to the Highway Safety Manual [\(HSM, 2010](#page-5-0)). The identified HCLs were considerably different from the rankings based on other metrics, such as crash frequency and crash rate. The is because the EB
+
+<span id="page-4-0"></span>**Table 2**  Model Estimates for Motorcycle Crash SPFs along Rural Multilane Roadway Segments.
+
+| Variable                        | Pre-Pandemic (2015–2019) |               |            |             |               |            | Post-Pandemic (2020–2022) |               |            |             |               |            |  |
+|---------------------------------|--------------------------|---------------|------------|-------------|---------------|------------|---------------------------|---------------|------------|-------------|---------------|------------|--|
+|                                 | CMP Model                |               |            | HTCMP Model |               |            | CMP Model                 |               |            | HTCMP Model |               |            |  |
+|                                 | Coef.                    | Std.<br>Error | P<br>Value | Coef.       | Std.<br>Error | P<br>Value | Coef.                     | Std.<br>Error | P<br>Value | Coef.       | Std.<br>Error | P<br>Value |  |
+| Mean-parameter count part       |                          |               |            |             |               |            |                           |               |            |             |               |            |  |
+| Constant                        | − 7.200                  | 1.221         | <0.001     | − 1.248     | 0.189         | <0.001     | − 8.450                   | 2.242         | 0.001      | − 8.476     | 2.459         | 0.001      |  |
+| Posted Speed Limit (≥55<br>mph) | − 0.873                  | 0.408         | 0.032      | N/S         | N/S           | N/S        | − 1.624                   | 0.743         | 0.029      | − 1.991     | 0.743         | 0.007      |  |
+| Undivided                       | 1.202                    | 0.370         | 0.001      | 0.904       | 0.337         | 0.007      | 1.234                     | 0.541         | 0.023      | 1.247       | 0.515         | 0.015      |  |
+| Right Shoulder Width            | N/S*                     | N/S           | N/S        | N/S         | N/S           | N/S        | 0.275                     | 0.084         | 0.001      | 0.390       | 0.098         | <0.001     |  |
+| Presence of Horizontal Curve    | 0.673                    | 0.226         | 0.003      | 0.839       | 0.235         | <0.001     | N/S                       | N/S           | N/S        | N/S         | N/S           | N/S        |  |
+| Ln(AADT)                        | 0.691                    | 0.124         | <0.001     | N/S         | N/S           | N/S        | 0.786                     | 0.248         | 0.002      | 0.710       | 0.248         | 0.004      |  |
+| Median Barrier: Cable           | N/S                      | N/S           | N/S        | N/S         | N/S           | N/S        | − 0.945                   | 0.457         | 0.039      | N/S         | N/S           | N/S        |  |
+| Dispersion part                 |                          |               |            |             |               |            |                           |               |            |             |               |            |  |
+| Constant                        | 1.131                    | 0.086         | <0.001     | 4.041       | 0.683         | <0.001     | 1.652                     | 0.103         | <0.001     | 0.697       | 0.649         | 0.283      |  |
+| Posted Speed Limit (≥55 mph)    | –                        | –             | –          | 0.895       | 0.264         | 0.001      | –                         | –             | –          | N/S         | N/S           | N/S        |  |
+| Right Shoulder Width            | –                        | –             | –          | N/S         | N/S           | N/S        | –                         | –             | –          | 0.116       | 0.062         | 0.064      |  |
+| Terrain Type: Flat              | ––                       | –             | –          | − 0.221     | 0.101         | 0.030      | –                         | –             | –          | − 0.224     | 0.118         | 0.057      |  |
+| Presence of Roadside Guardrails | –                        | –             | –          | N/S         | N/S           | N/S        | –                         | –             | –          | − 0.273     | 0.116         | 0.020      |  |
+| Median Barrier: Cable           | –                        | –             | –          | N/S         | N/S           | N/S        | –                         | –             | –          | 0.264       | 0.129         | 0.041      |  |
+| Ln(AADT)                        | –                        | –             | –          | − 0.373     | 0.067         | <0.001     | –                         | –             | –          | N/S         | N/S           | N/S        |  |
+| Fixed Disp. Parameter           | 3.098                    | 0.266         | <0.001     | –           | –             | –          | 5.220                     | 0.538         | <0.001     | –           | –             | –          |  |
+
+|                                      | Pre-Pandemic (2015–2019)         |                         | Post-Pandemic (2020–2022)        |                         |  |
+|--------------------------------------|----------------------------------|-------------------------|----------------------------------|-------------------------|--|
+| CMP Model                            |                                  | HTCMP Model             | CMP Model                        | HTCMP Model             |  |
+| Goodness-of-Fit (GOF) Statistics     |                                  |                         |                                  |                         |  |
+| Number of observations               | 208                              | 208                     | 130                              | 130                     |  |
+| Number of parameters                 | 6                                | 7                       | 7                                | 10                      |  |
+| Log-likelihood at convergence (LLβ)  | − 252.316                        | − 245.120               | − 115.332                        | − 109.778               |  |
+| Log-likelihood at zero (LL0)         | − 278.138                        | − 278.138               | − 134.487                        | − 134.487               |  |
+| Deviance (D) ¼ (2*(LLβ − LL0))       | 51.64 (P-Value < 0.001)          | 66.04 (P-Value < 0.001) | 38.31 (P-Value < 0.001)          | 49.42 (P-Value < 0.001) |  |
+| LRT (HTCMP vs CMP)                   | 14.392 (df = 1, P-Value < 0.001) |                         | 11.108 (df = 3, P-Value = 0.011) |                         |  |
+| McFadden pseudo R2                   | 0.093                            | 0.119                   | 0.142                            | 0.184                   |  |
+| AIC (Akaike Information Criterion)   | 516.63                           | 504.24                  | 244.66                           | 239.56                  |  |
+| BIC (Bayesian Information Criterion) | 536.66                           | 527.60                  | 264.74                           | 268.23                  |  |
+| MAD (Mean Absolute Deviance)         | 0.577                            | 0.515                   | 0512                             | 0.490                   |  |
+| MSPE (Mean Square Prediction Error)  | 0.578                            | 0.566                   | 0.505                            | 0.493                   |  |
+
+<sup>\*</sup> N/S = not statistically significant at 10 % level.
+
+![](_page_4_Figure_6.jpeg)
+
+**Fig. 2.** Right Shoulder Width (in "feet") versus Motorcycle Crashes per Segment.
+
+method accounts for variations in traffic volume and other risk factors that traditional metrics overlook. As a result, the EB method provides a more precise assessment of crash risk, highlighting locations that may not appear as critical using only crash frequency and crash rate metrics.
+
+Furthermore, an in-depth investigation of the motorcycle crashes at the top ten HCLs, pre- and post-pandemic, was conducted to identify the most frequent crash characteristics and facilitate targeted countermeasures. The analysis revealed that motorcycles were the most frequent atfault vehicles, with "single-vehicle" and "rear-end" collisions being the most common manner of collision at the HCLs. Furthermore, examining the crash causes from the police crash narratives revealed that "lost
+
+motorcycle control" and "failed to slow down or merge" were the most frequent crash causes. These findings highlight the susceptibility of motorcycles to crashes due to their inherent instability and emphasize the need for advanced driver assistance systems (ADAS) such as brake assist and lane change assist systems in motorcycles for improved rider safety ([Anaya et al., 2017](#page-5-0)).
+
+## **6. Conclusions and recommendations**
+
+This study investigated motorcycle crash frequency along rural multilane roadway segments in Kentucky by developing motorcyclespecific SPFs. Motorcycle-related crashes from 2015 to 2022 and various road-specific characteristics (e.g., posted speed limits, shoulder width, and AADT) were retrieved from KYTC and analyzed. To account for the effect of COVID-19, separate SPFs were fitted for each of the prepandemic (2015–2019) and post-pandemic (2020–2022) periods. CMP and HTCMP models were developed and compared while accounting for motorcycle crash under-dispersion during pre- and post-pandemic periods.
+
+For both pre- and post-pandemic periods, the HTCMP model consistently outperformed its CMP counterpart across various goodnessof-fit measures and prediction capabilities. The superiority of the HTCMP model is likely attributed to the varying dispersion, expressed as a function of site-specific characteristics, compared to the CMP model, which has a fixed dispersion parameter. From the developed SPFs, for both pre- and post-pandemic, AADT and undivided roadways were significantly associated with increased motorcycle crash frequencies, while posted speed limits (≥55 mph) were associated with reduced motorcycle crash frequency along rural multilane roadway segments. <span id="page-5-0"></span>Post-pandemic, the presence of cable median barrier was associated with reduced crash frequency, whereas wider right shoulders were associated with increased crash frequency. Furthermore, the predicted crash frequencies from the fitted HTCMP models were used to identify higher-than-expected motorcycle crashes along rural multilane segments in Kentucky using the EB method.
+
+Based on the study findings, proposed safety countermeasures to enhance motorcyclists' safety along rural multilane roadway segments in Kentucky are suggested, as follows:
+
+- Providing right shoulder widths within 6 feet (i.e., ≤ 6 ft) along rural multilane segments since significant increase in motorcycle crashes was noticed with right shoulder widths 6 feet or wider.
+- Since single-vehicle collisions were the most common crash type from the in-depth motorcycle HCL investigation, it is suggested to ensure provision of centerline grooved rumble strips along undivided rural multilane roadway stretches.
+- From the in-depth motorcycle HCL investigation, it is encouraged to adopt driver assistance systems in motorcycles, including brake assist and lane change assist systems to enhance rider safety.
+
+#### **7. Study limitations and future scope**
+
+It is worth mentioning that the insights drawn from the motorcyclespecific SPFs along rural multilane roadway segments in this study mainly pertain to motorcycle crashes in Kentucky. Thus, caution is advised when transferring these findings to a broader context. Also, the study's focus on rural multilane roadways means that its findings might not fully apply to other roadway types, e.g., urban roads. Moreover, this paper only dealt with under-dispersed crash data (i.e., crash variance is less than its mean), which may limit the generalizability of the results to scenarios involving over-dispersed crash data (i.e., when crash variance is greater than its mean). Nevertheless, the study findings can be transferable to states/regions that exhibit similar traffic and motorcycle crash characteristics as in Kentucky. To expand upon this study, future research can compare the SPF model outcomes to rural multilane roadways in different U.S. states to assess the transferability of the proposed models.
+
+### **CRediT authorship contribution statement**
+
+**Bharat Kumar Pathivada:** Data curation, Formal analysis, Software, Writing – original draft. **Arunabha Banerjee:** Validation, Writing – review & editing. **Kirolos Haleem:** Conceptualization, Supervision, Writing – review & editing, Funding acquisition. **Tathagatha Khan:**  Validation. **Dylan Justice:** Data curation, Formal analysis.
+
+## **Declaration of competing interest**
+
+The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
+
+## **Acknowledgment**
+
+The authors would like to acknowledge the Kentucky Transportation Cabinet (KYTC) - Highway Safety Improvement Program (HSIP) for the grant provided to conduct this research and for providing the necessary crash and roadway data used in this research. The opinions, findings, and conclusions in this paper are those of the authors and not necessarily those of KYTC.
+
+#### **Data availability**
+
+The authors do not have permission to share data.
+
+#### **References**
+
+- [Alnawmasi, N., Mannering, F., 2019. A statistical assessment of temporal instability in](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0005) [the factors determining motorcyclist injury severities. Anal. Methods Accid. Res 22,](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0005)  [100090](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0005).
+- [Anaya, J.J., Ponz, A., García, F., Talavera, E., 2017. Motorcycle detection for ADAS](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0010) [through camera and V2V communication, a comparative analysis of two modern](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0010) [technologies. Expert Syst. Appl. 77, 148](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0010)–159.
+- [Asal, H.I., Said, D.G., 2019. An approach for development of local safety performance](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0015) [functions for multilane rural divided highways in Egypt. Transportation Research](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0015)  [Record: Journal of the Transportation Research Board 2673 \(10\), 510](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0015)–521.
+- [Banihashemi, M., 2015. Is horizontal curvature a significant factor of safety in rural](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0020) [multilane highways? Transportation Research Record: Journal of the Transportation](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0020)  [Research Board 2515 \(1\), 50](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0020)–56.
+- [Chand, S., Yee, E., Alsultan, A., Dixit, V.V., 2021. A descriptive analysis on the impact of](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0025)  [COVID-19 lockdowns on road traffic incidents in Sydney, Australia. Int. J. Environ.](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0025)  [Res. Public Health 18 \(21\), 11701](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0025).
+- [Cheng, W., Gill, G.S., Sakrani, T., Dasu, M., Zhou, J., 2017. Predicting motorcycle crash](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0030)  [injury severity using weather data and alternative Bayesian multivariate crash](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0030) [frequency models. Accid. Anal. Prev. 108, 172](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0030)–180.
+- [Farid, A., Ksaibati, K., 2021. Modeling severities of motorcycle crashes using random](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0035)  [parameters. Journal of Traffic and Transportation Engineering \(English Edition\) 8](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0035) [\(2\), 225](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0035)–236.
+- [Flask, T., Schneider IV, W.H., Lord, D., 2014. A segment level analysis of multi-vehicle](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0040)  [motorcycle crashes in Ohio using Bayesian multi-level mixed effects models. Saf. Sci.](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0040)  [66, 47](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0040)–53.
+- [Haque, M.M., Chin, H.C., Huang, H., 2010. Applying Bayesian hierarchical models to](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0045) [examine motorcycle crashes at signalized intersections. Accid. Anal. Prev. 42 \(1\),](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0045)
+- [Hauer, E., Council, F.M., Mohammedshah, Y., 2004. Safety models for urban four-lane](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0050)  [undivided road segments. Transportation Research Record: Journal of the](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0050)
+- [Transportation Research Board 1897 \(1\), 96](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0050)–105. Highway Safety Manual (HSM). American Association of State Highways and Transportation Officials (AASHTO), Washington, D.C., 2010.
+- [Hosseinpour, M., Love, R., Williams, B., Haleem, K., 2022. Comprehensive investigation](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0060)  [of commercial motor vehicle crashes along roadway segments in Kentucky.](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0060) [Transportation Research Record: Journal of the Transportation Research Board 2676](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0060)  [\(7\), 655](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0060)–671.
+- [Islam, M.R., Abdel-Aty, M., Islam, Z., Zhang, S., 2022. Risk-compensation trends in road](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0065)  [safety during COVID-19. Sustainability 14 \(9\), 5057](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0065).
+- Kentucky Transportation Cabinet (KYTC). Interactive Statewide Traffic Counts Map. https://maps.kytc.ky.gov/trafficcounts/. Accessed March 9, 2024.
+- Kentucky Transportation Center (KTC), Kentucky Traffic Collision Facts: 2021 Report, 2023. CollisionFacts2021.pdf (ky.gov). Accessed April 15, 2024.
+- [Khaddar, S., Pathivada, B.K., Perumal, V., 2023. Modeling over speeding behavior of](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0080) [vehicles using a random parameter negative binomial approach: a case study of](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0080)  [Mumbai India. Transportation Research Interdisciplinary Perspectives 18, 100790.](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0080)
+- [Lord, D., Mannering, F., 2010. The statistical analysis of crash-frequency Data: a review](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0085)  [and assessment of methodological alternatives. Transp. Res. A Policy Pract. 44 \(5\),](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0085) 291–[305](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0085).
+- [Lord, D., Guikema, S.D., Geedipally, S.R., 2008. Application of the](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0090)  Conway–Maxwell–[Poisson generalized linear model for analyzing motor vehicle](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0090) [crashes. Accid. Anal. Prev. 40 \(3\), 1123](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0090)–1134.
+- Manan, M.M.A., Jonsson, T., Varhelyi, ´ [A., 2013. Development of a safety performance](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0095) [function for motorcycle accident fatalities on Malaysian primary roads. Saf. Sci. 60,](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0095)  13–[20](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0095).
+- National Highway Traffic Safety Administration (NHTSA), Motorcycle: Traffic Safety Facts 2021 Data, 2023. 2021 Data: Motorcycles (dot.gov). Accessed April 15, 2024.
+- Ospina-Mateus, H., Jim´[enez, L.A.Q., Lopez-Valdes, F.J., Sana, S.S., 2021. Prediction of](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0105) [motorcyclist traffic crashes in Cartagena \(Colombia\): development of a safety](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0105) [performance function. RAIRO-Operations Research 55 \(3\), 1257](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0105)–1278.
+- [Pathivada, B.K., Banerjee, A., Haleem, K., 2024. Impact of real-time weather conditions](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0110)  [on crash injury severity in Kentucky using the correlated random parameters logit](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0110) [model with heterogeneity in means. Accid. Anal. Prev. 196, 107453.](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0110)
+- [Saeed, T.U., Hall, T., Baroud, H., Volovski, M.J., 2019. Analyzing road crash frequencies](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0115)  [with uncorrelated and correlated random-parameters count models: an empirical](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0115)  [assessment of multilane highways. Anal. Methods Accid. Res 23, 100101](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0115).
+- [Savolainen, P., Mannering, F., 2007. Probabilistic models of motorcyclists](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0120)' injury
+- [severities in single-and multi-vehicle crashes. Accid. Anal. Prev. 39 \(5\), 955](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0120)–963. [Shaheed, M.S., Gkritza, K., 2014. A latent class analysis of single-vehicle motorcycle](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0125) [crash severity outcomes. Analytic Methods in Accident Research 2, 30](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0125)–38.
+- [Shaik, M.E., Ahmed, S., 2022. An overview of the impact of COVID-19 on road traffic](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0130)  [safety and travel behavior. Transp. Eng. 9, 100119.](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0130)
+- [Shmueli, G., Minka, T.P., Kadane, J.B., Borle, S., Boatwright, P., 2005. A useful](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0135) [distribution for fitting discrete data: revival of the Conway](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0135)–Maxwell–Poisson [distribution. J. R. Stat. Soc. Ser. C. Appl. Stat. 54 \(1\), 127](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0135)–142.
+- [Yasin, Y.J., Grivna, M., Abu-Zidan, F.M., 2021. Global impact of COVID-19 pandemic on](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0140)  [road traffic collisions. World Journal of Emergency Surgery 16 \(1\), 1](http://refhub.elsevier.com/S0001-4575(25)00171-X/h0140)–14.