category: literaturenote citekey: kroyerpedestrianbicyclistflowsaccident2016 title: Pedestrian and bicyclist flows in accident modelling at intersections. Influence of the length of observational period authors: "Kröyer, Höskuldur R.G." year: 2016 date: 2016-00-00 2016 doi: 10.1016/j.ssci.2015.09.015 publication: Safety Science url: "https://doi.org/10.1016/j.ssci.2015.09.015" zotero_key: RKC8BE9B zotero_storage: EB4R9S65 collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Kröyer, Höskuldur R.G."

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Safety Science

journal homepage: www.elsevier.com/locate/ssci

Pedestrian and bicyclist flows in accident modelling at intersections. Influence of the length of observational period

Höskuldur R.G. Kröyer ⇑

Department of Technology and Society, Lund University, Sweden

article info

Article history: Received 14 April 2015 Received in revised form 2 September 2015 Accepted 11 September 2015 Available online 22 October 2015

Keywords: Safety performance function Pedestrians Bicyclists Accidents Observation period Safety in numbers

abstract

Safety performance functions are frequently used to describe the relation of various factors to the number of accidents, often focusing on exposure of vulnerable road users. Those studies require reliable data regarding exposure, but collecting such data is resource demanding, often resulting in short observational periods. This might produce measurement errors, influencing the reliability of the models.

The aim of this work is twofold: to create a safety performance functions for pedestrian and bicyclist accidents at urban intersections, and to analyze the reliability of accident models based on short observational periods, i.e. how does random variation, resulting from short observational periods influence the models? This provides an aid for choosing between increasing the number of sites and the length of the observational period at each site. Accident data were compiled and traffic counting was conducted at 113 urban intersections in Sweden. Multiple samples were created from the counting sessions, facilitating tests of the models' reliability based on lengths of observational periods between 15 and 180 min. Four safety performance functions (accident types) were created. All models showed a safety in numbers effect, including the model for single pedestrian accidents, which might suggest that maintenance and infrastructure quality constitute an important factor for the safety in numbers effect. A sensitivity analysis showed geometric factors, describing the infrastructure quality, influencing the safety in numbers effect, further supporting this hypothesis. The safety performance functions based on short observational periods showed low reliability, indicating that those models are subjected to a considerable measurement error.

1. Introduction

Many cities aim to increase their modal share of pedestrians and bicyclists because of the expected positive influence on the traffic environment, sustainability, and public health. But since exposure is strongly related to the number of accidents, such increases might result in a higher number of accidents involving pedestrians and bicyclists. Therefore, models that focus on these relations are of great importance.

Several known factors influence the risk of an accident and thereby the number of accidents, e.g. road design (Chang, 2005; Elvik and Vaa, 2004), road user behavior, temporal environment factors such as weather (Eisenberg, 2004; Shankar et al., 1995) and speed environment (Nilsson, 2004). Another influential factor is the traffic volume (i.e. exposure); it has been shown to have a strong relation to the number of accidents, for both pedestrians and bicyclists (Elvik, 2009; Geyer et al., 2006; Jonsson, 2005). This relation is often approximated with the mathematical model presented in Eq. (1) (Elvik, 2009), where N is the number of expected accidents, Ei is the exposure (e.g. traffic volume) for different travel modes, b⁰ is the intercept of the model, and bⁱ is a constant. This relation seems to be non-linear, where the estimated bⁱ parameters are usually between 0 and 1 for traffic volume variables; hence, the accident risk per road user is lower when the traffic volumes, or exposure, of vulnerable road users is higher (Brüde and Larsson, 1993; Elvik, 2009; Geyer et al., 2006; Jacobsen, 2003; Jonsson, 2005, 2013; Turner et al., 2006). This phenomenon is frequently referred to as safety in numbers (e.g. Jacobsen, 2003; Jonsson, 2013).

$$N = e^{\beta_0} \prod E_i^{\beta_i} \tag{1}$$

Mathematical models in the form of Eq. (1) are often referred to as safety performance functions. Some studies apply Poisson regression for the modelling process (Geyer et al., 2006; Ye et al., 2013). Poisson distribution assumes equal mean and variance, and since accident data are often over-dispersed, this might influence the significance level of the parameters (Cameron and Trivedi, 1990; Poch and Mannering, 1996; Washington et al., 2013).

^⇑ Tel.: +46 46 222 9198; fax: +46 46 222 9100. E-mail address: hoskuldur.kroyer@tft.lth.se

Negative binomial regression (based on negative binomial distribution) is better able to handle this over-dispersion (Poch and Mannering, 1996; Washington et al., 2013).

Accident data frequently demonstrate an unusually high proportion of units (in this case, the unit is sites) with zero accidents compared to the mathematical distributions the model is based on; some authors have therefore suggested using zero inflated Poisson or zero inflated negative binomial models to improve the models (Chin and Quddus, 2003; Jang et al., 2010; Miaou, 1994; Yan et al., 2012). This results in a better fit of the model to the data (Lord et al., 2005). Those models assume, however, that the unit can be in either one of two states: the site is either 'safe' or the number of accidents follows a Poisson or negative binomial distribution (Lord et al., 2005; Washington et al., 2013). This might not be an accurate description, since no traffic site is 'safe' (Lord et al., 2007), and the extreme number of zero accidents likely results from low exposure (Lord et al., 2005). Therefore, the negative binomial regression continues to be frequently used for constructing safety performance functions.

The negative binomial distribution is described by Eq. (2) (Washington et al., 2013), where k is the mean, j-¹ is the overdispersion parameter and the variance is k + k² /j. The resulting safety performance function takes the form shown in Eq. (3), where N is the number of expected accidents, E¹ and E² are the exposure variables and b0, b1, b² and b^k are constants. Additional explanatory variables (Ek) are sometimes included in the model (e.g. geometric variables).

$$P(Y = y) = \left(\frac{\kappa}{\kappa + \lambda}\right)^{\kappa} \frac{\Gamma(\kappa + y)}{\Gamma(y)\Gamma(\kappa)} \left(\frac{\lambda}{\kappa + \lambda}\right)^{y}$$ (2)

$$N = e^{\beta_0} \cdot E_1^{\beta_1} \cdot E_2^{\beta_2} \cdot e^{\Sigma \beta_k E_k} \tag{3}$$

Some studies include only geometric variables and exclude exposure variables for pedestrians and bicyclists (Chin and Quddus, 2003; Hosseinpour et al., 2013). A possible explanation for this is that traffic volumes are not always available, and it is demanding to collect them. However, since there seems to be a non-linear relation between exposure and risk, this limits the models' validity, and they can be improved by including traffic volumes (Jonsson, 2005). Some authors apply proxy variables to account for exposure, e.g. bicycle lane kilometers (Wei and Lovegrove, 2013), share of cycling and walking as travel mode (Jacobsen, 2003), the mean distance travelled per day by a typical cyclist or pedestrian (Jacobsen, 2003; Schepers, 2012), and simulated exposure data (Geyer et al., 2006). This is a step in the right direction and provides interesting aspects; however, the relation between those proxy variables, the actual traffic volumes and the accident location is in some cases unknown.

Traffic counts are in some ways preferable; though they have some limitations (see Elvik, 2013a, 2015). Several studies (e.g. Ekman, 1996; Jonsson, 2005, 2013; Wang and Nihan, 2004) have used exposure values from counting as the basis for models that are location specific (i.e. accident data and exposure data are related to certain locations). For practical reasons, those studies usually apply average traffic volumes (e.g. average daily traffic), but it can be argued that this is also a proxy variable, where the exposure variable of choice would be the number of interactions or occurrences that can, and have, given probability of resulting in an accident (Elvik, 2013a), or focus on the exposure at the time of the accident (Mensah and Hauer, 1998).

Anyhow, when creating a safety performance function, the researcher must weigh the benefits of including many sites, which will produce a highly statistically significant model and might allow for the inclusion of several explanatory variables, against having longer observational periods, which will provide a more reliable estimation of the actual traffic volumes. Since collecting exposure data is demanding, short observational periods are often used, as brief as 15–20 min per site (e.g. Jonsson, 2013; Schepers et al., 2011). This might influence the reliability of the exposure estimation, and, consequently, the reliability of the safety performance function itself (Davis, 2000; Maher and Summersgill, 1996).

The aims of this study are (1) to create a safety performance function for accidents involving pedestrians and bicyclists at urban intersections, based on three hours of observations, and (2) to analyze the reliability of accident models based on short observational periods, i.e. explore how random variation, resulting from short observational periods at each site influences the models, and thereby provide an aid for choosing between increasing the number of sites and extending the length of the observational period per site.

2. Method

Creating a safety performance function requires a number of steps: site selection, compilation of accident data, determination of potential explanatory variables, data collection and statistical modelling.

2.1. Process of site selection and collection of accident data and geometric variables

The first step was to decide on sites to be included in the study. For a city to be eligible it had to fulfill three criteria: (a) the municipalities' population was between 50 and 200 thousands, (b) the traffic accident data for the period 2008–2012 had been recorded by the Swedish Traffic Accident Data Acquisition (STRADA) and (c) at least 10 main street intersections existed within city limits, that were eligible for the study. Six cities were chosen: Eskilstuna (population 99,729), Halmstad (94,084), Helsingborg (132,989), Kalmar (63,887), Kristianstad (81,009) and Västerås (142,131; SCB, 2014). The next step was choosing which intersections to include in the study. Eligible intersections fulfilled 4 criteria: (a) two main streets crossed there, (b) they were not roundabouts, (c) they exhibited at least some potential for bicyclist and pedestrian flows, and (d) nothing suggested that any change had occurred in the physical layout between 2008 and 2012 that would have significant influence on the accidents or the bicyclist or pedestrian flows. The final dataset included 113 intersections, 16 in Eskilstuna, 25 in Halmstad, 27 in Helsingborg, 12 in Kalmar, 20 in Kristianstad and 13 in Västerås.

Accident data were compiled from STRADA, which contains police and hospital records for traffic accidents that occur in Sweden. That the accident data includes hospital reports is essential to this work, since single pedestrian and single bicyclist accidents are frequently missing from police reports (Elvik and Mysen, 1999; Jonsson et al., 2011). All accidents registered in STRADA that occurred at each intersection or incoming links between 2008 and 2012 were studied and those who belonged to the intersections were included in the dataset. Basing the analyses on accident data for five years was a compromise. If the data period is too long, it might involve a time trend bias (i.e. the traffic is constantly changing; hence, older accident data might not reflect the current situation), while if it is too short it might be influenced by extreme random variability (Wei and Lovegrove, 2013). Four accident types were included in the dataset: (i) single pedestrian accidents, (ii) single bicyclist accidents, (iii) collisions between a motorized vehicle and a pedestrian, and (iv) collisions between a motorized vehicle and a bicyclist (single accidents involve only one road user – observe that those models exclude accidents between pedestrians and bicyclists; and between two or more bicyclists).

Several variables were collected, in order to include the quality of the infrastructure in the modelling process: the city, the traffic environment, the intersection's control mechanism, speed limit, whether it was a three- or four-leg intersection, the level of bicyclist integration with motorized traffic, the number of intersection-legs crossable by pedestrians and/or bicyclists and whether there was a traffic refuge, speed hump or a marked crosswalk. Henceforth, those variables are referred to as geometric variables.

2.2. Exposure data and the modelling process

Traffic volumes of motorized traffic were acquired from the municipalities. The measurement point was usually along an approach to the intersection. In some cases, however, the flow was estimated based on traffic volumes on nearby links, and if no data were available, flow was estimated based on the number of houses, land use and road network. Most data were less than five years old, but the traffic volume was scaled for traffic changes to render it representative for the year 2012 (Trafikverket, 2013).

Pedestrian and bicyclist volumes were observed for three hours at each site, either before or after noon. The measurements were carried out between 8:30 and 12:00 or 13:30 and 17:00 (with two 15 min breaks in each period: 9:30, 10:45, 14:30 and 15:45). The observation periods were conducted during off-peak hours, because the exposure values had to be scaled to average daily values in order to render them compatible for the modelling process. The peak hour for pedestrian and bicyclist flows is often short and extreme (Jonsson, 2005; Turner et al., 2006). Including the peak hours in the observations might cause problems in the scaling process, particularly if the time of the peak hour observed at an individual site is not the same as that of the distribution that the scaling values are based on.

Each observation at an individual site was divided into twelve 15 min periods. Pedestrians were divided into two groups: (a) pedestrians who did not cross a street and (b) pedestrians who crossed a street. The former group was intended to provide a suitable exposure variable for the single pedestrian accidents. The second variable was collected to facilitate the modelling of accidents between motorized vehicle and pedestrians. It was based on an earlier approach (Brüde and Larsson, 1993). Each time a pedestrian crosses a street, some risk of collision exists; therefore, crossing two streets should result in a higher probability of collision. The number of times a given pedestrian crosses a street might therefore be more descriptive for the model than the number of pedestrians simply entering the intersection. The bicyclists were counted as one group, mainly because the interactions between bicyclists and motorized traffic vary so extensively that it would have been impractical to divide them into groups.

Traffic volumes vary throughout the day; therefore, an unscaled observation from one time period is not compatible with an unscaled observation from another time period. For this reason, an exposure distribution curve had to be created, i.e. a curve that shows what proportion of the daily traffic occurs during each 15 min period. This curve can then be used to scale the exposure data for each 15 min period to an average daily volume so that the exposure data are compatible between sites. (1) The first step was to use external data (Ahlström, 2013; Indebetou and Quester, 2007; Jonsson, 2005) to determine what proportion of the daily traffic volume occurs during the entire observational period, i.e. what proportion of the pedestrian and bicycle traffic occurs between 8:30 and 12:00 and 13:30 and 17:00. (2) Next, the observations were divided into groups based on the type of traffic environment (a: central and residential areas, and b: arterials, industrial and rural areas) and observational time (a: before noon, and b: after noon). (3) From these divisions, twelve different exposure distribution curves were created (for each observation group there are three road user groups, a: pedestrians not crossing, b: pedestrian crossings and c: bicyclists, hence 2 2 3 = 12), and each curve is a mean of all the sites that fit the specific group's criteria, see Fig. 1 (the two 15 min breaks that were not measured were approximated as the mean of the 15 min periods before and after the break). Even though this procedure has limitations, it is the best data available and should provide an acceptable basis for scaling the exposure data.

A safety performance function based on 180 min of observations (scaled with the appropriate exposure distribution curve) was created (henceforth referred to as base models). This is the most reliable data, and for the purpose of this study it will be assumed that this is a 'correct' estimate of the traffic exposure so that it may be used for comparison. This is, of course, not a fully accurate description, since in reality exposure may vary between days, weeks and years (Ahlström, 2013; Esawey et al., 2013). Four different accident types were modelled: i: single pedestrian, ii: single bicyclist, iii: motorized vehicle – pedestrian, and iv: motorized vehicle – bicyclist. If these accident types are correlated, the analysis might be improved by applying multivariate analysis (Mannering and Bhat, 2014). However, when controlled for exposure, the accident types showed low correlation, suggesting that traditional analysis suffice.

In order to estimate the reliability of the models', based on shorter observational periods, related to the 'correct' model, it was necessary to create several new datasets based on shorter observational periods of various lengths. First, 10 datasets were created, each based on 15 min of observations per site. At each individual site, the 15 min observational period used was chosen at random (theoretically allowing for 12113 different combinations, given 113 intersections and 12 periods per site), i.e. each dataset can be thought of as representing an 'independent' study that conducted 15 min of observation at 113 sites. The next step was to create 10 dataset for 30 min of consecutive observations. This procedure was repeated for 45, 60, 90 and 120 min of consecutive observations for each of the sites. This yields 10 datasets for each observational period lengths, where each dataset might represent an 'independent' study. The reason for having several datasets for each length was to see how the models might vary due to random selection of observational periods.

The statistical modelling applied negative binomial regression. Even though it has some limitations (Miaou, 1994), this approach accounts adequately for the unequal mean and variance (Maher and Summersgill, 1996) and is frequently used for accident modelling (Lord and Miranda-Moreno, 2008). The influence of observational periods' length on this model type is therefore important.

The modelling started with the natural logarithm of the primary exposure variable (for model i, the number of pedestrians not crossing a street, for ii & iv, the number of bicyclists, and for iii, the number of pedestrian crossings). Next, for models iii and iv, the natural logarithm of the exposure variable for motorized vehicles was included. Including the exposure variable for motorized vehicles in the models is somewhat troublesome because it correlates with the primary exposure variable (CorrLN(Ped. crossing) – LN(Mot. veh.) = 0.41, CorrLN(Bicyclists) – LN(Mot. veh.) = 0.55). It was decided, however, that doing so was acceptable given that these accident types should be related to both of those exposure variables, and that this is a 'common practice' when creating safety performance functions (Elvik, 2009).

The next step was to add the geometric variables; however, none of those variables was statistically significant (marked crosswalks had to be excluded because these were highly correlated with traffic volumes). Excluding geometric variables can result in a biased estimate of the models (Mannering and Bhat, 2014). It was therefore decided to use two modelling approaches: (a)

Fig. 1. Exposure distribution curves that describe the proportion of daily traffic that occurs during each 15 min period. These curves can be used to scale the exposure data for each 15 min period to an average daily volume. The exposure distribution curves are based on the following number of sites: (a) central and residential areas nmorning = 26, nafternoon = 40, and (b) industrial, arterials and rural areas nmorning = 24, nafternoon = 23.

8:00 8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00 16:30 17:00 17:30

Time of day

models that are based on only the exposure variable (henceforth referred to as parsimonious models); and (b) models that include geometric variables based on a sensitivity analysis (henceforth referred to as fully specified models). The modelling started with the exposure variables, and then each geometric variable was added to the model and investigated how much it influenced the exponents of the traffic volumes. The final model included all geometric variables that had an influence greater than 5% on the exponents of the traffic volumes. Finally, the models for the shorter observational periods were created based on the variables included in the base models.

In order to compare the models based on shorter observational periods to the base models, five different statistical measures were applied: (a) the parameters of the safety performance function, i.e. bⁱ (elasticity factors are frequently used to evaluate the impact of 1% change in independent variable on the expected frequency of accidents (Washington et al., 2013); however, since the elasticity factor is directly proportional to the parameter of the models, the effect on those will be similar); (b) the Akaike information criterion (AIC value); this measure is frequently used to compare different models, where lower value suggest a better model; (c) the significance level of the models' parameters (the p values); (d) the standard error of the models' parameters; and (e) the mean absolute residuals of the model (see Eq. (4)).

Mean absolute residuals = $$\frac{\sum |Residuals|}{Number of sites}$$ (4)

3. Results

The intercept and the primary exposure factor were statistically significant in all four base models, see Table 1. The traffic flow of motorized vehicles was statistically significant for model iv, collisions between bicyclists and motorized vehicles (p = 0.021), it was however not statistically significant for model iii, i.e. collisions between pedestrians and motorized vehicles (p = 0.221). To maintain consistency between the models (iii and iv), it was decided to include the motorized flow even in model iii. As expected, all exposure parameters, bi, are between 0 and 1. The resulting accident models are presented in Eqs. (5)–(8).

Industrial areas, arterials and rural areas

$$N{\text{Ped. single}} = e^{-5.992} \cdot E{\text{Ped. not crossing a street}}^{0.589}$$ (5)

$$N{\text{Cycl.single}} = e^{-7.282} \cdot E{\text{Cycl.}}^{0.676}$$ (6)

$$N{\text{Ped.-M.veh.}} = e^{-12.937} \cdot E{\text{Ped. crossings}}^{0.554} \cdot E_{\text{M.veh.}}^{0.652}$$ (7)

$$N{\text{Cycl.-M.veh.}} = e^{-11.273} \cdot E{\text{Cycl.}}^{0.427} \cdot E_{\text{M.veh.}}^{0.687}$$ (8)

The parameters of the two single accident types (Eqs. (5) and (6)) and the two accident types involving motorized vehicles (Eqs. (7) and (8)) are similar, and the parameters fall in almost all cases within the 95% confidence interval of the parameters from the other models. Those accident types were however kept separate because of the fundamental differences between those groups (e.g. different interaction between pedestrians/bicyclists and motorized vehicles; and different travelling speed for pedestrians and bicyclists in single accidents).

Geometric variables were added to the models based on the sensitivity analysis. Since those variables were not statistically significant to begin with, it was not unexpected that the significance levels were poor. Also, it had a negative effect on the statistical significance level of the exposure variables. Even so, including those variables allows controlling for their influence, and confirms that the influence the observational period's length per site has on the models' reliability is not simply a result of excluding those variables.

To include the geometric variables had influence on the exponents of the exposure variables, resulting in the models showing an enhanced or reduced safety in numbers effect (compared to the parsimonious models), thereby influencing the predictions and residuals at individual sites (even though the actual data the

Table 1 Parameter estimations for the parsimonious and fully specified base models.

Variables	Parsimonious models				Fully specified models
	Model i	Model ii	Model iii	Model iv	Model i	Model ii	Model iii	Model iv
Intercept Pedestrian flow Bicycle flow Motorized vehicle flow	5.99 (1.40)a 0.59 (0.25)a	7.28 (1.44)a 0.68 (0.20)a	12.94 (4.58)a 0.55 (0.27)a 0.65 (0.53)*	11.27 (2.45)a 0.43 (0.18)a 0.69 (0.30)a	4.75 (1.16)a 0.43 (0.20)c	5.18 (1.53)a 0.37 (0.23)c	12.78 (6.72)b 0.30 (0.36) 0.64 (0.77)	11.48 (3.31)a 0.36 (0.18)a 0.71 (0.37)b
City** Helsingborg/Kristianstad Eskilstuna/Västerås Kalmar/Halmstad					0 1.14 (0.68) 1.47 (0.81)	0 0.31 (0.53) 0.27 (0.42)
Traffic environment Arterials Central Rural/industrial Residential						0.36 (0.45) 0.06 (0.65) 1.73 (1.00)b 0	0.48 (0.86) 0.57 (0.97) 1.09 (1.33) 0
Traffic control Traffic signal Right of way/yield/stop sign						0.33 (0.46) 0	0.94 (0.88) 0	0.22 (0.36) 0
Speed limit 30–40 km/h 50 km/h 60–70 km/h							1.73 (1.40) 1.22 (1.33) 0	0.39 (0.49) 0.60 (0.43)c 0
Sight conditions Good Average/poor						0.28 (0.39) 0		0.43 (0.30)c 0
Bicyclist integration with car traffic Fully integrated Partly integrated Fully segregated								0.47 (0.57) 0.16 (0.37) 0
Pearson v2 /df	1.85	0.88	1.14	0.88	1.78	0.87	1.37	0.839

^* This variable was included in the parsimonious model even though it was not statistically significant, in order to maintain compatibility between models iii and iv and to observe the interaction between the parameters (standard error in parenthesis).

models are based on have not changed), see Table 1. Since none of the geometric variables were statistically significant, we should be cautious about drawing too many conclusions regarding the influence of individual geometric variables.

3.1. Parameter estimation and the length of an observational period

The analysis of each dataset, each for the different observational period length, resulted in its own set of parameters. Fig. 2 shows the resulting parameters for the parsimonious models, where each cross represents one model (based on one of the datasets) and each gray circle is around the mean value of the parameter of all the datasets that had that specific length of observational period.

Fig. 2 shows three important tendencies. (1) The confidence intervals of the parameters of the base models are wide, and therefore most of those values fall within those intervals. (2) The parameters vary considerably within each observational period length. For example, the 10 batches of datasets for model i and 15 min observational period lengths resulted in intercepts (b0) between 6.00 and -3.95 and b¹ between 0.23 and 0.619, when the base model yielded b⁰ ¼ -5:99 and b¹ ¼ 0:589. A similar pattern can be seen for models ii, iii and iv. This variance in the resulting parameters seems to decrease as the observational periods gets longer, i.e. the reliability of the models is lower when they are based on short observational periods, but it improves with longer observational periods. (3) The parameters for the primary flow variable (pedestrian and bicyclist flows) seems to be slightly underestimated when based on short observational periods in all four models; hence, the models based on short observational periods overestimate the safety in numbers effect slightly. Also, the intercept seems to be overestimated in the models for single accidents based on short observational periods. For models iii and iv, the parameter for the motorized vehicle flows shows some tendencies towards overestimation. The fully specified model shows similar effects, i.e. even if the geometric variables are included, the intercept and exponents for traffic volumes show the same tendencies.

The results presented here demonstrate that the model is influenced by measurement error (even though it is within the confidence intervals of the base models in most cases). Therefore, it is worth investigating whether this measurement error, or the improvement associated with longer observational periods can be observed in the models' statistical properties. The first parameters tested are the AIC values. For the parsimonious models, the longer observational periods results in none or only a slight improvements in the AIC values. The difference is only marginal and much smaller than the variation of AIC values within the models based on the same length of observational period, see Table 2. The p values of the models parameters (bi) show in most cases considerable improvement as the observational periods became longer (those values, however, vary somewhat within each group of observational period lengths), however, the standard error was usually unchanged or increased. The mean absolute residuals show only a marginal improvement. The fully specified models show similar effects.

^** The cities were grouped based on their geographic location.

^a p < 0.05.

^b p < 0.10.

^c p < 0.20.

Fig. 2. The model parameters for models i, ii, iii and iv. Each cross represents one model (based on one dataset), the circles surround the means of all models with that specific observational length, the black line represents the model parameter estimated for the base model, and the gray lines marks the 95% confidence interval for the parameters of the base model.

Table 2 Statistical properties of the models for observational periods of different lengths. Each value is a mean of the 10 models within the group of a given observational period length (standard deviation in parenthesis).

| | Length |-------------------|------------ | | 15 | Model i | | b0 | 4.60 (0.55)
| p-value b0 | <0.001 | Standard error b0 | 0.80 (0.14) | b1 | 0.35 (0.11) | p-value b1 | 0.025 (0.016) | Standard error b1 | 0.15 (0.02) | AIC | 130.69 (6.01) | Mean residuals | 0.38 (0.020) | Model ii | | b0 | 5.70 (1.00)
| p-value b0 | <0.001 | Standard error b0 | 1.13 (0.17) | b1 | 0.46 (0.14) | p-value b1 | 0.014 (0.027) | Standard error b1 | 0.16 (0.02) | AIC | 173.63 (3.93) | Mean residuals | 0.47 (0.019) | Model iii | | b0 | 12.91 (0.38)
| p-value b0 | 0.008 (0.007) | Standard error b0 | 4.76 (0.50) | b1 | 0.32 (0.10) | p-value b1 | 0.157 (0.106) | Standard error b1 | 0.21 (0.02) | b2 | 0.82 (0.10) | p-value b2 | 0.129 (0.044) | Standard error b2 | 0.53 (0.05) | AIC | 113.42 (1.99) | Mean residuals | 0.28 (0.005) | Model iv | | b0 | 11.00 (0.41)
| p-value b0 | <0.001 | Standard error b0 | 2.43 (0.05) | b1 | 0.28 (0.09) | p-value b1 | 0.034 (0.033) | Standard error b1 | 0.11 (0.04) | b2 | 0.77 (0.10) | p-value b2 | 0.010 (0.008) | Standard error b2 | 0.28 (0.01) | AIC | 255.70 (2.30) | Mean residuals | 0.71 (0.02) of observational period (min) | | | | | | | | | | | | --------------------------|------------------|------------------|------------------|------------------|------------------|-----------|--|--|--|--|--| | 30 | 45 | 60 | 90 | 120 | 180 | | | | | | | | | | | | | | | | | | | 5.49 (0.31)
| 5.67 (0.24)
| 5.88 (0.30)
| 6.02 (0.23)
| 5.97 (0.12)
| 5.99
| | | | | | | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | | | | | | | 1.12 (0.07) | 1.19 (0.09) | 1.30 (0.13) | 1.39 (0.12) | 1.37 (0.05) | 1.40 | | | | | | | 0.51 (0.06) | 0.54 (0.04) | 0.57 (0.05) | 0.59 (0.04) | 0.58 (0.02) | 0.59 | | | | | | | 0.013 (0.009) | 0.012 (0.005) | 0.014 (0.002) | 0.015 (0.002) | 0.016 (0.001) | 0.018 | | | | | | | 0.20 (0.01) | 0.21 (0.01) | 0.23 (0.01) | 0.25 (0.02) | 0.24 (0.01) | 0.25 | | | | | | | 127.56 (2.40) | 128.05 (2.00) | 127.70 (1.75) | 127.48 (1.22) | 128.10 (0.85) | 128.71 | | | | | | | 0.37 (0.007) | 0.37 (0.006) | 0.37 (0.006) | 0.37 (0.005) | 0.37 (0.003) | 0.37 | | | | | | | | | | | | | | | | | | | 6.47 (0.52)
| 6.82 (0.23)
| 6.74 (0.22)
| 6.98 (0.23)
| 7.00 (0.14)
| 7.28
| | | | | | | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | | | | | | | 1.28 (0.08) | 1.35 (0.06) | 1.34 (0.03 | 1.38 (0.02) | 1.40 (0.03) | 1.44 | | | | | | | 0.56 (0.07) | 0.61 (0.03) | 0.60 (0.03) | 0.64 (0.03) | 0.64 (0.02) | 0.68 | | | | | | | 0.002 (0.001) | 0.001 (0.001) | 0.001 (0.001) | 0.001 (0.001) | 0.001 (<0.001) | 0.001 | | | | | | | 0.18 (0.01) | 0.19 (0.01) | 0.19 (0.01) | 0.19 (<0.01) | 0.20 (<0.01) | 0.20 | | | | | | | 172.66 (2.05) | 171.87 (1.24) | 172.29 (1.04) | 171.87 (1.00) | 171.99 (0.46) | 171.18 | | | | | | | 0.47 (0.011) | 0.47 (0.007) | 0.47 (0.005) | 0.47 (0.004) | 0.47 (0.004) | 0.46 | | | | | | | | | | | | | | | | | | | 12.95 (0.29)
| 12.89 (0.27)
| 12.99 (0.31)
| 13.00 (0.15)
| 12.90 (0.08)
| 12.94
| | | | | | | 0.006 (0.004) | 0.005 (0.002) | 0.004 (0.001) | 0.004 (0.001) | 0.005 (0.001) | 0.005 | | | | | | | 4.61 (0.30) | 4.54 (0.18) | 4.55 (0.10) | 4.54 (0.06) | 4.55 (0.07) | 4.58 | | | | | | | 0.46 (0.11) | 0.52 (0.07) | 0.52 (0.05) | 0.52 (0.04) | 0.54 (0.03) | 0,55 | | | | | | | 0.084 (0.059) | 0.050 (0.029) | 0.048 (0.018) | 0.043 (0.012) | 0.042 (0.010) | 0.039 | | | | | | | 0.25 (0.02) | 0.26 (0.01) | 0.26 (0.01) | 0.26 (0.01) | 0.27 (<0.01) | 0.27 | | | | | | | 0.73 (0.10) | 0.67 (0.06) | 0.68 (0.07) | 0.68 (0.03) | 0.65 (0.02) | 0.65 | | | | | | | 0.177 (0.058) | 0.203 (0.033) | 0.198 (0.047) | 0.195 (0.027) | 0.216 (0.013) | 0.221 | | | | | | | 0.53 (0.03) | 0.52 (0.02) | 0.53 (0.01) | 0.52 (0.01) | 0.53 (0.01) | 0.53 | | | | | | | 112.40 (1.86) | 111.68 (1.10) | 111.89 (0.90) | 111.90 (0.56) | 111.73 (0.48) | 111.67 | | | | | | | 0.27 (0.008) | 0.27 (0.005) | 0.27 (0.004) | 0.27 (0.003) | 0.27 (0.001) | 0.26 | | | | | | | | | | | | | | | | | | | 11.14 (0.18)
| 11.16 (0.22)
| 11.21 (0.16)
| 11.29 (0.12)
| 11.24 (0.09)
| 11.27
| | | | | | | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | | | | | | | 2.45 (0.05) | 2.44 (0.03) | 2.44 (0.02) | 2.45 (0.02) | 2.44 (0.01) | 2.45 | | | | | | | 0.34 (0.05) | 0.39 (0.04) | 0.40 (0.03) | 0.39 (0.03) | 0.42 (0.02) | 0.43 | | | | | | | 0.034 (0.023) | 0.019 (0.012) | 0.015 (0.006) | 0.019 (0.007) | 0.014 (0.004) | 0.014 | | | | | | | 0.15 (0.01) | 0.16 (0.01) | 0.16 (<0.01) | 0.17 (<0.01) | 0.17 (<0.01) | 0.17 | | | | | | | 0.75 (0.06) | 0.71 (0.04) | 0.70 (0.03) | 0.71 (0.02) | 0.69 (0.02) | 0.69 | | | | | | | 0.012 (0.006) | 0.017 (0.005) | 0.017 (0.005) | 0.015 (0.003) | 0.020 (0.004) | 0.021 | | | | | | | 0.29 (0.01) | 0.29 (0.01) | 0.29 (0.01) | 0.29 (<0.01) | 0.30 (<0.01) | 0.30 | | | | | | | 256.09 (1.66) | 255.31 (1.22) | 255.07 (0.76) | 255.59 (0.73) | 255.04 (0.53) | 255.15 | | | | | | | 0.71 (0.01) | 0.70 (0.01) | 0.70 (0.01) | 0.70 (0.005) | 0.70 (0.01) | 0.70 | | | | | |

4. Discussion and conclusions

4.1. The relation between traffic volumes and the number of accidents

The safety performance functions show that the number of accidents increases with an increase in the number of pedestrians and bicyclists (for respective accidents in models i to iv) and in the volume of motorized vehicles (in models iii and iv). The parsimonious model i resulted in intercept b⁰ ¼ -5:992 and b¹ ¼ 0:589 and the fully specified model resulted in intercept b⁰ ¼ -4:749 and b¹ ¼ 0:434, indicating that there is some safety in numbers effect. It is interesting to compare the models to earlier work. In that comparison the intercept can be expected to differ between different studies since the intersections those studies are based on differs. However, the exponents for the exposure parameters can be expected to show some similarities, if they are reflecting a common phenomenon. Only one earlier study was found that investigated this relation for single pedestrian accident. Jonsson (2013) analyzed two-lane and four-lane urban streets with a speed limit of 50 km/h, resulting in intercepts of -14.4 and -16.4 and exponents about 0.9 and 1.1, i.e. the intercept differs from our models and the exponent was about 1, indicating no safety in numbers effect.

The parsimonious model ii resulted in an intercept of b⁰ ¼ -7:282 and exponent of b¹ ¼ 0:676 for the bicyclist flow, and the fully specified model resulted in the intercept b⁰ ¼ -5:183 and an exponent of b¹ ¼ 0:371 for the bicyclist flow. Again, this model differed considerably from the model in Jonsson (2013), which had an intercept between -7.9 and -9.3 and exponents between 0.1 and 0.34. Schepers (2012) showed similar exponents; however, his model used kilometers travelled as the exposure measure. A possible reason why the results in Jonsson (2013) differ so much from models i and ii is that he modelled accidents on links, while our models are for intersections.

The parsimonious model iii resulted in an intercept of -12.937, an exponent for pedestrian flows of b¹ ¼ 0:554 and an exponent for motorized vehicle flows of b² ¼ 0:652, whereas the fully specified model yielded an intercept of -12.776, an exponent for pedestrian flows of b¹ ¼ 0:296 and an exponent for motorized vehicle flows of b² ¼ 0:644. Several studies have estimated safety performance functions for these types of accidents and the results are 'similar', even though the parameter values vary somewhat (Brüde and Larsson, 1993; Geyer et al., 2006; Maher and Summersgill, 1996). A weighted mean of the exponent for pedestrian exposure from several studies (the definition of exposure varies) was 0.58 (Elvik, 2009). The parameter for motorized vehicles in Maher and Summersgill (1996) was not compatible with this study, and our estimate differed considerably from that of Geyer et al. (2006). This difference might be attributed to the fact that the parameter for motorized vehicle flows was neither statistically significant in this study or in Geyer et al. (2006). The weighted mean exponent for motorized flow from Elvik (2009) was 0.46, i.e. similar to the one in our study.

The parsimonious model iv resulted in an intercept of -11.273, an exponent for bicyclist flows of b¹ ¼ 0:427 and an exponent for flows of motorized vehicles of b² ¼ 0:687, while the fully specified model resulted in an intercept of -11.477, an exponent for bicyclist flows of b¹ ¼ 0:359 and an exponent for flows of motorized vehicles of b² ¼ 0:712. This is similar to findings in previous studies, such as Schepers and Heinen (2013), Brüde and Larsson (1993). The weighted mean exponent from Elvik (2009) was similar: 0.5 for bicyclists and 0.63 for motorized vehicles. We encountered only one study (Kim et al., 2012) that showed different results (Poisson regression for signalized intersections), in which the parameters were far above 1 (1.64 for bicyclist volumes). No explanation was found for why these results differed from those of similar studies.

It was somewhat unexpected that the motorized vehicle flow was not statistically significant in model iii (p = 0.221). It is reasonable to expect that the number of pedestrians and bicyclists, and even the number of motorized vehicles (in non-single accidents) relate to the number of accidents. The presence of more road users increase the probability that two road users will cross paths at the same point of time, an interaction that should have some probability of resulting in an accident. A possible explanation for this is that the number of sites was simply too small to reach significance or that there is a correlation between the pedestrian flows and the flow of motorized vehicles. Lee et al. (2015) encountered similar problem (though a much higher correlation); they therefore combined the traffic volumes with potential pedestrian volumes (they used a proxy variable for pedestrian volumes: population). This approach, however, makes it impossible to study the influence of traffic volumes on individual road user groups, so it was considered unsuitable for the present study. This is not the only study that has been unable to show a statistically significant relation for vehicle volumes. A model from Geyer et al. (2006) reported similar results. Other studies have shown the flow of motorized vehicles to have statistically significant relation to the number of accidents between pedestrians and motorized vehicles (Hosseinpour et al., 2013) and between bicyclists and motorized vehicles (Wang and Nihan, 2004).

A comparison between the parsimonious models and the fully specified models showed some differences in the parameters. We can therefore expect that the parameters in the parsimonious models are reflecting components of variation that arise not only from differences in exposure but also from geometric differences. It is also interesting that some individual geometric variables influence only some accident types; however, since the geometric variables were not statistically significant and many had wide confidence intervals, no conclusion should be drawn regarding how those variables influence the accident models.

4.2. Safety in numbers

The safety in numbers effect, i.e. the phenomenon that the risk per vulnerable road user is lower if the number of vulnerable road user is higher, is apparent in all four models. This aligns with previous studies (Brüde and Larsson, 1993; Elvik, 2009), even though Jonsson (2013) did not find this effect for single pedestrian accidents. The model also shows the risk per pedestrian or bicyclist to be higher if the volume of motorized vehicles is higher; this, again, concurs with prior studies (Brüde and Larsson, 1993; Elvik, 2009).

There are at least five possible explanations for this safety in numbers effect. Greater traffic volumes of pedestrians and bicyclists (a) might influence the behavior of the drivers of motorized vehicles, i.e. the road users might become be more careful, gain experience and learn safer traffic behavior (Brüde and Larsson, 1993; Elvik, 2013a, 2015; Jacobsen, 2003; Jonsson, 2005, 2013); (b) might increase the efforts of municipalities to create a safe road infrastructure (Brüde and Larsson, 1993; Jonsson, 2013; Schepers, 2012); (c) might have complex relation to the number of interactions between different road users (the number of interaction is not always proportional to the number of road users (Elvik et al., 2009), therefore, if we were to apply number of interactions instead of number of road users, that might influence the safety in numbers effect); (d) there might be some additional, currently unknown explanation that influences this relation (for example, the relation might move in the other direction, safer infrastructure could result in more road users (Bhatia and Wier, 2011)) and/or (e) this might be partly a statistical phenomenon or a spurious correlation (Brindle, 1994; Elvik, 2013b).

There is some logical support for explanation (a). For example, a driver on a road where bicyclists are rare is probably not as prepared for encountering a cyclist on the road as if he/she were driving on road with dense bicycle traffic, which could influence the risk of an accident occurring. Also, if there is high exposure at the site, the exposure might be working as a proxy variable for the experience of the respective road users. There might be correlation between the exposure and the experience of the average road user. If explanation (a) is 'correct', then the safety in numbers effect could be expected for models iii and iv. It is also possible that more cyclists on the street or more cycling might cause behavioral adaptation (Schepers, 2012), resulting in safer riding of bikes (possibly lower speeds of the cyclists), and hence yielding a safety in numbers effect for single bicyclist accidents (model ii). But, the same effect is observed for single pedestrian accidents (model i). The logical link in this case is not as clear as it is for the other three accident types. Why should the presence of more pedestrians influence the number of single pedestrian accidents through behavior changes? It is possible that denser pedestrian traffic would cause greater awareness of the pedestrian resulting in safer behavior (reason (a)); however, we would expect that to primarily influence the interaction with other road users. For behavioral adaptation to influence the number of single accidents would require the pedestrian to focus more on him/her selves and the traffic environment/ pavement than on other road users. Of course this possibility cannot be dismissed; however, this might also be because of correlation between the number of accidents, the exposure and the quality of the infrastructure and maintenance (reason (b)). If the exposure is generally greater at locations that are safer in regards to single accidents, then the exposure might be working as a proxy variable for the geometrics of the site. Since we did not include pavement quality as geometric variable, nor maintenance, we were not able to test this hypothesis. However, we tested how the individual geometric factors (included in this study) influence the exponents for the traffic volumes (by removing that variable from the fully specified model). Even though one cannot draw any firm conclusions from those results regarding how individual geometric variables influence the safety in number effect (because they were not statistically significant), the findings suggest that to include a geometric factor into the model could influence the exponents of the models, i.e. the safety in numbers effect, supporting the hypothesis that the infrastructure influences the magnitude of the safety in numbers effect. The overall effect of including all the geometric variables enhanced the safety in numbers effect, see Table 1.

The third possible explanation, that there is a complex relation between the number of interactions between different road users and the number of accidents deserves some attention. It is possible that the exposure (as number of road users) is simply working as a proxy variable for the number of events, i.e. that it is not the number of road users that is most important (related to some given risk of an accident occurring), but rather number of events, such as the number of encounters, and number of times a road user crosses an uneven pavement. Since the relation between the number of road users, and the number of events is not necessarily linear (Elvik et al., 2009), this might influence the form of the safety performance function, i.e. it is possible that if the exposure variable would be defined as number of events, then that would result in different magnitude of the safety in numbers effect. This is only a hypothesis and requires further studies. The fourth explanation (d) must be considered likely to contribute to this effect. It is unlikely that we fully understand the underlying mechanisms of the accident process. Finally, Elvik (2013b) and Brindle (1994) discussed the possibility that if a randomly generated data (i.e. if the number of accidents and the exposure are independent of each other) is used to estimate a model for risk per individual road user that can automatically result in hyperbolic relation, i.e. lower risk per road user as the exposure increases. Elvik (2013b) concluded that models on the form in Eq. (3) are not as sensitive to this effect.

Since the safety in numbers effect is apparent in model i, is it possible that greater part of the safety in numbers effect for the other accident types is due to the other reasons (b, c, d and e) than previously assumed? This requires further research.

4.3. Reliability of safety performance functions based on short observational periods

The regression assumes that there is no measurement error in the explanatory variables and that the estimation of the exposure is accurate (Maher and Summersgill, 1996). If a model, based on short observational periods, is reliable, then the parameters of the different datasets should not vary extensively within each observational period length and should fit the base model well. Fig. 2 shows this is not the case. The model parameters for shorter observational periods (15–30 min per site) vary considerably. This means that an independent study, based on 15 min of observations per site, might result in any of the results in Fig. 2, without the researcher realizing how much the model parameters (and consequently their confidence intervals) are influenced by the selection of the observational time. Even though the parameters in most cases fall within the confidence interval of the base models, it is important to be well aware of this fact. Especially since interpretation of accident models often focuses on the absolute estimated value of the models parameters, and the confidence intervals for those parameters. If the study had been based on the data resulting in the lowest value of b¹ in model iv, that might have resulted in wrong conclusions (since the confidence interval of that parameter would be centered around that value, not the value of the base model). Furthermore, there seems to be a systematic underestimation of the parameters for pedestrian and bicyclist flows and an overestimation of the parameter for the flow of motorized vehicles compared to the base model when the observational periods are short (again, the parameters are within the confidence interval of the base model). This fits well with Maher and Summersgill's (1996) conclusion that ''(a) Randomizing one flow variable leads to a bias (underestimate) in its coefficient. (b) Randomizing one variable has a small but non-zero ''cross-over" effect leading to a bias in the coefficient associated with other flow variables. (c) The effect of randomizing two variables simultaneously is the sum of the two separate effects" (Maher and Summersgill, 1996, pp. 293–294).

Finally, this unreliability of shorter observational periods is not apparent in all the statistical measures frequently used to estimate the reliability of safety performance functions. In fact, improvement from longer observational periods was found only for the p values of the parameters. The reason for this might be that improved estimation of the exposure could reduce random variation, hence allowing a better fit of the relation that the parameter is meant to describe. The researcher will, however, not see the model's unreliability in the other statistical measures, which might lure him or her into false security regarding the results' reliability.

4.4. Limitations

There are some limitations to this work that need to be acknowledged. (1) This study was based on intersections in Sweden. The traffic situation varies between countries and is constantly changing; it is therefore possible that the influence of the observational period's length might also differ between countries. Even so, these results provide some indication regarding how great this influence might be and indicate that more attention should be paid to obtaining a reliable estimation of the exposure variable. (2) The study was based on intersections from municipalities with population between 50 and 200 thousand inhabitants. This criterion was aimed at middle sized Swedish cities. Even though this might influence the transferability of the results, this is not expected to have any considerable influence on the models. The number of accidents is probable more related to the exposure at site and the layout of the intersections, rather than if the intersection belongs to large/small city. (3) This study, like all accident related studies, is subjected to underreporting, especially for single pedestrian and single bicyclist accidents (Elvik and Mysen, 1999; Jonsson et al., 2011). This might influence the difference between models for single accidents and models involving motorized vehicles. Furthermore, different injury severities have different reporting degrees (Elvik and Mysen, 1999; Jonsson, 2013). Schepers (2012) showed that the exponent of the safety performance function varied depending on the injury severity. (4) This study is based on only 113 sites. (5) The traffic volumes vary between time periods, days, years, etc. (Esawey et al., 2013). Even though this study assumes that the three hour exposure values are 'correct' in order to approach the research question, they might give imprecise values for long term exposure; hence, the convergence of the parameters might take longer if the base model relies on longer time periods. Moreover, one can argue that the traffic volume, as a mean daily value, might be a proxy variable for the exposure of importance for the accident mechanisms, i.e. the real risk of an accident relate to the number of events that occurs that could result in an accident (Elvik, 2013a, 2015) and/or perhaps the exposure to consider should rather be for the time of the accident and that an average exposure over longer periods could even be contra productive (Mensah and Hauer, 1998).

4.5. Concluding remarks

To sum up, even though there are some limitations to these results, it seems that basing a safety performance functions on short observational periods might result in a biased estimate of the model parameters and hence in low reliability. It seems that the models are unreliable when the observational period per site is short. The researcher must weigh the advantages of including many sites (that will allow the inclusion of many explanatory variables) against the reliability of those variables for the modelling process. Finally, the observed safety in numbers effect for single pedestrian accidents raises some questions regarding the source of that effect. This requires further studies.

Acknowledgements

I would like to express my gratitude to Thomas Jonsson (Norwegian University of Science and Technology, NTNU) and to András Várhelyi (Lund University, Sweden) for their help during preparations for this study, data collection and the writing process. I would also like to thank the reviewers for their contribution. Finally, I would also like to acknowledge the financiers of this project, The Swedish Transport Administration (Grant Number: TRV 2013/12520).

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