category: literaturenote citekey: shibayesianridgeregressionanalysis2016 title: "A Bayesian ridge regression analysis of congestion's impact on urban expressway safety" authors: "Shi, Qi; Abdel-Aty, Mohamed; Lee, Jaeyoung" year: 2016 date: 2016-00-00 2016 doi: 10.1016/j.aap.2015.12.001 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2015.12.001" zotero_key: CAMUCXUF zotero_storage: XW5N3JW5 collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Shi, Qi"

status: converted

Contents lists available at ScienceDirect

Accident Analysis and Prevention

jour nal home page: www.elsevier.com/locate/aap

A Bayesian ridge regression analysis of congestion's impact on urban expressway safety

Qi Shi ^∗, Mohamed Abdel-Aty, Jaeyoung Lee

Department of Civil, Environmental and Construction Engineering, University of Central Florida, Engineering II-215, Orlando, FL 32816, United States

a r t i c l e i n f o

Article history: Received 15 August 2014 Received in revised form 23 November 2015 Accepted 1 December 2015 Available online 4 January 2016

Keywords: Urban expressway safety Congestion Bayesian inference Ridge regression

a b s t r a c t

With the rapid growth of traffic in urban areas, concerns about congestion and traffic safety have been heightened. This study leveraged both Automatic Vehicle Identification (AVI) system and Microwave Vehicle DetectionSystem(MVDS)installed onanexpressway inCentral Florida to explorehow congestion impacts the crash occurrence in urban areas. Multiple congestion measures from the two systems were developed. To ensure more precise estimates of the congestion's effects, the traffic data were aggregated into peak and non-peak hours. Multicollinearity among traffic parameters was examined. The results showed the presence of multicollinearity especially during peak hours. As a response, ridge regression was introduced to cope with this issue. Poisson models with uncorrelated random effects, correlated random effects, and both correlated random effects and random parameters were constructed within the Bayesian framework. It was proven that correlated random effects could significantly enhance model performance. The random parameters model has similar goodness-of-fit compared with the model with only correlated random effects. However, by accounting for the unobserved heterogeneity, more variables were found to be significantly related to crash frequency. The models indicated that congestion increased crash frequency during peak hours while during non-peak hours it was not a major crash contributing factor. Using the random parameter model, the three congestion measures were compared. It was found that all congestion indicators had similar effects while Congestion Index (CI) derived from MVDS data was a better congestion indicator for safety analysis. Also, analyses showed that the segments with higher congestion intensity could not only increase property damage only (PDO) crashes, but also more severe crashes. In addition, the issues regarding the necessity to incorporate specific congestion indicator for congestion's effects on safety and to take care of the multicollinearity between explanatory variables were also discussed. By including a specific congestion indicator, the model performance significantly improved. When comparing models with and without ridge regression, the magnitude of the coefficients was altered in the existence of multicollinearity. These conclusions suggest that the use of appropriate congestion measure and consideration of multicolilnearity among the variables would improve the models and our understanding about the effects of congestion on traffic safety.

1. Introduction

Providing motorists with efficient and safe traffic system has long been considered a priority of traffic professionals. With the growth in traffic demand outpacing construction of road infrastructure, congestion and safety concerns arise. In urban areas, many traffic authorities have turned to toll/turnpike facilities and efficient use of Intelligent Transportation Systems (ITS) techniques as remedies for congestion and to improve safety. Whether

This paper aims at identifying the relationship between congestion and crashes for urban expressway State Road (SR) 408 in Central Florida based on the existing ITS facilities. The 21.4-mile expressway of interest is managed by Central Florida Expressway Authority (CFX). The toll expressway accommodates large commuter traffic and travels through the downtown Orlando area. Consequently, recurrent congestion during peak hours is observed on the expressway. For more efficient operation, CFX has installed Automatic Vehicle Identification (AVI) system for Electronic Toll

ameliorating congestion or safety would have positive effects on the other depends on the relationship between congestion and safety. Itis widely acknowledged that crashes lead to non-recurrent congestion while the effects of congestion on crashes seem inconclusive.

^∗ Corresponding author.

E-mail addresses: shiqi@knights.ucf.edu (Q. Shi), m.aty@ucf.edu (M. Abdel-Aty), jaeyoung@knights.ucf.edu (J. Lee).

Collection (ETC) and travel time estimation. In 2013, Microwave Vehicle Detection System (MVDS) was introduced for traffic monitoring on the expressways. Based on the real-time data derived from the detection systems, CFX provides motorists with travel time and other information via the Dynamic Message Signs (DMS). The multiple ITS instruments deployed on the expressway give the authority a favorable margin for better understanding of the traffic conditions and potential for improvement. In this study, traffic parameters from the MVDS and AVI systems along with roadway geometric characteristics were collected to identify their relationship with crash occurrence. Compared with previous crash frequency studies exploring congestion's effects on safety, the current work applied ridge regression within Bayesian framework to avoid the problem of multicollinearity. Through this method, it is expected that more accurate parameter estimates will be achieved.

2. Background

Traffic congestion especially in urban areas has been receiving increasing attention during the past few decades. Congestion is not only costly in terms of time losses and fuel waste, but also causes unstable traffic flow because of the stop-and-go traffic pattern. Whether congestion has an impact on traffic safety has been researched in many previous studies. One early study by Shefer and Rietveld (1997) stated that congestion leads to lower numbers of fatalities because of lower speed in congestion. Baruya (1998) investigated crash frequencies in four European countries and declared that congestion is positively associated with crash frequency. Golob and Recker (2003) in their study of Southern California freeways found rear-end crashes are generally associated with high variations in relatively low speeds which are commonly observed in congested traffic. Kononov et al. (2008) developed safety performance functions (SPFs) using Colorado, California, and Texas data. Their conclusion suggested that safety deteriorates with the degradation of congestion expressed through level of service but segments with the highest volumes experienced lower crash rates. Wang et al. (2009) explored the relationship between crash and congestion for an orbital motorway around London using a spatial analysis approach. They found that congestion imposes little or no impact on the frequency of road accidents. In a recent review of previous works about the congestion's effect on road safety, Wang et al. (2013) concluded that existing research has not yet reached an agreement on the impact oftraffic congestion. To understand the complex phenomenon, several factors should be taken into account; (1) how is congestion measured. With the development of traffic detection systems, new congestion measurements become available.Whether these detection systems reflectthe congestion accurately is crucial in the congestion-safety studies. (2) Congestion could be time and location specific in urban area. Freeways/expressways in urban areas carry large commuter traffic in morning and evening peak hours. The necessity to consider peak and non-peak traffic hours is worth investigation. (3) The complex nature of crashes often requiresmultiple explanatory variables included in safety analysis. The traffic speed, volume, and density could be correlated which would introduce multicollinearity. Multicollinearity could affect accurate interpretation of the variables' effects (Berry and Feldman, 1985). Yet existing studies haven't addressed the problem adequately. The conflicting results in previous congestion-safety studies might be resulting from this issue.

2.1. Congestion performance measures

Congestion is easy to be observed and interpreted. Nevertheless, how to measure congestion could be a challenging task. Whether congestion estimates are accurate and useful largely depends on the available data. For a long time, Volume-to-Capacity Ratio (V/C Ratio) and the Level of Service (LOS) have been the primary congestion indicators since continuous data collection was not feasible (National Research Council, 2010). However, they lack the flexibility to capture the dynamics of congestion. With the development of ITS equipment, more direct measurement of speed, travel time, and volume becomes available. New approaches to measure congestion are proposed (Lomax et al., 1997; Schrank and Lomax, 2007). Current congestion measures can be broken down into three general categories: density-based, travel-time-based and travelspeed-based. Traffic occupancy has become more available to reflect the traffic density on roadways thanks to the deployment of ITS facilities. The numerous types of ITS traffic detectors also promote the creation of other congestion measures. Travel Time Index (TTI) is a widely used travel-time-based congestion measure. It is defined as the ratio of the travel time during the peak period to the time required to make the same trip at free-flow speeds (Schrank and Lomax, 2007). TTI can be measured by the AVI system which identifies vehicles at different detecting locations and calculates the traveltime. In the meantime, speed-based congestion measures are also applied. Christidis and Rivas (2012) in their report of European road congestion defined the intensity of congestion as the ratio of the average to free flow speeds. Hossain and Muromachi (2012) used the rate of reduction in speed caused by congestion from the free-flow speed condition as congestion index. In this study, the expressway has both AVI and MVDS systems. Multiple congestion measures were tested to investigate their feasibility to be used in traffic safety studies.

2.2. Time-specific nature of congestion

Another issue related to the congestion in crash frequency studies is the problem of averaging. Mensah and Hauer (1998) discussed the potential problems resulting from averaging traffic flows over a period of time. They argued that the cause-effect relationship is between crashes and the flows prevailing near the time of crash occurrence and there are two (daytime and nighttime) and perhaps many more cause-effect functions in the course of the study period. Wang et al. (2009) in their study suggested that in reality, the level of congestion varies over time and averaging may have an impact on the effect of congestion on crashes. In this paper, as it is still a crash frequency study, aggregating the traffic flow data is unavoidable. Nevertheless, we divided the traffic into peak and non-peak hours to testifthere is significant difference ofthe effects of traffic variables during these two time periods.

2.3. Multicollinearity

Multicollinearity occurs when the explanatory variables are correlated among each other. The inherent collinearity won't affect the overall model fitting (Burnham and Anderson, 2002). However, some consequences could result from multicollinearity as pointed out by Washington et al. (2010) such as high standard errors of the estimated parameters of the collinear variables, large sampling variability which means estimated parameters based on different sample could vary widely, and inaccurate interpretation of the effects of explanatory variables because the change of one variable would inevitably lead to the change of another variable. In crash frequency studies, the traffic data are observational data in most cases and not controlled. Correlations exist among speed, density and volume. Nevertheless, existing literature shows that this issue is not adequately addressed. This might be a reason why researchers haven't reached a conclusive statement about the congestion's effect on traffic safety. Solutions to the multicollinearity problem were generalized by Congdon (2007) which include: (1) the introduction of extra information; (2) the multivariate

Fig. 1. (a) AVI speed distribution; (b) MVDS speed distribution.

reduction to a smaller set of uncorrelated predictors; (3) ridge regression. In this study, both roadway geometric characteristics and traffic flow parameters are incorporated and no extra information regarding driver characteristics and weather conditions is available at this stage. Multicollinearity is present mainly among the traffic flow parameters. Despite that the focus of this research is congestion's effect, we still kept volume as it is an important exposure variable. As a result, ridge regression is adopted to deal with the multicollinearity issue and to provide accurate parameter estimation.

3. Data description and preparation

Three types of data were prepared to uncover the effects of congestion on traffic safety: (1) traffic flow data from AVI and MVDS systems on the expressway provided by CFX; (2) roadway geometric characteristics data downloaded from Florida Department of Transportation (FDOT) Roadway Characteristics Inventory (RCI) database; (3) crash data on the expressway of interest from Signal Four Analytics database.

Two ITS traffic detection systems are currently deployed on the expressway. AVI system which was originally installed for Electronic Toll Collection (ETC): when a vehicle equipped with encrypted transponder (tag) passes an AVI detector, the tag ID and the timestamp of detection are recorded. After matching the tag ID of vehicles, the travel time between adjacent AVI detectors can be determined. These detectors continuously monitor and archive the traffic status along the expressway. Based on these records, TTI as the congestion measure for each segment can be calculated as

$$TTI = \frac{\text{actual travel time}}{\text{free flow travel time}}$$ (1)

where the free flow travel time information for each segment was provided by CFX. The higher TTI indicates higher intensity of congestion. There were in total two years of raw (uncapped) AVI traffic data collected for this research from September 2012 to August 2014.

In 2013, the authority introduced MVDS for the purpose of more precise and timely traffic detection and future applications. Different from AVI system, the MVDS detectors do not differentiate vehicles with or without tags. They measure the traffic flow parameters (i.e., volume, speed, and occupancy per lane) at their installed locations and return the aggregate traffic information at one-minute interval. Multiple congestion measures can be developed from the MVDS traffic data. Occupancy is defined as the percent of time a point on the road is occupied by vehicles (Hall, 1996). Gerlough and Huber (1975) referred to occupancy as a surrogate for density. Compared with traditional V/C Ratio or LOS, occupancy has the advantage that it could be monitored in realtime. The speed detected by MVDS detector is spot speed and speed-based congestion index (CI) is expressed as

$$CI = \frac{\text{free flow speed} - \text{actual speed}}{\text{free flow speed}} \times 100\% \quad \text{when CI} > 0;$$

$$= 0 \quad \text{when CI} \leq 0$$ (2)

The CI is a continuous congestion indicator ranging from 0 to 1. The free flow speed is the 85th percentile speed at the corresponding detection point. From Eq. (2) above it can be seen that when the actual speed is above free flow speed, CI will be recorded as 0. When CI increases, the congestion becomes more severe. The MVDS traffic data were collected from July 2013 to August 2014. A preliminary comparison was conducted to check how comparable the MVDS data is to the AVI data. Fig. 1 shows that the speed distributions are similar for the two types of data. MVDS average daily traffic volumes were compared with expressway AADT for 2012 and no significant changes in traffic volume were found in the two years. Therefore we believe that the MVDS traffic volume could reflect the traffic volume on the expressway in the last two years.

Consequently,trafficdata were collectedfromthe two ITSdetection systems. Three congestion measures (i.e., TTI, occupancy, CI) were developed and fed to the crash frequency models. The congestion measures were aggregated at 5-minute interval each day and then averaged during the study time period. Speed information from the two sources was also prepared. As for the traffic volume, only the MVDS daily traffic volumes were used. Although AVI detectors can capture vehicles equipped with tags, they are not able to report the total volume on the expressway. As an alternate measure of traffic volume for the AVI segments, the average volume at the MVDS detection points within an AVI is calculated. In addition, allowing for the peak-hour traffic effects, the traffic data were aggregated into two groups: peak hours (7:00–9:00 and 17:00–19:00 on weekdays) and non-peak hours (the rest time period during 6:00–21:00 on weekdays). The weekend and the time period of 21:00–6:00 on weekdays were excluded considering the lighttraffic during this period. Itis more likely that crashes occurred during the discardedtimeperiod are affected by other factors rather than traffic flow parameters.

FDOT RCI online database stores comprehensive roadway characteristics information. When multiple geometric characteristic

Table 1 Statistics summary of variables.

Variable	Number	Mean	Std Dev	Minimum	Maximum
Dependent variable
Crasha	198	1.899	3.511	0.000	23.000
Crashb	198	1.662	2.979	0.000	31.000
Independent geometric variables
Segment length (mile)	198	0.218	0.128	0.100	0.762
Number of lanes	198	–	–	2.000	5.000
2 lanes: 0; 3 lanes: 1; 4 or 5 lanes: 2
Existence of auxiliary lanes: 0 – no auxiliary lanes; 1 – auxiliary lanes	198	–	–	0.000	1.000
Horizontal degree of curvature	198	0.480	0.889	0.000	5.250
Shoulder width (ft)	198	10.030	0.245	10.000	12.000
Median width (ft)	198	35.374	18.636	20.000	64.000
Inside shoulder width (ft)	198	6.909	3.476	0.000	12.000
Distance to the nearest ramp	198	0.219	0.222	0.001	1.269
Pavement: 0 – good pavement condition; 1 – very	198	–	–	0.000	1.000
good pavement condition
Speed Limit 65 mph: 0 – speed limit of 55 mph; 1 – speed limit of 65 mph	198	–	–	0.000	1.000
Independent traffic variables
MVDS average volumea	104	12,398.200	4295.600	4911.070	23,482.940
MVDS average volume per lane a	104	4318.420	1214.3	2455.540	9061.800
MVDS average occupancya	104	7.321	2.118	3.887	14.393
MVDS average speeda	104	63.282	3.603	51.123	74.796
MVDS standard deviation of speeda	104	3.194	2.305	1.331	12.017
MVDS Congestion Indexa	104	8.862	3.352	0.390	21.908
AVI average speeda	42	62.491	5.041	47.926	72.083
AVI standard deviation of speeda	42	6.854	2.728	2.930	13.573
AVI Travel Time Indexa	42	1.001	0.146	0.808	1.432
AVI average volumea	42	12,645.000	4190.000	5523.000	21,347.000
AVI average volume per lane a	42	4318.550	1013.96	2761.520	7250.970
MVDS average volumeb	104	21,958.130	8119.830	8522.470	42,252.130
MVDS average volume per lane b	104	7582.050	2008.770	4261.230	14,983.790
MVDS average occupancyb	104	4.312	1.116	2.396	8.264
MVDS average speedb	104	64.777	2.791	57.672	74.799
MVDS standard deviation of speedb	104	1.646	0.451	1.232	4.158
MVDS Congestion Indexb	104	6.695	1.428	1.284	10.561
AVI average speedb	42	63.677	5.076	53.003	75.662
AVI standard deviation of speedb	42	5.411	1.885	2.675	10.393
AVI Travel Time Indexb	42	0.970	0.155	0.739	1.510
AVI average volumeb	42	22,689.000	8303.000	9011.000	39,352.000
AVI average volume per laneb	42	7653.510	1666.570	4505.250	11,945.260

^a Peak hours.

variables are selected, the roadway will be automatically divided into homogeneous segments by the system. The segments are specified as RCI segments. To avoid the segments with length too short for analysis, it was decided in this study that the smallest segment length is 0.1 mile. If a RCI segment length is below this threshold, it will be combined with adjacent segment with higher geometric similarity. The motivation and an example of combining these small segments were addressed in a previous study (Ahmed et al., 2011). The roadway geometric information included the number of lanes, existence of auxiliary lanes, horizontal degree of curvature, speed limit, pavement conditions, and speed limit, distance of the segment to the nearest ramp, etc. as shown in Table 1. In total, there are 99 segments generated for EB and 99 segments for WB. The number of lanes per direction on the studied expressways ranges from 2 to 5. Consequently, a categorical variable for number of lanes was created with 2 lanes as the base level. Since only a very small proportion of the segments contained 5 lanes, 4 and 5 lanes were combined as one category. On the studied expressway, there are two speed limit posts (55 and 65 mph) and thus the speed limit is treated as a binary variable. For the pavement conditions, they were denoted as visual interpretation of the condition of the road surface in FDOT RCI Features and Characteristics Handbook (2012). On SR 408, only pavement conditions with scores from 3 to 5 were found. According to the handbook, scores between 3 and 4 indicates good pavement condition and 4–5 as very good condition. Since the pavement score is based on visual interpretation which does not possess real quantitative meaning, it is also treated as a binary categorical variable.

The crashes in Florida are archived in Signal Four Analytics database. Given the two years traffic data available, the crash data during the same period were collected. To reveal the relationship between safety and congestion, two factors were considered: (1) the data should reflect traffic conditions for the days when recurrent congestion occurs; (2) the crashes should be more likely to be influenced by traffic flow. Taking the two points into account, we retained the mainline crashes occurring from 6:00 to 21:00 on weekdays for further analysis. Based on these criteria, 705 crashes were identified. The 705 crashes weremeasured at 5 severity scales, including 1 fatal crashes (K), 8 incapacitating injury crashes (A), 91 non-incapacitating crashes (B), 119 possible injury crashes (C) and 486 property damage only crashes (O/PDO).

After the three data sets were prepared, they were merged together. The crashes were aggregated into the RCI segments on which they took place during peak and non-peak hours. The total number of observations is (99 + 99) × 2 =396. Then traffic data were assigned to the RCI segments. Since the AVI data reflect traffic conditions for segments on the expressway, each RCI segment is matched with the traffic information of the AVI segment that

^b Non-peak hours.

includes the RCI segment. On SR 408, there are 22 AVI segments on the Eastbound (EB) and 20 AVI segments on the Westbound (WB). In contrast, the MVDS detectors keep record of traffic flow parameters at the installed points. In this study, 110 MVDS detectors (55 for EB and 55 for WB) were utilized on the expressway. For each RCI segment, if it has no MVDS detector within it, then its traffic states will be represented by the nearest downstream MVDS detector. If the segment contains one MVDS detector, it will use the traffic data from that detector. In cases that one segment has multiple MVDS detectors, the traffic flow parameters will be averaged. Table 1 summarizes the information of all variables involved in the data preparation process.

4. Methodology

4.1. Diagnostics of multicollineairity

Total independence between explanatory variables is rare in observational data analysis. However, when the multicollinearity problem is severe, issues with coefficient estimates will arise. Therefore, detection of multicollinearity is a necessary step especially when highly correlations among explanatory variables are suspected. In crash frequency studies analysis, congestion could be related with traffic volume and speed. To examine the presence of multicollinearity, several procedures could be applied: correlation test, comparing the coefficients of determination, calculating tolerance (TOL) and the Variance Inflation Factor (VIF) (Kalatzis et al., 2011).

In this study, correlations between variables were tested using the Pearson's correlation test. Klein (1962) suggested that if the determination coefficients of auxiliary regressions are greater than that of the general regression, multicollinearity is statistically significant. The coefficient of determination of a auxiliary regression model is calculated as

$$R^{2} = \frac{\sum (\hat{y}{i} - \bar{y})^{2}}{\sum (y{i} - \bar{y})^{2}}$$ (3)

In crash frequency modeling, the Poisson model and its variations (i.e., Negative Binomial which is also known as Poisson-gamma, Poisson-lognormal) are the fundamental modeling structures. The models belong to the generalized linear model framework. Thus the determination coefficient of general regression calculated in this study is the pseudo R² ^p = 1 −

$$\frac{\sum_{i=1}^{n} \left[ (y_i - \hat{\lambda}_i) / \sqrt{\hat{\lambda}i} \right]^2}{\sum{i=1}^{n} \left[ (y_i - \hat{y}) / \sqrt{\hat{y}} \right]^2}$$ following Washington et al. (2010). The deter-

mination coefficient of auxiliary regression is expressed as R² ^k for regression of the kth explanatory variable on all the other explanatory variables (Greene, 2003). TOL and VIF are two other measures that assess multicollinearity. TOL of the kth variable is equal to ¹ ⁻ ^R² ^k TOL ranges from 0 to 1 and higher TOL value indicates less severe multicollineairty. VIF is the reciprocal of TOL, which shows how multicollinearity has increased the instability ofthe coefficient estimates (Freund et al., 2000).As a rule ofthumb, Neter et al.(1996) thought that multicollinearity is only severe at VIFs > 10. However, recommend maximum VIF value of 5 (Rogerson, 2001) or even 4 (Pan and Jackson, 2008) could be found in the literature.

4.2. Bayesian ridge regression

Full Bayesian inference using Markov Chain Monte Carlo (MCMC) algorithm was used to construct the models. Bayesian modeling framework has been praised for its capability to deal with hierarchical data structure (Huang and Abdel-Aty, 2010). In this study, the hierarchical data structure exist as multiple RCI segments are nested within one AVI or MVDS segment and sharing the same traffic data. In addition, for each segment, crashes were aggregated into peak hours and non-peak hours, hence introducing another level. The hierarchical crash frequency model can then be set up as:

$$Y{ijt} \sim Poisson(\lambda{ijt})$$ (4)

Yijt is the observed crash frequency at RCI segment i(i = 1, 2, . . ., 198) belonging to AVI or MVDS segment j(j = 1, 2, . . . 42 for AVI segments and j = 1, 2, . . . 104 for MVDS segments) during peak and non-peak hour t(t = 1, 2) It follows Poisson distribution with parameter ijt However, one limitation of Poisson model that has been discussed in numerous previous studies is the lack of ability to handle over-dispersion (Lord and Mannering, 2010). Two types of random effectterms were included to cope with the issue as shown below:

$$\log(\lambda{ijt}) = a{jt[i]} + \mathbf{X}{ijt} \mathbf{\beta}{ijt} + \delta1 \varepsilon{it} + \delta_2 \varepsilon_i$$ (5)

where Xijt are geometric characteristics variables at RCI level and ijt are the coefficients for random parameters. If the estimated standard deviations of the parameter distribution are insignificant from zero, parameters will be fixed across observations (Anastasopoulos and Mannering, 2009). εit is segment-peak-hour uncorrelated random effects and εⁱ is the segment correlated random effects. The random effects follow normal distribution εit ∼ N(0, 1/it) and εⁱ ∼ N(0, 1/i) respectively. ajt[i] is the AVI or MVDS level effects. In normal regression, it is defined as ajt = Ujt^t . Ujt are traffic parameters for peak and non-peak hours ^t are vectors of coefficients.

In this paper, ridge regression was implemented to deal with multicollinearity. Early applications of ridge regression trace back to Hoerl's work (1962, 1970). When we have a function with the form like ajt = Ujt^t and multicollinearity among the Ujt, traditional analysis method such as OLS would lead to incorrect inference of variables. By ridge regression, small bias will be induced, however, more precise regression parameter estimates will be yielded (Congdon, 2007). The ridge regression approach can be viewed as a version of standard posterior Bayes regression with exchangeable prior distribution on the elements of the regression vector (Lindley and Smith, 1972). Nevertheless, to make this assumption plausible, a preliminary standardization is needed (Congdon, 2007). Following the suggestions of Ntzoufras (2002) and Crainiceanu (2004),the variables and their coefficients are standardized as in Eq. (6),

$$\begin{cases} z{jt} = \frac{(u{jt} - \bar{u}{jt})}{\operatorname{sd}(u{jt})} \ \gamma_t = \frac{bt}{\operatorname{sd}(u{jt})} \end{cases} (6)$$

And ajt becomes ajt = Zjtbt.

The models were calibrated in the WinBUGS software (Lunn et al., 2000). ijt was assigned with prior normal distribution N(, 2) or each observation, where as non-informative prior distribution N(0, 106) and ¹ ² ^∼gamma(10−3, ¹⁰⁻3) Ridge regression further restricts bt to be exchangeable with bt ^∼ ^N(0, −1) and ^∼ gamma(10−3, ¹⁰−3). The random effects terms it and ⁱ follow gamma(10−3, 10−3). When (ı1, ı2) = (1, 0) the model is with uncorrelated random effect; when (ı1, ı2) = (0, 1) it is model with correlatedrandomeffect.Intotalthree types ofmodels were tested: (1) uncorrelated random effects model; (2) correlated random effects model; 3) correlated random effects with random parameters varying across observations. Each model was simulated with three chains of 20,000 iterations. The first 5000 iterations were discarded as burn-in period. Model convergence was visually checked according to that the trace plots of the three chains overlap one another (Spiegelhalter et al., 2003). Deviance Information Criterion

Table 2 Summary of multicollinearity test.

| Peak | Correlation between variables | | | | | | | | | | | | |---------------------------|-------------------------------|-----------------------|--------------------------|--------|---------------------------------|-------|----------------|--|--|--|--|--| | | Log(MVDS volume
per lane) | MVDS average
speed | MVDS std dev
of speed | CI | Coefficient of
determination | TOL | VIF | | | | | | | Log(MVDS volume per lane) | 1.000 | −0.095 | 0.325 | 0.324 | 0.148 | 0.852 | 1.174 | | | | | | | MVDS average speed | −0.095 | 1.000 | −0.591 | −0.716 | 0.537 | 0.463 | 2.162 | | | | | | | MVDS std dev of speed | | −0.591 | 1.000 | 0.864 | 0.759 | 0.241 | 4.155 | | | | | | | CI
VIF mean | 0.324 | −0.716 | 0.864 | 1.000 | 0.827 | 0.173 | 5.786
3.319 | | | | | | | R2 general | | | | | 0.229 | | | | | | | | | Non-peak | Correlation between variables | | | | | | | | | | | | | | Log(MVDS volume
per lane) | MVDS average
speed | MVDS std dev
of speed | CI | Coefficient of
determination | TOL | VIF | | | | | | | Log(MVDS volume per lane) | 1.000 | −0.023 | 0.170 | 0.274 | 0.088 | 0.912 | 1.097 | | | | | | | MVDS average speed | | 1.000 | −0.311 | −0.383 | 0.177 | 0.823 | 1.215 | | | | | | | MVDS std dev of speed | 0.170 | −0.311 | 1.000 | 0.438 | 0.223 | 0.777 | 1.287 | | | | | | | CI | 0.274 | −0.383 | 0.438 | 1.000 | 0.312 | 0.688 | 1.453 | | | | | | | VIF mean | | | | | | | 1.263 | | | | | | | | 0.325
−0.023
R2 general | | | | 0.212 | | | | | | | |

(DIC) is widely used to measure the goodness of fit in Bayesian inference. DIC consists of two elements, the model fitting part D¯ and the number of effective variables pD Smaller DIC indicates better model. In addition, the pseudo adjusted R² values were also calculated to facilitate model comparisons. Parameter significance was investigated using Bayesian Credible Interval (BCI). If 0 is not contained within the 95% BCI, the effect of the variable is significant. t-statistics are another helpful indicator used to interpret the effects of variables and therefore included. With the selected best model, the feasibility of different congestion measures for safety evaluation was compared in the same model setting. In the end, traditional crash frequency model with ridge regression and model without specific congestion indicator were conducted to test the necessity to account for multicollinearity and to incorporate the congestion variable when exploring congestion's impact on traffic safety for urban expressways.

5. Modeling results and discussion

5.1. Presence of multicollinearity

To examine whether multicollinearity exists among the traffic data, correlation coefficients, coefficients of determination, and TOL/VIF values were calculated. Among the three candidate congestion measures, CI from MVDS data was used to represent the congestion conditions on the expressway. Logarithmic average traffic volume per lane, average speed, standard deviation of speed and average CI were used to test multicollinearity. In crash frequency models for highway segments, traffic volume and segment length are often treated as important exposure variables. However, many existing studies have shown that crash count is not proportional to the volume and segment length. Therefore logarithimic transformation was adopted for traffic volume and segment length in this study. Table 2 shows that during peak hours, the traffic parameters have much higher correlations compared with nonpeak hours. As expected, logarithmic volume per lane is positively related with congestion and negatively related with speed. The VIFs for peak hours are less than 10 but higher than 5 for CI, indicating potential multicollinearity. Based on the other standard, the coefficients of determination of the auxiliary regressions are higher than that of general regression, which also means the presence of multicollinearity. Therefore, when the objective is to quantify congestion's effect on traffic safety, multicollinearity is worth consideration.

5.2. Model comparison

Three types of ridge regression models were compared. Table 3 illustrates the parameter estimates for each model and the overall performances of the models. RCI level geometric variables and MVDS level traffic variables were evaluated in the hierarchical model framework. The mean, standard deviation (SD), the 95% BCI and t-value of the coefficients of estimated variables are given. DIC and pseudo adjusted R² adjusted ⁼ ¹ ⁻ ⁿ−¹ n−p (1 ⁻ ^R² ^p) here n is the number of observations (396 in this study) and p is the effective number of parameters pD in the model are also displayed.

The effects of geometric variables in the three models are comparable. Logarithmic segment length affects the crash frequency with a positive sign, which implies that crash frequency is likely to increase under larger exposure. In the random parameter model, it was found that the logarithmic length and horizontal degree of curvature should be assigned with random parameters. By assigning horizontal curve with random parameter, it became significant at 95% BCI with a negative sign. The effects of horizontal curve on safety should be interpreted from two perspectives. One is engineering effects and the other driver reaction. Existing studies haven't reached a consistent conclusion aboutits effects. Abdel-Aty and Radwan (2000), Hauer (1999) and Bonneson et al.(2007)found positive relationship between horizontal curve and crash frequencies while Milton and Mannering (1998), Haynes et al. (2007) and Ahmed et al.(2011) reached opposite conclusions. In this study,the negative association between horizontal degree of curvature and crash frequency might indicate that drivers are more careful and hence drive with appropriate speed on curves in the urban area.

When a segment contains 4 or 5 lanes, it could significantly increase crashes compared when there are 2 lanes. With more lanes on a cross-section, there would be higher likelihood of lanechanging behavior and increased probability of sideswipe crashes. Thefindings aboutnumber oflanes was consistent withseveralprevious studies (Abdel-Aty and Radwan, 2000; Kononov et al., 2008). As for the safety effects of speed limit on the expressway, the modeling results from all of the three models showed that when the speed limit was 55 mph, the segments tended to have more crash occurrences. This indirectly suggested expressway segments with lower speed contributed to crashes on access-controlled high speed facilities. The existence of auxiliary lanes was found to significantly contribute to crashes in model type (3) at 95% BCI and model type (2) at 90% BCI. Auxiliary lanes are the lanes adjacent to through lanes and provide turning movements where speed changes are

Table 3Parameter estimates andmodel fitting for model comparison.

ble Va ria s	e ( 1): Typ Un	d r and effe cts om		e ( 2): Typ	rel d r Cor ate	and effe cts om		e ( 3): rel d r and effe Typ Cor ate cts om and and ete r om par am rs
	Me an	SD	95% BC I	t-V alu e	Me an	SD	95% BC I	t-V alu e	Me an	SD	95% BC I	t-V alu e
RC le I vel
Int ept erc	2.1 66	0.3 27	( 1.4 83, 2.7 90 )	6.6 32	2.1 09	0.3 76	( 1.3 51, 2.8 30 )	5.6 06	2.1 88	0.3 45	( 1.4 83, 2.8 60)	6.3 38
( len h) Log gt	1.0 39	0.1 48	( ) 0.7 50, 1.3 25	7.0 44	1.0 44	0.1 82	( ) 0.6 83, 1.4 08	5.7 33	1.0 95	0.1 53	( 87) 0.8 03, 1.3	7.1 71
Sta nda rd d evi atio f o n r d istr ibu tio f ete par am n o log ( len h) gt	–	–	–	–	–	–	–	–	0.2 28	0.1 32	( 0.0 42, 0.5 17)	1.7 24
riz al de f cu Ho ont tur gre e o rva e	−0 .11 7	0.0 84	(− 0.2 83, 0.0 35 )	−1 .39 9	−0 .12 4	0.1 04	(− 0.3 29, 0.0 82)	−1 .19 4	−0 .22 0	0.1 45	(− 0.5 37, −0 .00 6)	−1 .52 3
Sta nda rd d evi atio f o n r d ibu f istr tio ete par am n o hor izo l nta cu rve	–	–	–	–	–	–	–	–	0.4 02	0.1 89	( 0.0 49, 0.7 37 )	2.1 24
Seg ith 4 o r 5 lan nt w me e	0.3 52	0.2 13	* (− 0.0 58, 0.7 74)	1.6 57	0.3 62	0.2 45	(− 0.1 50, 0.8 35 )	1.4 80	0.4 35	0.2 38	* (− 0.0 35, 0.9 06 )	1.8 26
ed lim f 6 h Spe it o 5 m p	−0 .52 2	0.1 89	(− 7) 0.8 91, −0 .15	−2 .76 4	−0 .44 8	0.2 23	(− 3) 0.8 89, −0 .00	−2 .00 9	−0 .40 3	0.2 17	* (− ) 0.8 31, 0.0 39	−1 .86 0
Exi f au xili lan ste nce o ary e	0.2 28	0.1 52	(− ) 0.0 73, 0.5 22	1.5 05	0.3 22	0.1 84	* (− 80) 0.0 33, 0.6	1.7 45	0.3 71	0.1 70	( ) 0.0 48, 0.6 98	2.1 77
Me dia idt h w n	−0 .00 9	0.0 05	* (− 0.0 19, 0.0 01 )	−1 .74 4	−0 .00 9	0.0 06	* (− 0.0 20, 0.0 02 )	−1 .64 4	−0 .01 0	0.0 05	* (− 0.0 20, 0.0 00 1)	−1 .81 2
MV DS le vel
ion Ind MV DS C est ong ex	0.0 81a	0.0 55	* (− 12) 0.0 10, 0.2	1.4 81	0.0 71a	0.0 39	* (− 54) 0.0 02, 0.1	1.8 11	0.1 00a	0.0 23	( ) 0.0 55, 0.1 45	4.4 11
	34b 0.0	0.0 57	(− 0.0 68, 0.1 61)	0.5 94	25b 0.0	0.0 47	(− 0.0 54, 0.1 34 )	0.5 39	46b 0.0	0.0 53	(− 0.0 50, 0.1 62)	0.8 74
MV DS / AV I St and ard dev iati on of eed sp	0.0 33a	0.0 72	(− 0.1 15, 0.1 78)	0.4 55	0.0 54a	0.0 50	(− 0.0 41, 0.1 53)	1.0 93	0.1 36a	0.0 37	( 0.0 63, 0.2 11)	3.6 64
	20b 0.1	0.1 77	(− 0.1 68, 0.5 12)	0.6 80	77b 0.0	0.1 35	(− 0.1 63, 0.3 84)	0.5 73	44b 0.0	0.1 29	(− 0.2 00, 0.3 40 )	0.3 44
Mo del rfo pe rm anc e
¯ D			100 0.9 00				994 .35 6				997 .86 1
pD			173 .11 6				128 .13 8				126 .47 1
DIC			117 4.0 16				112 2.4 94				112 4.3 32
2 ud dju d R ste pse o a			0.7 00				0.7 73				0.7 72

Significant at 90% BCI.

Peakhours.

Non-peak hours.

Fig. 2. Auxiliary lanes along expressway mainline (FDOT RCI Features and Characteristics Handbook, 2012).

required. On freeways/expressways, they provide space for turning movements and speed changes on the mainline near ramps as shown in Fig. 2. Vehicles merging in or diverging from the mainline can cause disturbance in traffic flow and increase crash likelihood. What's worth mentioning is that the definition and diagram of the auxiliary lanes in Fig. 2 are based on FDOT RCI Features and Characteristics Handbook (2012) which is in accordance with the study period and data source. Nevertheless, according to the Freeway Management and Operations Handbook (Neudorff et al., 2003) and FDOT's 2009 Quality/Level of Service Hanbook (FDOT, 2009), auxiliary lane was indicated as an additional lane on a freeway connecting an on-ramp to the off-ramp of downstream. The different defnitions should thus be acknowledged when interpretating the effects of existence of auxiliary lanes in this study. In case of this study, the existence of auxiliary lanes implies the proximity of mainline segments to ramps on the expressway.

Wider median width could significantly reduce the safety risks on the expressway. This result is expected since wider median width provides more space for correcting driving errors or for temporary stop during emergency. Also, in the case of potential hazardous conditions, drivers could use the median to avoid collisions.

The effects of traffic variables were found to vary across peak and non-peak hours. During peak hours, several segments on the expressway experience congestion and the Congestion Index is significant with a positive sign. The result indicates that crashes are more likely to occur on those congested segments. According to Table 3, standard deviation of speed in addition to Congestion Index also increases number of crashes during peak hours. The variation of speed on the same traveling lane could lead to rear-end crashes under congested conditions while the variation of speed between adjacent lanes might result in sideswipe crashes.

Regarding the performances of the three models, DIC and pseudo adjusted R² values were compared. The models with correlated random effects have much lower DIC and higher pseudo adjusted R² values. For each segment, the crash counts were observed during peak and non-peak hours,thus leading to repeated observations per segment. As a result, the unobserved heterogeneity portion of the disturbance term is correlated within segments (Washington et al., 2010). Several studies have proved that by introducing the correlated random effects, models for repeated measures would provide better model fitting (smaller DIC) (Ntzoufras, 2011; Yu et al., 2013). In this study, it is confirmed that by applying correlated random effects,the model fitting significantly improved. Another finding from the model comparison shows that by introducing random parameters varying across observations, horizontal degree of curvature become significant at 90% BCI, the existence of an auxiliary lane, Congestion Index and speed standard deviation are significant at 95% BCI. In conclusion, correlated random effects combined with random parameters model is chosen as the best model based on model fitting and parameter estimates.

5.3. Congestion measure selection

According to the model comparison from the above section, the random parameters model with correlated random effects was adopted to evaluate the performances of the three candidate congestion measures. Of the three congestion measures, occupancy and CI were derived from MVDS data while TTI was calculated from AVI data. The results are shown in Table 4.

Comparing the performances of the three measures, it can be seen that the parameter estimates in the model with occupancy and CI share higher similarity than those with the TTI. The effects of geometric variables in the three models agree with each other.

For the congestion variables, Congestion Index and occupancy are significant during peak hours and insignificant during non-peak hours. For the model with Travel Time Index, standard deviation of speed during peak hours is significant but TTI is insignificant. One potential issue contributing to this result might be that the free flow travel time used to calculate the TTI is defined based on average speed limit by the authority. However, Fig. 1 shows that about half ofthe speed is distributed above the 65 mph speed limit. Consequently, congestion might be underestimated by TTI and thus their effect is insignificant. According to the goodness-of-fit and variable significance shown in Table 4, CI and occupancy were selected for further comparison to see their feasibility in studying crash frequency.

Regarding the performance of CI and occupancy, the average weekday congestion distributions in a random month are shown below in Figs. 3 and 4. According to the figures, the congestion conditions evaluated by these two measures are highly comparable. Both measures reflect the spatial-temporal distribution of congestion at a detailed level. Congestion is confirmed to occur during morning or evening peak hours and also location specific. During non-peak hours, the congestion readings are low without significant variation, which might help in understanding their effects during non-peak hours. As a result, they should be regarded as adequate indicators with high precision. As for which congestion measure could be better used in safety evaluation to explore the effects of congestion on crashes, it should be judged based on their performance in the models. In this study, the model with CI has slightly lower DIC and higher pseudo adjusted R² than the model with occupancy. Although this difference is not significant, the model using CI could identify more significant variables, which is another advantage. Therefore, CI could be selected as an appropriate measure to evaluate the effects of congestion on safety. In case it is not available, occupancy should be considered.

5.4. Congestion and crash severity

As discussed earlier, existing studies haven't reached an agreement about how congestion lead to crashes. In addition, the question that whether congestion is related to more severe crashes or it would lower crashseverity also remains. For a single crashcase, the severity of crash heavily depends on the speed ofthe vehicles at the crashing time. However, the answer is not that straightforward when it comes to crash frequency studies since the average conditions instead of real-time traffic conditions immediately prior to crashes were used. Of the 705 crashes in this study, only 9 crashes involved incapacitating injury or fatality,the property damage only crashes (486) and possible injury crashes (119) are the majority of the sample. Given the sample size in each severity scale, the effects of congestion on PDO (486) and KABC (219) crashes were evaluated as shown in Table 5. Random parameter multilevel ridge regression models with correlated random effects were applied. Congestion Index was selected to indicate the effects of congestion on crashes at different crash severity scales.

Table 4Parameter estimates andmodel fitting for congestion measure selection.

ria ble Va	Con tio ges			Oc cup anc	y				vel Tim dex Tra e In
	Me an	SD	95% BC I	t-V alu e	Me an	SD	95% BC I	t-V alu e	Me an	SD	95% BC I	t-V alu e
RC le I vel
Int ept erc	2.1 88	0.3 45	( 60) 1.4 83, 2.8	6.3 38	2.3 56	0.3 20	( ) 1.7 34, 2.9 42	7.3 72	2.2 89	0.3 77	( ) 1.5 39, 2.9 91	6.0 68
( len h) Log gt	1.0 95	53 0.1	( 03, 1.3 87) 0.8	7.1 71	73 1.1	0.1 42	( 30 ) 0.8 68, 1.4	8.2 49	31 1.1	63 0.1	( 32, ) 0.8 1.4 40	6.9 56
Sta nda rd d evi atio f o n r d ibu f istr tio ete par am n o log ( len h) gt	0.2 28	0.1 32	( 0.0 42, 0.5 17)	1.7 24	0.3 18	0.1 43	( 0.0 63, 0.5 80)	2.2 15	0.2 21	0.1 45	( 0.0 28, 0.5 00 )	1.5 17
Ho riz al de f cu ont tur gre e o rva e	−0 .22 0	0.1 45	(− 0.5 37, −0 .00 6)	−1 .52 3	−0 .19 5	0.1 41	(− 0.5 07, 0.0 70)	−1 .37 6	−0 .20 5	0.1 38	(− 0.4 89, 0.0 51)	−1 .48 3
Sta nda rd d evi atio f o n r d istr ibu tio f ete par am n o hor izo l nta cu rve	0.4 02	0.1 89	( 0.0 49, 0.7 37 )	2.1 24	0.3 99	0.1 87	( 0.0 38, 0.7 58)	2.1 32	0.3 27	0.1 81	( 0.0 35, 0.6 75)	1.8 08
Seg ith 4 o r 5 lan nt w me e	0.4 35	0.2 38	* (− 0.0 35, 0.9 06 )	1.8 26	0.3 52	0.2 23	(− 0.0 77, 0.8 17)	1.5 77	0.5 39	0.2 46	( 0.0 74, 1.0 45 )	2.1 87
ed lim it o f 6 h Spe 5 m p	−0 .40 3	0.2 17	* (− ) 0.8 31, 0.0 39	−1 .86 0	−0 .41 4	0.2 09	(− 2) 0.8 08, −0 .01	−1 .98 2	−0 .32 4	0.2 44	(− 60) 0.8 16, 0.1	−1 .32 6
Exi f au xili lan ste nce o ary e	0.3 71	0.1 70	( 0.0 48, 0.6 98 )	2.1 77	0.3 62	0.1 73	( 0.0 0.7 08 ) 17,	2.0 97	0.3 55	0.1 66	( 0.0 14, 0.6 80)	2.1 41
Me dia idt h w n	−0 .01 0	0.0 05	* (− 0.0 20, 0.0 00 1)	−1 .81 2	−0 .01 1	0.0 06	(− 0.0 22, −0 .00 06 )	−1 .94 8	−0 .01 3	0.0 06	(− 0.0 25, −0 .00 2)	−2 .22 3
le vel MV DS
ion Ind MV DS C est ong ex	00a 0.1 46b 0.0	0.0 23 0.0 53	( ) 0.0 55, 0.1 45 (− 0.0 50, 0.1 62)	4.4 11 0.8 74	– –	– –	– –	– –	– –	– –	– –	– –
MV DS O ccu pan cy	–	–	–	–	0.1 54a	0.0 43	( ) 0.0 68, 0.2 35	3.5 74	–	–	–	–
	–	–	–	–	55b 0.0	0.0 68	(− ) 0.0 63, 0.1 98	0.8 03	–	–	–	–
AV Tr I l Ti In dex ave me	–	–	–	–	–	–	–	–	0.8 74a	0.8 64	(− 0.4 57, 2.6 88)	1.0 11
	–	–	–	–	–	–	–	–	2b −0 .14	0.6 08	(− 1.4 21, 1.0 41 )	−0 .23 4
/ and ard dev iati MV DS AV I St on of eed sp	0.1 36a	0.0 37	( 11) 0.0 63, 0.2	3.6 64	0.1 39a	0.0 36	( ) 0.0 67, 0.2 06	3.8 40	0.1 73a	0.0 41	( 53) 0.0 91, 0.2	4.1 86
	44b 0.0	0.1 29	(− 0.2 00, 0.3 40 )	0.3 44	48b 0.0	0.1 30	(− 0.2 08, 0.3 34 )	0.3 72	29b 0.0	0.0 49	(− 0.0 55, 0.1 37 )	0.5 96
del rfo Mo pe rm anc e
¯ D			997 .86 1				999 .42 2				100 6.3 20
pD			126 .47 1				131 .30 4				130 .58 8
DIC			112 4.3 32				113 0.7 26				113 6.9 08
2 ud dju d R ste pse o a			0.7 72				0.7 65				0.7 11

Significant at 90% BCI.

Peakhours.

Non-peak hours.

Fig. 3. Congestion conditions on SR 408 EB (left) and WB (right) by Congestion Index.

Fig. 4. Congestion conditions on SR 408 EB (left) and WB (right) by Occupancy.

The modeling results indicate that higher congestion and speed variation are related to both higher number of PDO and KABC crashes during peak hours. While during non-peak hours, they are not major contributing factors. By checking the crash data, it was found that rear-end crashes were the leading crash type in each of the severity scale: 271 out of 486 PDO/O scale crashes, 78 out of 119 C scale crashes, 51 out of 91 B scale crashes, 6 out of 8 A scale crashes, and 1 out of 1 K scale crash. Such high proportion indicated that rear-end crashes could have a significant impact on crashes at all severity scales. Since congestion is more likely to cause rearend crashes (Shi and Abdel-Aty, 2015), the effects of congestion

on crash severity in this study is understandable. In some previous studies, the relationship between congestion and fatal and serious injury crashes were explored (Shefer and Rietveld, 1997; Wang et al., 2013). Nevertheless, given serious injury crashes were rare (9 for combined K and A crashes)in the current research, no statement on such issues could be reached at this stage.

5.5. Model fitting without ridge or without congestion indicator

After we constructed the crash frequency model using Bayesian ridge regression, two issues remain: (1) Is it necessary to use

Table 5 Parameter estimates of congestion on PDO and KABC crashes.

| Variables | PDO | | | | KABC | | | | | | |----------------------------------------------------------------|------------|-------|------------------|---------|---------|-------|-------------------|---------|--|--| | | Mean
SD | | 95% BCI | t-Value | Mean | SD | 95% BCI | t-Value | | | | RCI level | | | | | | | | | | | | Intercept | 1.686 | 0.347 | (1.000, 2.382) | 4.864 | 2.046 | 0.393 | (1.281, 2.860) | 5.205 | | | | Log(length) | 1.088 | 0.167 | (0.750, 1.419) | 6.523 | 1.384 | 0.202 | (0.999, 1.833) | 6.848 | | | | Standard deviation of parameter
distribution of log(length) | 0.237 | 0.160 | (0.027, 0.555) | 1.480 | 0.459 | 0.148 | (0.054, 0.670) | 3.103 | | | | Segment with 4 or 5 lane | 0.444 | 0.236 | (−0.034, 0.896)* | 1.883 | – | – | – | – | | | | Speed limit of 65 mph | −0.580 | 0.220 | (−1.028, −0.151) | −2.637 | −0.673 | 0.248 | (−1.159, −0.166) | −2.715 | | | | Existence of auxiliary lane | 0.396 | 0.183 | (0.011, 0.753) | 2.166 | – | – | – | – | | | | Median width | – | – | – | – | −0.013 | 0.007 | (−0.027, 0.0005)* | −1.940 | | | | MVDS level | | | | | | | | | | | | MVDS Congestion Index | 0.079a | 0.023 | (0.035, 0.123) | 3.402 | 0.099a | 0.034 | (0.033, 0.173) | 2.910 | | | | | 0.077b | 0.058 | (−0.036, 0.194) | 1.319 | 0.053b | 0.073 | (−0.070, 0.220) | 0.728 | | | | Standard deviation of speed | 0.119a | 0.038 | (0.042, 0.192) | 3.107 | 0.087a | 0.046 | (−0.002, 0.179)* | 1.869 | | | | | 0.128b | 0.154 | (−0.159, 0.451) | 0.829 | −0.085b | 0.179 | (−0.512, 0.239) | −0.475 | | | | Model performance | | | | | | | | | | | | D¯ | | | 943.422 | | | | 654.004 | | | | | pD | | | 119.848 | | | | 82.987 | | | | | DIC | | | 1063.270 | | | | 736.991 | | | | | pseudo adjusted R2 | | | 0.669 | | | | 0.553 | | | |

^* Significant at 90% BCI.

^a Peak hours.

^b Non-peak hours.

Table 6Comparison for regression without congestion indicator, without ridge and with both congestion indicator and ridge regression.

ble Va ria	tho Wi ut con	nde tio n I ges	h v olu onl x & wit me y		tho Wi ut	rid ge			th Wi con	nde tio n i ges	nd rid x a ge 95% BC I ( 1.4 83, 2.8 60) ( 87) 0.8 03, 1.3 ( 17) 0.0 42, 0.5 (− 0.5 37, −0 .00 6) ( ) 0.0 49, 0.7 37 (− 0.0 35, 0.9 06 ) (− ) 0.8 31, 0.0 39 ( 0.0 48, 0.6 98 ) * (− 0.0 20, 0.0 00 1) – – ( ) 0.0 55, 0.1 45 (− 62) 0.0 50, 0.1 ( 0.0 63, 0.2 11) (− 0.2 00, 0.3 40 ) 997 .86 1 126 .47 1 112 4.3 32
	Me an	SD	95% BC I	alu t-V e	Me an	SD	95% BC I	alu t-V e	Me an	SD		alu t-V e
RC le vel I
Int ept erc	2.2 64	0.3 69	( 1.5 79, 2.9 77)	6.1 29	1.4 17	0.5 58	( 0.4 84, 2.6 61)	2.5 38	2.1 88	0.3 45		6.3 38
( len h) Log gt	1.0 96	0.1 65	( ) 0.7 81, 1.4 33	6.6 26	1.1 66	0.1 55	( 76) 0.8 36, 1.4	7.5 27	1.0 95	0.1 53		7.1 71
nda rd d evi atio f p Sta ter o n ara me dis trib uti f lo ( len h) gt on o g	0.3 14	0.1 52	( 59) 0.0 32, 0.5	2.0 67	0.2 75	0.1 58	( ) 0.0 28, 0.5 44	1.7 46	0.2 28	0.1 32		1.7 24
Ho riz al de f cu ont tur gre e o rva e	−0 .20 5	0.1 34	(− 0.4 84, 0.0 30 )	−1 .53 1	−0 .23 9	0.1 46	* (− 0.5 42, 0.0 21 )	−1 .64 4	−0 .22 0	0.1 45		−1 .52 3
nda rd d evi atio f p Sta ter o n ara me dis trib uti f h ori tal on o zon cur ve	0.3 51	0.1 81	( 80) 0.0 43, 0.6	1.9 43	0.4 30	0.1 79	( 70) 0.0 54, 0.7	2.3 96	0.4 02	0.1 89		2.1 24
Seg ith 4 o r 5 lan nt w me e	0.5 22	0.2 33	( 0.0 83, 0.9 66)	2.2 35	0.2 82	0.2 46	(− 0.2 14, 0.7 77)	1.1 43	0.4 35	0.2 38		1.8 26
ed lim it o f 6 h Spe 5 m p	−0 .44 1	0.2 12	(− 5) 0.8 55, −0 .02	−2 .08 0	−0 .39 0	0.2 12	* (− ) 0.8 20, 0.0 29	−1 .84 4	−0 .40 3	0.2 17		−1 .86 0
Exi f au xili lan ste nce o ary e	0.3 61	0.1 67	( 0.0 14, 0.6 85)	2.1 26	0.3 85	0.1 75	( 0.0 35, 0.7 15)	2.2 05	0.3 71	0.1 70		2.1 77
Me dia idt h w n	−0 .01 2	0.0 06	* (− 0.0 24, 0.0 01 )	−1 .95 8	−0 .01 1	0.0 05	(− 0.0 22, −0 .00 1)	−2 .10 9	−0 .01 0	0.0 05		−1 .81 2
le vel MV DS
( vol er l ) Log MV DS um e p ane	97a 0.4 2b −0 .11	0.4 05 0.3 04	(− 1.3 ) 0.1 79, 25 (− 0.7 29, 0.4 56)	1.2 26 −0 .36 9	– –	– –	– –	– –	– –	– –		– –
Ind MV DS C ion est ong ex	–	–	–	–	0.0 83a	0.0 69	(− ) 0.0 53, 0.2 07	1.2 04	0.1 00a	0.0 23		4.4 11
	–	–	–	–	92b 0.0	0.0 53	* (− 86) 0.0 09, 0.1	1.7 43	46b 0.0	0.0 53		0.8 74
Sta nda rd d evi atio f sp eed o n	–	–	–	–	0.0 61a	0.0 77	(− 0.0 87, 0.2 16)	0.7 89	0.1 36a	0.0 37		3.6 64
	–	–	–	–	98b 0.1	0.2 34	(− 0.2 64, 0.6 37 )	0.8 44	44b 0.0	0.1 29		0.3 44
del rfo Mo pe rm anc e
¯ D			997 .93 1				997 .81 8
pD			138 .40 6				131 .10 5
DIC			113 6.3 37				112 8.9 23
2 ud dju d R ste pse o a			0.7 56				0.7 69				0.7 72

Significant at 90% BCI.

Peakhours.

Non-peak hours.

specific congestion measure when exploring congestion's impact on safety? Is volume itself alone could be a good predictor of the effects of congestion on safety? (2) How does the crash frequency model with ridge regression differ from the model without ridge regression. Consequently, one model without ridge regression and another model without congestion indicator were constructed under the same model structure.

Table 6 summarizes the results of the two models compared with the ridge regression model with congestion indicator. The results show that in the model without congestion indicator and with only logarithmic volume per lane, the effects of volume were not significant. Such results indicate although higher traffic volume could lead to congestion, traffic volume per lane itself could not replace the use of a direct congestion measure when investigating the effects of congestion on traffic safety. In addition,the goodnessof-fit is much lower than the model that incorporates Congestion Index. These findings imply that when the aim is to evaluate how congestion leads to crashes, it is suggested to use direct measure as congestion indicator.

As for the model without ridge regression, both the goodness of fit and variable significance are comparable to the ridge regression model. As discussed above, multicollinearity does not significantly affect model fitting. Thus the difference of DIC values between the two models are small. For geometric variables that are not affected by the multicollinearity, parameter estimates for models with and without ridge are also similar. It is only the traffic variables within which ridge regression was used that show the difference between these two models. First of all, the significance of the traffic variables during peak hours was affected in the model without ridge. Although Washington et al. (2010) stated that highly correlated variables do not always cause estimation problems. If the standard errors of the estimated parameters are low, it is still possible to get well-defined parameter estimates. However, in this study, Congestion Index and standard deviation of speed are highly correlated during peak hours. Furthermore, the standard errors of parameter estimates for logarithmic volume and congestion indicator are relatively high. Consequently, without considering the multicollinearity, the significance of the traffic variables during peak hours became insignificant.

Except for the variable significance of traffic variables, we could still detect substantial differences in the parameter estimates. The estimated parameters of Congestion Index during peak and non-peak hours are 0.083 and 0.092, respectively without ridge regression and 0.100 and 0.046 with ridge regression. The same effects were found for standard deviation of speed. This means that without taking multicollinearity into account, the effect of traffic variables during non-peak hours was exaggerated. Under such condition, the traditional elasticity and marginal effects of variables could not be directly applied since the change of one variable will cause the change of the correlated variable. In conclusion, if multicollinearity issue is severe enough that requires a response, ridge regression along with other measures listed in Section 2.3 could be considered. If the multicollinearity issue is not severe and the priority is goodness-of-fit and variable significance, multicollinearity might be a secondary issue for consideration. In other cases thatthe elasticity or marginal effects of the variables are the major concern, it is better to apply remedial countermeasures.

6. Conclusions

Traffic congestion and safety are the priorities of traffic professionals. In urban areas, the growing traffic has raised an urgent call for traffic authorities to provide both efficient and safe transportation system to the motorists. With the rapid development of ITS techniques, more proactive traffic management can be achieved. To improve the congestion and safety conditions simultaneously, a better understanding of their relationship is required. In this study, the effects of congestion on crash frequency based on an urban expressway in Central Florida were investigated. The 21.4-mile expressway ofinterestis equipped with 42AutomaticVehicle Identification (AVI) segments and 110 Microwave Vehicle Detection System (MVDS) traffic detectors. Comprehensive traffic information was extracted from the ITS systems.

To properly evaluate the effects of congestion on crash frequency, the presence of multicollinearity was checked. Multicollinearity is caused by the correlation among independent variables and will interfere with parameter estimates in regression. In traffic safety analysis, the traffic parameters are often related. It was found in this study that during peak traffic hours (7:00–9:00 and 17:00–19:00), high correlations among traffic variables were verified. Since the coefficients of determination of auxiliary regressions are greater than the value of general regression and VIF value of CI during peak hours is higher than 5, multicollinearity was regarded necessary to be taken into consideration.

Three models were constructed using Bayesian ridge regression. The longer segmentlength indicates higher exposure and increases the crash frequency. Horizontal curvature was found to contribute to reduction in crashes.When a segment contains 4 or 5 lanes, more crashes are more likely to occur. Also, lower speed limit is related to higher crash frequency. Wider median width could also effectively improve safety on the studied expressway. It was also found that auxiliary lanes could pose potential risks for mainline traffic. On the segments with auxiliary lanes, more frequent merging and diverging behaviors are expected. These movements can raise the likelihood of crashes. Regarding the effects of traffic parameters, their influences on crash occurrences were better revealed by assigning them with random parameters taking into account the peak and non-peak hours. Congestion and speed variation were proved to be significantly related with crashes only during the peak hours. In sum, to better capture the congestion's effects on urban expressway safety, distinguishing the time period could result in more straightforward interpretation. As for the modeling technique, correlated random effects could significantly improve model fitting performance when repeated observations were involved. Random parameters model taking into account the unobserved heterogeneity was also proven helpful in revealing the effects of contributing variables.

Previous studies have made substantial effort to discover the effect of congestion on traffic safety but failed to reach a consistent conclusion. One of the possible explanations is the selection of congestion measures. In this study, Travel Time Index (TTI) from AVI data, occupancy and Congestion Index (CI) from MVDS data were developed and tested in the crash models. The results using these different congestion measures generally agree with each other. Current TTI on the studied system tends to underestimate congestion and their effects were not significant in the model. CI and occupancy reflect the congestion more objectively and thus their performances in the models were slightly better in both model fitting andparameter estimates.As a conclusion,to identify the effects of congestion on crashes, the selection of appropriate congestion measure is crucial. This statement does not imply that there is a fixed congestion measure that performs the bestin crash frequency models. With the development of ITS systems and real-time traffic data, both congestion and safety management could benefit.

The effects of congestion on crash severity were also investigated in this research. Considering the limited sample size in each crash severity scale, the analyses were conduct on PDO crashes and KABC crashes respectively. On the studied expressway, congestion was found to increase both PDO and KABC crashes during peak hours. To understand the effects, crash types at each severity scale were examined. For all of the 5 scales, rear-end crashes were the majority type of crashes. Since congestion could cause stop-and-go traffic conditions on expressways, rear-end crashes at all severity levels are more like to occur. Therefore the findings in this study were understandable. In the future, alleviating congestion might have a positive effects on safety improvement for not only PDO crashes but also crashes involving injury and fatality.

In this paper, multicollinearity was introduced because of the correlation between congestion measures and speed variations in the model. The necessity of doing so was evaluated by using the same model with only traffic volume as a surrogate measure of congestion. Ridge regression was proposed to deal with multicollinearity. How this method alters our understanding about the model results was also calibratedusingmodel without ridge regression. In the model without congestion indicator, traffic volume per lane was not significant during peak hours. Also, the model with only volume as an indirect congestion indicator was proven inadequate in model fitting. This result means that congestion is a more direct and important traffic factor influencing the safety condition during peak hours on urban expressways. Also, to understand how congestion affects traffic safety, it is suggested that a congestion indicator be used in the modeling process. The model without ridge regression did not have a significant impact on the model fitting. However, given the high correlation between congestion and standard deviation and the relatively large standard error of estimated parameters, the significance of the traffic variables was altered. Furthermore, the consequence of multicollinearity was found in the magnitude of estimated parameters for Congestion Index and standard deviation of speed, which posed certain challenges to understand the effects ofthe variable.Whether to take any countermeasure in face of multicollinearity should be based on its severity and the objective of the study. Significant but less severe multicollinearity might not be an issue if the aim is to have general good model performance and to identify important variables. Instead, if multicollinearity is severe or precise estimation of marginal effects of variables is required, corresponding remedial measures should be taken.

Although this study tried to make comprehensive analysis about the effects of congestion on traffic safety, there are still several importantissues that could not be covered in one paper. They serve as motivations for further analysis ofthe relationship between congestion and safety. In the current study, the relationship between congestion and crashes involving fatality and serious injury was beyond evaluation due to the limited sample size. In addition, congestion might have distinct effects on different types of crashes. Given the highly instrumented expressway in this study, the issues of crash severity and crash type under congestion could be further explored. In crash frequency studies, since the traffic information needs to be aggregated, much information is lost. For example, the intensity of congestion is highly affected by the location and time period. Averaging it might cause the true effect of congestion not properly revealed. With this disadvantage of crash frequency model, one alternative method is real-time crash prediction in which the pre-crash condition for each crash case will be used. Last but not the least, crashes are still random in nature. Therefore including more information might also help in a better understanding of crash mechanisms. Depending on the availability of data, weather and driver related factors should also be considered in the crash modeling in the future.

Acknowledgments

The authors wish to thank the Central Florida Expressway Authority (CFX) and the UTC National Center for Transportation Systems Productivity and Management (NCTSPM) for funding this research and CFX for providing the data. The authors also wish to thank Charles R. Lattimer, PE, PMP Sr. Project Manager, Intelligent Transportation Systems at ATKINS for his suggestions.