Examination of the severity of two-lane highway traffic barrier crashes using the mixed logit model

Journal of Safety Research 70 (2019) 223–232

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Examination of the severity of two-lane highway traffic barrier crashes using the mixed logit model Mahdi Rezapour, a,⁎ Shaun S. Wulff, b Khaled Ksaibati c a b c

Department of Civil & Architectural Engineering, University of Wyoming, Office: EN 3084, 1000 E University Ave, Dept. 3295, Laramie, WY 82071, United States of America Department of Mathematics and Statistics, University of Wyoming, Office: Ross Hall 336, Laramie, WY 82071, United States of America Wyoming Technology Transfer Center, 1000 E. University Avenue, Department 3295, Laramie, WY 82071, United States of America

a r t i c l e

i n f o

Article history: Received 18 February 2019 Received in revised form 3 May 2019 Accepted 16 July 2019 Available online 25 July 2019 Keywords: Random parameter Traffic barrier Guardrail Crash severity Interaction terms

a b s t r a c t Introduction: Vehicles in transport sometimes leave the travel lane and encroach onto natural or artificial objects on the roadsides. These types of crashes are called run-off the road crashes, which account for a large proportion of fatalities and severe crashes to vehicle occupants. In the United States, there are about one million such crashes, with roadside features leading to one third of all road fatalities. Traffic barriers could be installed to keep vehicles on the roadways and to prevent vehicles from colliding with obstacles such as trees, boulder, and walls. The installation of traffic barriers would be warranted if the severity of colliding with the barrier would be less severe than colliding with other fix objects on the sides of the roadway. However, injuries and fatalities do occur when vehicle collide with traffic barriers. A comprehensive analysis of traffic barrier features is lacking due to the absence of traffic barrier features data. Previous research has focused on simulation studies or only a general evaluation of traffic barriers, without accounting for different traffic barrier features. Method: This study is conducted using an extensive traffic barrier features database for the purpose of investigating the impact of different environmental and traffic barrier geometry on this type of crash severity. This study only included data related to two-lane undivided roadway systems, which did not involve median barrier crashes. Crash severity is modeled using a mixed binary logistic regression model in which some parameters are fixed and some are random. Results: The results indicated that the effects of traffic barrier height, traffic barrier offset, and shoulder width should not be separated, but rather considered as interactions that impact crash severity. Rollover, side slope height, alcohol involvement, road surface conditions, and posted speed limit are some factors that also impact the severity of these crashes. The effects of gender, truck traffic count, and time of a day were found to be best modeled with random parameters in this study. The effects of these risk factors are discussed in this paper. Practical applications: Results from this study could provide new guidelines for the design of traffic barriers based upon the identified roadway and traffic barrier characteristics. © 2019 National Safety Council and Elsevier Ltd. All rights reserved.

1. Introduction Every year, about 1.3 million people die due to traffic crashes worldwide. In addition, 50 million more are severely injured on roadways (OECD. Publishing, 2017). In the United States in 2017, more than 37,000 people died as a result of road crashes (National Highway Traffic Safety Administration, 2017). About 8000 people died in 2016 in collisions with fixed objects, which is 3% higher than the 2015 crashes (FARS and National Highway Traffic Safety Administration, 2016). Motorcycles (5.1%), vans (8.4%), and pickup trucks (1.5%) are the vehicles at the highest risk of fatality from fixed object collisions (National Highway Traffic Safety Administration, 2016). Roadway departure ⁎ Corresponding author. E-mail addresses: [email protected] (M. Rezapour), [email protected] (S.S. Wulff), [email protected] (K. Ksaibati).

https://doi.org/10.1016/j.jsr.2019.07.010 0022-4375/© 2019 National Safety Council and Elsevier Ltd. All rights reserved.

crashes tend to be hazardous, especially in rural and mountainous areas like Wyoming. While run off the road crashes accounted for 16% of all crashes, they resulted in 31% of fatal crashes (US Department of Transportation, 2015). Various mitigating strategies can be used to reduce the severity of these types of crashes. This includes the removal of hazards such as trees or transferring culverts out of the clear zones. Another more practical method is interposing traffic barriers. A traffic barrier itself is hazardous, so the use of the barrier is only recommended when it redirects a driver from more hazardous objects. A variety of factors might be associated with run off the road (ROR) crashes that relate to traffic barrier crash severity. These include environmental related factors, road environments, and barrier geometry characteristics (National Highway Traffic Safety Administration, 2009). Identification of the contributory factors to crash severity is necessary for the improvement of traffic safety (Rezapour, Moomen, & Ksaibati, 2019).

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Many studies have been conducted that consider the role of roadway design, environmental characteristics, and various geometric design on the severity of traffic barrier crashes. However, few studies incorporated traffic barrier design characteristics. Traffic barrier design plays a crucial rule on the severity of traffic barrier crashes as the inappropriate design of traffic barriers can lead to either override or underride of a vehicle. Therefore, the main contribution of this study is to evaluate traffic barrier crash severity with respect to different predictors, especially traffic barrier geometry and roadside characteristics. This study incorporated all two-lane highway traffic barrier involved crashes in Wyoming. These crashes also include run-off the roadway and rollover crashes. Since only two-lane highway traffic barrier crashes were of interest in this study, median barriers were not included due the infrequent occurrence of such barriers and the differences in performance of these barrier types. This study provides guidelines for policy makers concerning roadside safety design and performance of traffic barriers by identifying important factors that impact crash severity.

deflection characteristics (stiffness) that could impact the effects of vehicle collisions. Other issues could arise involving guardrail height and offset from the side of the roads. The literature review also indicated that to identify the unique contributory factors associated with a particular crash type, that the crash type should be analyzed separately (Mashhadi, Wulff, & Ksaibati, 2017a; Mashhadi, Wulff, & Ksaibati, 2018a). Therefore, this study just focused on traffic barrier crashes. It is hypothesized that traffic barrier crashes vary in severity according to barrier height, barrier offset, and shoulder width. Thus, this study also focuses on traffic barriers for highway systems and excludes interstate traffic barriers. Although there have been studies that identify contributory factors of severe traffic barrier crashes, there have not been studies that have incorporated the geometric characteristics of traffic barriers. Thus, this study could contribute to the body of knowledge by incorporating different traffic barrier characteristics and the interactions between these factors.

1.1. Literature review

2. Data collection

There is a vast body of literature examining different barrier types. A review of the past studies indicated that these studies could be divided into two sections: studies with real world crash data and the ones with simulated data. A recent evaluation was conducted using a random parameter model to identify the factors that influence median crash frequency, severity, and barrier collision outcomes (Russo & Savolainen, 2018a). The results indicated that different roadway, traffic, environmental, and vehicle characteristics could impact crash frequency, severity, and barrier strike outcomes. Significant differences were also found across thriebeam, cable barrier, and concrete barrier. On the other hand, Zou, Tarko, Chen, and Romero (2014) used a binary logistic regression model with mixed effects to evaluate the effectiveness of different traffic barriers in reducing crash injury. Three types of traffic barrier were investigated. The offset of the barriers to the edge of the traveled-way was incorporated in the study. The results indicated that traffic barriers reduce crash severity compared to colliding with a roadside object. However, there was no confirmation that median concrete walls located at the traveled way edge (offset 7–14 ft.) could reduce crash severity compared with fixed objects. A nested logit model was used to evaluate the severity of median barrier crash severity (Hu & Donnell, 2010). The results indicated that increasing the median barrier offset is associated with a lower probability of severe crashes. Also, it was shown that foreslopes between 6H:1V are associated with an increase in severe crashes compared with slopes 10H:1V or less. On the other hand, many studies have been conducted by crash testing methods, such as the NCHRP Report 350 (Michie, 1981), and the AASHTO manual for assessing safety hardware (MASH) (AASHTO, 2009). Not much change has been made on W-beam, box-beam guardrail systems since the 1960s. For instance, the W-beam guardrail being used by some policy makers is characterized by a 533-mm (21 in.) center mounting and 75 in. post spacing (Nordlin, Stoker, & Stoughton, 1976). The results of this study were based on a four-vehicle impact test into a metal beam guardrail using three different types of posts. The guardrail characteristics proved to meet the NCHRP Report 350 performance criteria when tested for pickup trucks. However, the literature review showed there would be improvement for designs of this type of guardrail during light truck crashes (Ross, 2002). Statistical analyses also indicated that light trucks are at higher risk of crash severity due to rollovers than conventional vehicles (Mak, Bligh, & Menges, 1996). Despite this research on the evaluation of traffic barriers through standard crash testing procedures and under different scenarios, it should be noted that the testing procedures do not account for different confounding factors that might be at play on roadways. When the use of traffic barriers is warranted, policy makers need to decide which types of barriers are more appropriate since different barriers have different

Data from various sources were aggregated for this study. Traffic barrier crashes were obtained from the Wyoming Department of Transportation (WYDOT) through the critical analysis reporting environment (CARE) package from 2007 to 2017. A crash was extracted and incorporated into the dataset if the first harmful event column indicated that collision with a traffic barrier was a cause of the crash. A traffic barrier crash was excluded from the dataset if more than one vehicle was involved in a traffic barrier crash. The resulting dataset has a total of 1619 traffic barrier crashes. Roadway geometrics and traffic counts were obtained and aggregated from an inventory dataset maintained by WYDOT. This inventory was obtained from a field survey involving over 1.3 million linear feet of traffic barriers to measure geometric characteristics, length, offset, and side slope height. Over 7700 photos were provided to ensure the measurements were accurate (see Fig. 1). This study focused on two lane highway systems instead of aggregating interstate and highway systems. This disaggregation was due to the differences in the traffic barrier designs and traffic characteristics for these two highway systems in Wyoming. Lee and Mannering (2002) indicated that there are different human factors and driver behaviors due to roadside features for run-off roadway crashes, which provides further justification for separating different road systems. Based on the FHWA Safety Program (n.d.), a shoulder width of 2–8 ft is recommended for local roads with low traffic volume. The width of a shoulder is dependent on many factors, such as space for maintenance and enforcement activities, road geometrics, or space for drivers to maneuver. Shoulder width was kept as a continuous predictor in this study in terms of feet. Barrier offset (offset) in this study is defined as an offset from the edge of a barrier to the traveled way. Side slope height is defined as the height created by the cut or fill, expressed as a difference between the bottom of the road side, at the location of traffic barrier installation, and the top of the paved road surface elevation. A continuous raw predictor of barrier offset was grouped into three categories: zero side slope (level) as reference category, a value of 1 for fill-type side slopes, and a value of 2 if the side slope height was greater than the roadway edge height (cut-type side slope). Due to the low number of events for different categories of the KABCO scale on these rural two-lane highways in Wyoming, a binary response category was specified in order to have enough observations for modeling. Property Damage Only (PDO) was given a value of zero and a value of 1 was given for fatal, minor injury, and major injury crashes. Summary statistics for the categorical and continuous predictors are presented on Tables 1 and 2. An age of 35 was selected as a cutting point for driver age as this value divided drivers into two equal categories.

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W-beam with fill side slope, with some offset

W-beam with no offset and filled side slope

Concrete Brrier with no offeset

Box beam barrier with some offset, with fill side slope

Fig. 1. Examples of traffic barrier evaluation.

3. Research methodology The mixed logit model (MLM), also called the random parameters logit model, has been used in many recent traffic studies (Milton et al.,

2008; Ye & Lord, 2014). The MLM is an extension of the ordinary logit model (OLM). The OLM assumes that the unobserved variables are uncorrelated over the response outcomes, which is called the independence from irrelevant alternative (IIA) assumption. The IIA

Table 1 Data summary of important categorical variables. Variables

Fixed parameter Driver condition Rollover Driver restrain (improper) Road condition Side slope indicator

Citation record Vehicle type Posted speed limit Gender Driver age Time of a day

Categories

Normal* Others No* Yes No* Yes Dry* Others Level* Fill slope Cut slope No* Yes Heavy truck Others Greater or equal than 55 mph Others Male* Female Less than or equal 35 years old* Others Peak hours* Off peak hours

PDO

Injury/fatality

#

%

#

%

818 332 1130 20 1060 90 375 775 307 162 681 668 482 55 1095 757 393 742 408 622 528 913 237

50.5 21 70 1 65 6 23 48 19 10 42 41 30 3 68 46 24 46 25 38 33 56 15

208 261 425 44 365 104 300 169 101 71 297 201 268 20 449 320 149 317 152 225 244 342 127

13 16 26 3 23 6 19 10 6 4 18 12 17 1 28 20 10 20 9 14 15 21 8

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Table 2 Data summary of important continuous variables.

Guardrail height (in) Shoulder width (ft) Offset distance (in) Heavy truck traffic

Min

Mean

Max

St.Dev

0.25 quantile

0.75 quantile

18.00 0.00 0.00 7.00

29.17 4.65 10.47 294.77

42.00 21.00 69.50 1260.00

3.99 3.23 11.00 225.96

27.60 2.50 2.00 108.00

31.20 6.50 16.90 369.00

assumption can be viewed as a model restriction or as a reasonable assumption for a well-specified model that captures all sources of correlation over the alternatives (Train, 2009). However, crashes can be complex events involving a variety of factors that are accident-specific that might not be adequately modeled under the IIA assumption. In such cases, the MLM is used to account for heterogeneity across crashes by allowing the influence of predictors to vary by crash. A severity function determining the proportion of injury severities on a roadway segment is defined as (Train, 2009) Snj ¼ x0nj βnj þ εnj ;

ð1Þ

where j indexes the injury severity category (j = 1, …, J), n indexes the crash (n = 1, …, N), Snj is a severity function, xnj is a vector of observed predictors, βnj is a vector of unknown parameters, and εnj is the error term that is assumed to be independent and identically distributed (iid) with an extreme value distribution. Conditional on βn, the probability for alternative i is πni ðβni Þ ¼


 exp x0ni βni  : J ∑ j¼1 exp x0nj βnj

ð2Þ

The unconditional probability for alternative i is given by Z P ni ¼

πni ðβni Þf ðβni jφi Þdβni

ð3Þ

where f(βni | φi) is the probability density function (PDF) of βni with φi denoting a vector of parameters characterizing the PDF of βni (Milton et al., 2008). The PDF of βni is the assumed mixing distribution used to incorporate crash-specific variations of the effect of the predictors on the crash severity probabilities. The same PDF is assumed across the crashes indexed by n where a common choice for the PDF is the normal distribution. Furthermore, independent normal PDFs are also often assumed over the K predictors and J response categories so that the PDF of βni is the product of the individual PDFs corresponding to the distribution βnik


  Normal βik ; σ ik

ð4Þ

where βik is the unknown marginal mean and σik is the unknown marginal standard deviation of βnik across all crashes indexed by n. If σik = 0 for some i, k, then the distribution in (4) is degenerate so that βnik ¼ βik with probability 1. In this case, the unknown βnik is taken to be constant or fixed across crashes with a common value βik . Thus, it is of primary interest to estimate βik and σik. When J = 2, the index i is not necessary. Consider a given combination of predictors and condition on the mean parameter vector β. Using (2), the odds (Ο) of combination x is given by
 Ο βjx ¼



 π βjx
 ¼ exp x0 β : 1−π βjx

ð5Þ

The odds ratio (OR) is the ratio of the odds in (5) for combinations x1

relative to x2 expressed as


 Ο βjx1 OR βjx1 ; x2 ¼
 ¼ exp x01 β−x02 β : Ο βjx2

ð6Þ

The odds ratio is useful for interpreting effects in logistic regression. A common example is OR ¼ expðβk Þ which would denote the odds ratio associated with a unit increase in the main effect predictor k with all other predictors held constant conditional on the value βk . In this case, a positive parameter value is an indication that variable k increases the odds of crash severity and a negative parameter value is an indication that variable k decreases the odds of crash severity. The more general form in (6) is useful when it comes to working through the interpretation of interaction effects. As in the Ye and Lord (2014), the collection of predictors was selected based upon data availability, engineering judgment, and initial screening with a significance level of 0.05. Using this collection of predictors, the mixed logit regression model was developed by initially including all predictors as random using the distribution in (4). This classification was changed to fixed if it was found that any variable did not exhibit non-zero standard deviation based upon a z test (Ye & Lord, 2014). Predictors associated with fixed parameters were also tested to determine if they were different from zero using a z test. A predictor with fixed parameter determined to be zero at the 0.05 level was removed from the model. Fig. 1 shows the framework for the severity of traffic barrier crashes. The main outcome is whether a driver was injured or killed in a barrier crash. The traffic barrier features were predictors of interest for this study of traffic barrier crashes. Thus, interactions involving these predictors were incorporated as it was hypothesized that various combinations of these predictors would have different effects on crash severity. Interactions were considered between barrier height and different roadway characteristics, such as shoulder width and offset. The interaction terms could aid in the understanding of the combined impacts of traffic barriers and roadway characteristics on traffic barrier involved crash severity. 4. Results The previously described modeling approach was applied to data for traffic barrier crashes on two-lane highways in Wyoming during 2007– 2017. The fit of the resulting mixed logit model is shown in Table 3. The model contains an intercept, 3 random parameters (mean and standard deviation), 12 main effect fixed parameters, 3 two-way fixed interaction parameters, and 1 three-way fixed interaction parameter. The interactions were incorporated to assess the combined effects of the traffic barrier features. The model fit in Table 3 had log-likelihood −824.60 (df = 23). For comparison, the zero model with no random or fixed parameters had log-likelihood −974.43 (df = 1) resulting in a McFadden R2 of 0.154. The corresponding model fit without the 3 random parameters has log-likelihood −832.19 (df = 20). The corresponding model fit without the 4 interaction terms has log-likelihood −827.54 (df = 19) and the corresponding model fit without any of the 7 traffic barrier feature predictors has log-likelihood −831.50 (df = 16). The fit of the full model shown in Table 3 is used for interpretation as the effects of the traffic

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Table 3 Results of the mixed logit model fit for traffic barrier crash severity. Variable

Estimate

Odds ratio

Std.Err.

p-value

Fixed parameters Constant Driver condition (1 anger, anxious, 0 for normal driver condition) Rollover crash (1 if occurred, 0 otherwise) Driver restrain (1 is driver was not buckled up, 0 otherwise) Road conditions (1 is not dry, 0 otherwise) Side slope indicator (ft) (0 if it is flat, 1 otherwise) Fill-type side slope (1) Cut-type side slope (0) Citation record (1 is any, 0 otherwise) Vehicle type(1 heavy truck, 0 otherwise) Posted speed limit Driver age (1 for over 35, 0 otherwise) Shoulder width Traffic barrier height Offset distance Guardrail height–offset interaction Shoulder width – traffic barrier height interaction Offset-shoulder width interaction Offset-shoulder width-traffic barrier height interaction

−4.909 0.969 2.469 1.550 −1.456 1.765 1.711 0.341 −0.916 −0.453 0.414 0.742 0.121 0.190 −0.007 −0.024 −0.058 0.002

– 2.635 11.81 4.714 0.233 5.859 5.543 1.406 0.400 0.636 1.512 2.085 1.129 1.209 0.993 0.976 0.944 1.002

1.975 0.198 0.452 0.293 0.229 0.288 0.217 0.175 0.418 0.225 0.171 0.442 0.065 0.117 0.004 0.015 0.030 0.001

0.022 b0.001 b0.001 b0.001 b0.001 0.047 0.013 0.040 0.027 0.044 0.012 0.096 0.064 0.105 0.080 0.111 0.052 0.047

Random parameters Gender (1 if the driver was female, 0 otherwise) Standard deviation Truck traffic (continuous) Standard deviation Time of a day (1 is it is off peak, 0 otherwise) Standard deviation

−0.115 1.356 −0.002 0.003 −0.443 2.459

0.891 3.881 0.998 1.003 0.642 11.693

0.248 0.547 0.001 0.001 0.335 0.813

0.619 0.013 0.019 0.008 0.186 0.002

barrier features are of special interest in this study. From an engineering perspective, it is reasonable to expect that the effects of different traffic barrier features is dependent on roadway characteristics. For instance, consider a traffic barrier set up at the side of the roadway with zero shoulder width and zero offset compared with a barrier type set up with some shoulder width and with some offset. In such a scenario, it could expected that the impact of such variable combinations would differ on traffic barrier involved crash severity.

4.1.4. Road conditions Crashes occurring on less-than-optimal road conditions are less c = − 1.436). Although the results may seem likely to be severe (OR

4.1. Main effect fixed parameters

4.1.5. Side slope height Studies have been conducted on the importance of different roadway features such as side slope. Results have indicated that steeper side slope could result in an increase in crash frequency (Graham & Harwood, 1982). It has also been found that vehicles that leave the roads are less likely to be involved in severe crashes if the side slope is flat (Marquis & Weaver, 1976). The previous studies also showed that the presence of roadside obstacles, the offset from the roadway edge, and the degree of side slope can impact run off the road crash risk (Kennedy, Roberts, & McGennis, 1984; Wolford & Sicking, 1997). It should be noted that recovering vehicle control over side slope degrees over 1V:3 is rated as non-recoverable while flatter side slopes are classified as recoverable (Guide, and D.AASHTO Washington, DC, 2002). Although various studies have been conducted on the impact of side slope on crash frequencies and severities, there is little information about side slope for barrier crashes. The estimates in Table 3 agree with previous studies that flat side slopes have a lower crash severity compared to when the traffic barrier is installed either higher or lower than the road surface. Almost similar estimates were found for the different side slopes, for cut (sslope1) and for filled (sslope2), with estimated c = 5.86 and OR c = 5.54, respectively. odds OR

4.1.1. Driver condition The predictor non-normal driver condition pertains to the emotional state of the driver with conditions such as anger, sadness, or agitation. Results indicated that drivers under non-normal conditions are an estic = 2.53) more likely to experience severe crashes mated 2.53 times (OR when hitting a traffic barrier crashes than under normal conditions. This result is expected as non-normal driver conditions can lead to driving error (Dingus et al., 2016; Rezapour & Ksaibati, 2018). 4.1.2. Roll-over This predictor was found to have the highest estimated impact on c = 12.16). The result is in accordance traffic barrier crash severity (OR with previous studies that having a rollover before hitting a traffic barrier would increase the risk of crash severity (Gabler & Gabauer, 2007). Although this driver action occurred before hitting a barrier, traffic barriers can stop vehicles experiencing a rollover from going further off the roadway leading to even a higher level of crash severity. 4.1.3. Improper restraint Drivers in traffic barrier crashes who did not use safety restraints are c = 4.60) more likely to be injured or killed an estimated 4.60 times (OR than drivers who do use safety restraints. This predictor has the second highest impact on the traffic barrier crash severity. The result is expected and has been well established (Mashhadi, Wulff, & Ksaibati, 2018b).

counterintuitive, previous studies showed that this could result from the fact that drivers drive more cautiously on non-dry road conditions (Mashhadi, Wulff, & Ksaibati, 2017b; Rezapour, Wulff, & Ksaibati, 2018b).

4.1.6. Citation record The predictor citation record refers to a driver who has a clean traffic record (0) versus one who does not (1). This result agrees with previous studies that there is a positive correlation between crash severity and citations (Rezapour, Wulff, & Ksaibati, 2018b). Other researchers have found that prior enforcement is associated with a decrease of crash

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injuries and fatalities (Kockelman & Kweon, 2002; Rezapour, Wulff, & Ksaibati, 2018a).

is investigated instead of crash frequency. Given that a traffic barrier crash occurred, a higher crash severity is linked to older drivers.

4.1.7. Vehicle type The results indicated that heavy trucks (N16,000 lb) are less likely to be involved in severe crashes compared with all other vehicles when hitting a traffic barrier. This might result from the fact that these types of vehicles have greater ground clearance and structural strength than other vehicles. It should be noted that, while other vehicles could face underride when hitting a traffic barrier, this would be unlikely for heavy trucks. The estimate also could be due to the fact that truck drivers tend to drive more carefully compared with other drivers, and they are able to manage the vehicle more professionally when hitting a traffic barrier.

4.2. Random parameters

4.1.8. Posted speed limit A cutting point of 55 mph was chosen for this variable as the mean mph on these roadways was 52 mph and there were only a few observations for the 50 mph category. The results indicated that lower speed limit is associated with higher severity of traffic barrier crash severity. Although the results sound counterintuitive, this finding might result from the fact that lower speed limit is associated with more challenging roadway geometry, which would increase the severity of traffic barrier crashes, especially for a mountainous area like Wyoming. Although this result disagrees with some studies that increased posted speed limit increases the odds of severe traffic barrier crashes (Zou et al., 2014), it is in accordance with other studies (Russo & Savolainen, 2018b). Also, this impact could be linked to a higher standard deviation of vehicle traffic speed (Aarts & Van Schagen, 2006). 4.1.9. Driver age Driver age was treated as a binary predictor that divided the age of drivers into two equal sized groups. The results indicated that older drivers (older than 35 years old) are more likely to be involved in severe traffic barrier crashes compared to younger drivers. Previous studies have indicated that older drivers do not increase crash frequency (Baldock & McLean, 2005). However, for this case study, crash severity

The fit of the mixed logit model revealed random parameters for gender, time of day, and truck traffic. For these predictors, it was necessary to account for heterogeneity across individual traffic barrier crashes. This means that there are unknown uncertainties involved in these predictors that vary across these crashes when it comes to modeling crash severity. Fig. 2 shows plots of the estimated normal distributions in Eq. (4) for the three random parameters. The estimates of the random parameters in Table 3 correspond to the mean of the normal distribution in (4). The standard deviations for the random parameters in Table 3 correspond to the standard deviation of the normal distribution in Eq. (4). (See Fig. 2.) 4.2.1. Gender The predictor gender is an indicator variable for female (1) versus male (0). The estimate for this random parameter is −0.123 and the estimated standard deviation is 1.356. The top plot in Fig. 2 shows the estimated normal distribution. For this distribution, 53.6% of the crashes involving a traffic barrier have parameter estimates for gender that are less than 0. This means that a slight majority of traffic barrier crashes are more severe when the driver is male. It could be speculated that this effect could be related to the fact that female drivers drive more cautiously with lower speed. 4.2.2. Time of a day The predictor time of a day indicates whether a crash occurs at peak hours (0) or off-peak hours (1). The estimated normal distribution has mean −0.443 and standard deviation 2.459. The middle plot in Fig. 2 shows the associated estimated normal distribution. For this distribution, 57.1% of parameter estimates are less than 0. This means that for 57% of traffic barrier crashes were estimated to be more severe during peak hours.

Traffic barrier geometric characteristics

Environmental characteristics

Geometric characteristics

Risk of severe crashes, given a traffic barrier crash already occurred

Driver characteristics

Crash characteristics

Fig. 2. Traffic barrier crash severity frame work.

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Fig. 2. Plots of the estimated normal distributions of the random parameters; gender (top), time of day (middle), truck traffic (bottom). The number in the plot represents the probability that the random parameter is less than 0.

4.2.3. Truck traffic Truck traffic is a continuous predictor that was found to be random. The bottom plot in Fig. 2 shows the estimated normal distribution, which has mean − 0.002 and standard deviation 0.003. Even though the estimates are small in magnitude, 74.1% of parameter estimates are less than 0. This means that 74% of traffic barrier crashes were estimated to decrease in crash severity for a unit increase in truck traffic. 4.3. Interaction effects The assessment of traffic barrier features on barrier crash severity was of special interest. These predictors did not enter the model as random parameters since the associated standard deviations could not be declared to be different from 0 at the 0.05 level. However, these predictors entered the model as fixed parameters by way of a 3-way interaction term (p-value = .047). Interpretation of the 3-way interaction term can be difficult, but scenarios of interest can be investigated using Eq. (6). Based on the three term interaction, at least 27 scenarios could be set forward. However, this section describes a few of the effects that would be of interest involving this interaction term. 4.3.1. Scenario 1: shoulder width and traffic barrier height constant at mean and offset increases by one unit For the first scenario, a comparison is made when shoulder width and traffic barrier height are constant at the mean value while the offset distance of traffic barrier is increased by one unit (in). For this scenario, the estimated odds of crash severity is exp.(− 0.0130) = 0.987 times higher with a 95% confidence interval of (0.972, 1.003). Even though the decrease in estimated odds is small, the units of measurement is

in inches. However, it should be noted that this interval does contain the value 1 indicating this particular effect is not important.

4.3.2. Scenario 2: offset and barrier height at their means, shoulder width increases by one unit For this scenario, a change is made on shoulder width by an increase of 1 unit (ft) while barrier offset and barrier height remain constant at their means. The estimated odds is 1.034 times higher with this increase of 1 unit with a 95% confidence interval of (0.980, 1.091). Even though the decrease in estimated odds is small, the units of measurement is in feet. However, this interval does contain the value 1 indicating this comparison at the mean values is not important. The result indicates that a change in shoulder width does not change crash severity at the 0.05 level when barrier offset and barrier height are constant at their means. It could be worth investigating other values of these two predictors to see if changes in shoulder width could impact the severity of crashes.

4.3.3. Scenario 3: offset and shoulder at the mean, traffic barrier increases by one unit For this scenario, a change is made on traffic barrier height by an increase of 1 unit (in) while offset and barrier height remain constant at their means. The estimated odds is 1.034 times higher with this increase of 1 unit with a 95% confidence interval of (0.985, 1.085). Even though the increase in estimated odds is small, the units of measurement is in inches. However, this interval does contain the value 1 indicating no effect due to this increase when comparing at the mean values.

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4.3.4. Scenario 4: one unit decrease in shoulder width mean, one unit decrease in barrier height mean, and one unit increase in offset mean The previous scenarios examine the three-way interaction of the traffic barrier features in terms of changes in one variable at a time. While those comparisons help in understanding increases or decreases in odds, all associated 95% confidence intervals contained the value 1, which indicates no change in the odds for these predictor combinations at the 0.05 level. However, other comparisons involving changes in these three predictors can highlight the importance of the three-way interaction term. For example, consider a one unit (in) decrease in shoulder width (in), a one unit (in) decrease in traffic barrier height, and an increase of one unit (in) in offset. The estimated odds is 0.920 times higher with these changes in the traffic barrier feature means with a 95% confidence interval of (0.859, 0.986). Thus, the probability of crash severity is reduced by these changes in the traffic barrier features by an estimated amount of 8%.

corresponding marginal densities in Fig. 2. This is confirmed by comparing the means of the conditional estimates in Table 5 to the marginal estimates of the random parameters in Table 3. However, the spread of the densities for the conditional mean in Fig. 3 are notably smaller than the spread of the corresponding marginal distributions in Fig. 2. This is confirmed by the smaller standard deviations of the conditional means in Table 5 compared to the marginal standard deviations in Table 3. The variation in the conditional means captures roughly 25% of the total variation in these random coefficients where the remaining variation could b (Train, 2009). be attributable to variability associated with β −β nik

nik

This means that there is some variation between the true value of the random coefficients and the estimated conditional means across crashes. This is particularly evident in gender and time of day. Specifically, note the number of very large conditional estimates for gender in Fig. 3(top). Thus, while the random coefficient estimates for each crash may be accurate on average, it could be necessary to improve the modeling of the variations in these random parameters.

4.4. Insights into interaction terms A few more points should be made about the interaction terms. Although barrier types were not found to be statistically significant, barrier type could play a role in crash severity when considered in light of the barrier height. The results suggest that lower barrier height, along with different shoulder and offset dimensions, resulted in a lower traffic barrier crash severity (Scenario 4). As can be seen from Table 4, cable barrier and box beam barriers tend to have lower barrier heights compared to W-beam barriers and concrete barriers. In light of the results in Scenario 4, it might be expected that cable barrier and box beam barriers would have lower odds of traffic barrier crash severity, while concrete might be expected to have the highest severity of traffic barrier crashes. Such a conjecture has support in the literature (Russo & Savolainen, 2018b; Zou et al., 2014). This interaction term also indicated that providing enough offset from the end of shoulder could result in reduction of crash severity. These findings should be examined in future empirical studies. 4.5. Conditional distributions Eq. (4) describes the assumed distribution for the unknown regression coefficient βnik for observation n, severity category i, and variable k. The normal distribution is used with mean βik and standard deviation σik. These terms are marginal across the observations since they do not depend upon n. Estimates of these quantities are reported in Table 3 for this study. As pointed out by Train (2009), the conditional distribution of βnik given the values of the response, predictors, and βik and σik can be used as a model diagnostic tool since the expected value of this estimated conditional distribution should approximately correspond to the estimated marginal distribution. Consider the conditional mean, denoted by βnik which is specific to observation n. The conditional means can be readily computed and plotted. Fig. 3 shows the plot of the estimated conditional means for the random parameters gender, time of day, and truck traffic. Table 5 gives the means and standard deviations of these conditional means for gender, time of day, and truck traffic. Fig. 3 can be compared to Fig. 2. The densities of the conditional means have the same location as the

Table 4 Different barrier types height in descending order (American Association of State Highway and Transportation Officials, 2011; Guide, 1996). Barrier type

Recommended heights for different barrier types (in)

Box beam barrier Cable barrier W-beam barriers Concrete barrier

27–29 30 29–33 42

5. Conclusions Traffic barriers are safety countermeasures that are generally installed at locations where crash severity can be reduced. Previous studies are generally consistent about the fact that road traffic barrier installation tends to lower crash severity compared to hitting a fixed object. However, it is not guaranteed that redirecting vehicles through traffic barriers would be done without injuries. Redirection and deceleration of vehicles while hitting traffic barriers depend on many factors such as the type and size of traffic barriers. The three most widely used types of traffic barriers in Wyoming include: W beam, box beam, and concrete barriers. Different standards, such as traffic barrier height, have been discussed in the literature review. However, few, if any, studies have simultaneously considered roadway type, environmental factors, and traffic barrier characteristics when analyzing crash data. The lack of such studies is likely due to the associated costs of collecting and maintaining traffic barriers inventory. A comprehensive data set was collected and made available to this study to incorporate traffic barrier characteristics, such as traffic barrier height, shoulder width, and offset, for two-lane highways in Wyoming from 2007 to 2017. Mixed logit models were used to model crash severity using this combination of predictors and fitting associated fixed and random parameters. Many main effect fixed parameters were identified. The odds of a traffic barrier crash severity were lower when driving on not dry road conditions and when driving a heavy truck. On the other hand, the odds of a traffic barrier severe crash increased for rollover crashes, being improperly buckled, cut side slope other than level, non-normal driver condition, and being an older driver. Three predictors were found to have random coefficients in which the parameter estimates varied across traffic barrier crashes. These predictors included gender, time of day, and truck traffic count. These predictors demonstrated heterogeneity across the crashes that were unaccounted for. Further study of these predictors should examine whether or not it is possible to explain aspects of these heterogeneities across the traffic barrier crashes. This may require the incorporation of more complicated combinations of predictors, such as interaction terms. The effects of traffic barrier characteristics on crash severity in the presence of these other predictors was of primary interest in this study. These effects entered the mixed logit model as a three-way interaction term involving shoulder width, guardrail height, and guardrail offset. This term meant that the effects of these traffic barrier characteristics on traffic barrier crash severity changed depending upon the values of the other predictors. In other words, the effects of these predictors should be combined. This fact makes sense from an engineering perspective and the result makes this study unique among those that have examined traffic barrier crashes. Various combinations of these

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Fig. 3. Plots of the estimated conditional expectations of the random parameters; gender (top), time of day (middle), truck traffic (bottom).

predictors were examined in interpreting the odds ratio. In particular, it was found that a decrease in shoulder width, a decrease in guardrail height, and an increase in guardrail offset led to a decrease in the odds of traffic barrier crash severity. The results of this study provide guidelines for Wyoming policy makers in optimizing the upgrades of traffic barriers for the highway system in the state. Further studies are recommended to confirm the findings of the interaction terms, but a clear recommendation can be made to examine these traffic barrier characteristics using interaction terms to account for the combined effects of these predictors on crash severity. Some other predictors were considered, but not included in the modeling. Traffic barrier types were not found to be important in the mixed logit model. This could indicate that there were not differences in traffic barrier crash severity by traffic barrier type considering included predictors. However, further examination is needed to confirm this finding. There was also limited data to study the impacts of different

Table 5 Means and standard deviations of the conditional means for the random parameters. Random parameters

Mean

Standard deviation

Gender (1 if the driver was female, 0 otherwise) Truck traffic (continuous) Time of a day (1 is it is off peak, 0 otherwise)

−0.124 −0.002 −0.444

0.315 0.001 0.674

vehicles on barrier crash severity. For future studies, incorporation of vehicle type is recommended.

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Rezapour, M., & Ksaibati, K. (2018). Application of multinomial and ordinal logistic regression to model injury severity of truck crashes, using violation and crash data. Journal of Modern Transportation, 26(4), 268–277. Rezapour, M., Moomen, M., & Ksaibati, K. (2019). Ordered logistic models of influencing factors on crash injury severity of single and multiple-vehicle downgrade crashes: A case study in Wyoming. Journal of Safety Research, 68, 107–118. Rezapour, M., Wulff, S. S., & Ksaibati, K. (2018a). Effectiveness of enforcement resources in the highway patrol in reducing fatality rates. IATSS Research. Rezapour, M., Wulff, S. S., & Ksaibati, K. (2018b). Predicting truck at-fault crashes using crash and traffic offence data. The Open Transportation Journal, 12(1). Ross, H. E. (2002). Evaluation of roadside features to accommodate vans, minivans, pickup trucks, and 4-wheel drive vehicles. Transportation Research Board.. Russo, B. J., & Savolainen, P. T. (2018a). A comparison of freeway median crash frequency, severity, and barrier strike outcomes by median barrier type. Accident Analysis & Prevention, 117, 216–224. Russo, B. J., & Savolainen, P. T. (2018b). A comparison of freeway median crash frequency, severity, and barrier strike outcomes by median barrier type. Accident Analysis & Prevention, 117, 216–224. Train, K. E. (2009). Discrete choice methods with simulation. Cambridge University Press. US Department of Transportation (2015). Traffic safety facts 2005: A compilation of motor vehicle crash data from the fatality analysis reporting system and the general estimates system. Wolford, D., & Sicking, D. (1997). Guardrail need: Embankments and culverts. Transportation Research Record: Journal of the Transportation Research Board, 1599, 48–56. Zou, Y., Tarko, A. P., Chen, E., & Romero, M. A. (2014). Effectiveness of cable barriers, guardrails, and concrete barrier walls in reducing the risk of injury. Accident Analysis & Prevention, 72, 55–65. Mahdi Rezapour has a few years of working experience as an engineer and project manager. In August 2015, he received his Master of Science in civil engineering with an emphasis on pavement engineering from the University of North Dakota. He then moved to Wyoming to pursue his PhD in civil engineering with an emphasis on traffic safety at the University of Wyoming. His current research is focused on truck safety and he is currently working for Wyoming Technology Transfer Centre. Shaun S. Wulff is a professor of Statistics at the University of Wyoming. He received his MS in Statistics from Montana State University and his PhD in Statistics from Oregon State University. His research interests include mixed models, experimental design, Bayesian statistics, and applications to engineering. Khaled Ksaibati obtained his BS degree from Wayne State University and his MS and PhD degrees from Purdue University. Dr. Ksaibati worked for the Indian Department of Transportation for a couple of years prior to coming to the University of Wyoming in 1990. He was promoted to an associate professor in 1997 and full professor in 2002. Dr. Ksaibati has been the director of the Wyoming Technology Transfer Center since 2003.