A Random Parameters Multinomial Logit Model Analysis of Median Barrier Crash Injury Severity on Wyoming Interstates

Abstract

This paper investigated factors influencing injury severity of crashes involving median traffic barriers, including the impact of barrier characteristics and their geometric features in Wyoming. Combining field data of inventoried median barriers with crash data on Wyoming interstates highways, a random parameters multinomial logit (mixed logit) model of injury severity was estimated. This methodological approach allowed for the possibility of estimated model parameters to vary randomly across crash observations to account for heterogeneity with respect to driver characteristics, roadway attributes, and vehicle characteristics. The estimation results indicated concrete barriers installed on front side-slopes and box beam barriers were associated with severe injury crashes. It was also found that median barrier crashes involving sports utility vehicles, pickups, and improperly restraint vehicle occupants are complex and vary significantly across observations. Other statistically significant variables found to increase the likelihood of severe injury crashes were rural interstate roads, concrete barriers installed on a front side-slope, box beam barriers with lateral offset less than 2 feet, and rollover crashes. These parameters were fixed across observations. The findings of this research point to the need to further investigate the impacts of sport utility vehicles, pickups, and rollover crashes on median barrier crash injury severity.

1. Introduction

It has been reported that in 2017 over 37,000 fatalities attributed to motor vehicle crashes were recorded on highways in the United States [1]. About 20 percent of the motor vehicle crash deaths in 2017 were attributed to collisions with fixed objects alongside the road [2]. Interstates in the United States are designed to the highest geometric standards but do account for a high proportion of traffic deaths compared to other highways. Over 8000 fatal crashes that occurred in 2016 were attributed to interstates and freeways [3]. Federal highway statistics indicate that one cross-median crash (CMC) fatality occurs in every 200 freeway miles, resulting in approximately 250 fatalities annually [4]. A study by Donnell et al. found that over 17% of CMC on Pennsylvania Interstate highways were fatal with 67% being injury crashes [5]. Median barriers have been recommended as a cost-effective treatment to mitigate the incidence of CMC events on divided highways [6,7,8]. Guidelines for the types and installation of median barriers are available in the American Association of State and Highway Transportation Officials (AASHTO) Roadside Design Guide [9].

The geometric dimensions of traffic barriers have been found to play a role in the types of barrier-related crashes that occur on highways. Evaluation of barriers has usually been conducted using simulation or statistical approaches. A large number of past traffic barrier assessments were conducted by using simulation tools [10,11,12,13,14]. Julin et al. evaluated the performance of the midwest guardrail systems (MGS) in different height ranges [10]. Their study recommended that barrier heights of 36 inches should be considered as maximum for the MGS [10]. Barrier heights taller than 36 inches created a risk of underride crashes for vehicles hitting the MGS. Another study conducted a series of LS-DYNA simulation models to investigate the safety performance of a pickup truck impacting the 22 in and the 25 in W-beam barrier in terms of override potential [15]. It was found that both the W-beam barriers could create an override at a speed of 62 mph.

Statistical analysis of historical crashes has been another approach used in the past to evaluate traffic barriers. A majority of past statistical studies considered before–after evaluations to identify the performance of new traffic barrier segments in comparison to before period [16,17,18,19]. Villwock et al. studied the effect of installing new median cable barriers and found that new cable barriers increased the frequency of median crashes [19]. However, the severity of these crashes was identified to be less than before installing the new cable barriers. A study conducted by Chimba et al. estimated that a reduction of 82% and 76% in fatal and incapacitating injury crashes, respectively, were observed after the installation of new median cable barriers [18]. Some studies also considered statewide crash datasets for evaluating all traffic barrier crashes reported in a region (state) in a specific period [20,21,22,23,24]. Molan et al. studied traffic barrier crashes involving trucks in Wyoming between 2007 and 2016, and investigated whether semi-rigid barriers could increase the possibility of severe truck crashes [21]. Approximately, half of the truck crashes hitting semi-rigid barriers involved a truck rollover. Russo and Savolainen collected crashes hitting median traffic barriers in Michigan between 2009 and 2013 [23]. It was found that median cable barriers were associated with the least risk of fatal or severe injury crash.

Different statistical approaches have been used to analyze crash data based on the data type. Several analyses involving count data used Poisson and negative binomial regression models to analyze crash data [25,26]. However, Poisson regression models are unable to handle over- and under-dispersed data and can be affected by low sample-means [27]. Negative binomial models are also limited by their inability to handle under-dispersed data and difficulties in estimating the dispersion parameter in the presence of low sample-mean values and sample sizes that are small [28]. Other researchers have used zero-inflated (Poisson and negative binomial) [29,30] models, gamma models [31], and Conway–Maxwell–Poisson [32] models among others.

Discrete choice models that allow for the prediction of a choice from a set of alternatives have also been utilized extensively in crash severity modeling [33,34,35]. Such modeling approaches include binary probit, multinomial logit [36], nested logit [37], and ordered probit/logit models [38]. Some researchers have accounted for unobserved heterogeneity by allowing coefficient estimates to vary across observations by utilizing random parameters for both count and discrete choice modeling [33,34]. Over the last couple of decades, researchers have also used machine learning approaches in analyzing crash data such as support vector machines [39], decision trees [40], and neural networks [41].

As summarized above, past studies presented valuable results related to the safety performance of traffic barriers. Apart from adding to the current literature on injury-severity outcomes attributed to roadside objects, this study also aims to add new insights by evaluating the impact of median traffic barrier geometric variables on crash injury severity. This is a gap in past statewide statistical studies as they did not include the effect of the geometric variables of traffic barriers in their analysis. For this purpose, dimensional variables such as post-spacing, side-slope, height, post-spacing, and lateral offset of median traffic barriers were collected during a field survey undertaken in Wyoming. The data collected was then combined with the crash database from the CARE package to provide a comprehensive dataset, which was utilized for analysis.

2. Materials and Methods

This paper uses the random parameter multinomial logit (mixed logit) model to evaluate the effects of median traffic barrier variables on crash injury severities on interstate highways in Wyoming. Accident severity analysis is usually conducted by considering the number of crashes that have occurred on an entity over a specified period. The injury severities of the recorded crashes are then modeled as discrete outcomes (for example, no injury, possible injury, serious injury, fatality) [34]. However, some of the factors influencing the probability of a crash may not have been recorded and are thus unavailable for analysis. These factors, also known as unobserved or latent heterogeneity, can introduce bias in the estimated impact of the observed variables effect on crash injury severity. Such unobserved heterogeneity may include factors such as physique of driver, variations in driver response, vehicle kinematics during crashes, vehicle model among other unobserved factors [33,34]. With respect to traffic barriers, sources of heterogeneity may include barrier type (solid, semi-solid), angle and speed of impact, effects of median width, and conditions at time of crash [23,34].

Considering the above discussion, it is important to utilize an approach that permits the variables that affect traffic barrier injury severity to vary across observations. Accounting for unobserved heterogeneity in road safety has been undertaken in several studies through the use of latent-class (finite mixture) models [42,43], Markov switching count models [44,45], random-parameters models [23,46,47], among others. The random-parameter models developed in previous studies allowed parameters to vary randomly across observations based on a distribution, thereby accounting for the individual heterogeneity of factors impacting crash injury severity. The mixed logit is formulated by specifying a function that determines driver-injury severity. This is expressed in Equation (1) as [33]:

$$S_{in}={\beta }_{in}{X}_{in}+{\epsilon }_{in}$$

(1)

where $S_{in}$ is a function of severity determining the injury-severity category i (no injury, minor, and serious injury) for crash barrier n; $\beta$ is a vector of estimable parameters; X_in is a vector of independent variables (e.g., driver, geometric, road characteristics); and ${\epsilon }_{in}$ is an error term.

By allowing crash-specific unobserved heterogeneity, $\beta$ can have a continuous density function such that prob ($\beta$) = f(β|$\theta$), where $\theta$ refers to a vector of parameters of the density function. The resulting random multinomial logit injury-severity probabilities are expressed in Equation (2) as [34,48]:

$$P_n\left(i\right)=\frac{EXP({\beta }_{in}{X}_{in})}{{{\displaystyle \sum }}_i^{}EXP({\beta }_{in}{X}_{in})}f(\beta |\theta )\mathrm{d}\beta$$

(2)

where $P_n\left(i\right)$ defines the outcome probability of traffic barrier crash n, resulting in injury severity i. A simulated maximum likelihood approach is used to estimate the model. Revelt and Train explained the simulated likelihood estimation process of multinomial logit regression models by assuming a set of choices presented to a group of individuals [49]. Maximum likelihood estimation of the choice model of the individuals utilizes the probability of each individual’s sampled sequence of choice model observed. Assuming that i(n,t) represents the alternative that individual n chose within period t, the probability of individual n’s observed choice sequence may be estimated as the product of the standard logits, which is conditional to all possible values of ${\beta }_n$ [49]. An unconditional probability for the sequence of choices is defined as [50]:

$$P_n\left(\theta \right)={\displaystyle \int }{S}_n\left({\beta }_n\right)f\left({\beta }_n|\theta \right)d{\beta }_n$$

(3)

where ${\beta }_n$ in this equation is a parameter representing person n’s tastes. This approach allows for an estimation of $\theta$. The log-likelihood function is estimated as [51]:

$$LL\left(\theta \right)={\displaystyle \sum }_n\ln P_n\left(\theta \right)$$

(4)

Maximum likelihood is estimated through simulation because it is not possible to integrate and calculate Equation (4) analytically [50]. The probability is approximated through simulation and maximized by using the simulated maximum log-likelihood function. The probability function $P_n\left(\theta \right)$ is estimated by summing the chosen random values of ${\beta }_n$. For a given parameter value of $\theta ,$ a value of ${\beta }_n$ is drawn from the distribution. The product of standard logits are then estimated from the sequence of draws. This process is undertaken several times with the average of the resulting $S_n\left({\beta }_n\right)$’s calculated as the choice probability [50].

$$S{P}_n\left(\theta \right)=\left(\frac{1}{R}\right){{\displaystyle \sum }}_{r=1,\dots ,R}{S}_n\left({\beta }_n^{r|\theta }\right)$$

(5)

where R represents the number of repetitions, ${\beta }_n^{r|\theta }$ represents the r-th draw from $f\left({\beta }_n|\theta \right)$, and $S{P}_n\left(\theta \right)$ is the simulated probability of a person n’s choices. The estimated parameters are those that lead to a maximization of the simulated log-likelihood function defined as $SLL\left(\theta \right)={\displaystyle \sum }_n\ln (S{P}_n\left(\theta \right))$.

Halton draws are frequently used to estimate logit probabilities based on its ability of providing a more efficient distribution of draws used for numerical integration in comparison to pure random draws [52,53]. A 1000 Halton draws was used to estimate the mixed logit model for this study as recommended by Hensher et al. [53].

Elasticity was computed as an estimate of the sensitivity of the dependent variable Y when there is a change in the independent variable X. Elasticity was estimated as [51]:

$$e_i={\beta }_i\frac{X_i}{Y_I} \approx \frac{\partial Y_i}{\partial X_i}\times \frac{X_i}{Y_i}$$

(6)

To evaluate the performance of the model, F-measure and G-mean were calculated. F-measure is a weighted average of precision and recall [54]. It is defined as:

$$F=2\times \frac{Recall\times Precision}{Recall+Precision}$$

(7)

The F-measure provides a balance between precision and sensitivity and is used to assess the model’s accuracy. The F-measure ranges from 0 to 1, with higher values representing higher model performance.

G-mean provides a measure of the balance between classification of the performance of the majority and minority classes. The G-mean is used to assess the accuracy of predictions from unbalanced data with low G-mean values indicating a poor performance in classification. It is defined as:

$$G\text{-}Mean=\sqrt{Sensitivity\times Specificity}$$

(8)

The performance metrics were estimated using a k-fold out-of-sample cross validation method. This method utilized all the observations in the dataset to undertake a robust evaluation of the model with k set to be five. The data was randomly split into five subsets (4 × 375 + 1 × 380) with the model estimated with four of the five subsets (training data) and the performance metrics estimated with the remaining subset (testing data). The estimation was undertaken five times with each of the five subsets taken out and used as testing data. Again, a 1000 Halton draws was used to estimate the model. To simplify the evaluation process, the means of the random parameters were used in estimating the outcome probabilities. The highest predicted probability using the trained model that corresponded to an injury outcome was assigned to the severity level. For instance, if the highest predicted probability from the testing data using the training model corresponded with the no-injury category, then the predicted class was assigned to the no-injury outcome. The performance metrics were estimated by comparing these predicted injury outcomes to the actual injury categories from the test data subsets. These metrics were then weighted based on the number of observations per injury category from the test data and averaged to obtain the final evaluated performance of the model.

3. Data

A field data inventory was undertaken in the summer of 2016 through the summer of 2018 to collect traffic barrier data from approximately 600,000 linear feet (over 110 miles) of traffic barriers across interstate highways in Wyoming. The inventory indicated that 204 miles of median barriers have been installed on a total of 912 miles of interstate roads in Wyoming. The length of median barriers inventoried on interstate highways were 53.1 and 27.6 of box beam and W-beam barriers, respectively. For concrete and cable systems, 16.7, and 16.3 were inventoried, respectively. Figure 1 shows pictures of the traffic barrier types inventoried for this study.

The data from the traffic barriers was combined with the crash database from the Critical Analysis Reporting Environment (CARE) package. The CARE package contains a comprehensive database of historical crashes. It considers over 160 different variables associated with driver, environmental, and crash characteristics. A total of 1880 crashes involved median barriers while side barriers accounted for 1634 crashes between 2008 and 2017 in Wyoming. Crashes included in the analysis were because of direct impact with traffic barriers.

A limited information on cable barrier systems (24 out of the 200 segments) was collected statewide during the inventory since this barrier type is relatively new with its design based on new stands and policies. Again, this barrier type has been identified in previous studies as having the least injury risk [17,23,24]. The other traffic barrier types were therefore considered as priorities with needs for improvements in the near-term by stakeholders in Wyoming. In a few road segments where barrier geometric data was not recorded, the Pathweb web-based application [55], Google Earth [56], and AutoCAD [57] were used to augment the data collection process.

Table 1. Summary of crash characteristics by crash severity.

Crash Variables	High-Severity	Moderate-Severity	Low-Severity
Crashes Occurred in Summer	29	110	429
Crashes Occurred in Winter	27	199	1116
Crashes Occurred with Alcohol Involved	9	28	53
Crashes Occurred in Daylight	34	108	945
Crashes Occurred in Darkness Unlighted	22	201	600
Crashes Occurred on Dry Surface	36	109	367
Crashes Occurred on Non-Dry Surface	20	200	1178
Crashes Occurred with Passenger Cars Involved	25	126	600
Crashes Occurred with Heavy Vehicles Involved	26	181	945
Crashes Occurred with Motorcycles Involved	5	2	0
Female Driver	13	158	505
Male Driver	43	151	1040
Resident (WY) Driver	36	197	840
Non-Resident Driver	20	112	705
Same Direction of Force (Crash) within 15 degrees	55	243	1238
Non-Direction of Force (Crash) within 15 degrees	1	66	307

is a summary of the crash characteristics by severity, while

Table 2. Summary of features collected during field survey for this study.

Traffic Barrier Geometric Features		Box Beam Barriers	W-Beam Barriers	Concrete Barriers	Cable Barriers
Length (mile)	Mean	0.14	0.68	0.41	0.58
	Standard Deviation	0.17	0.53	0.67	0.67
	Maximum	0.93	2.20	1.94	1.92
	Minimum	0.01	0.01	0.01	0.11
Lateral Offset (ft)	Mean	4.00	4.80	0.60	8.80
	Standard Deviation	5.60	5.00	2.10	1.70
	Maximum	23.50	26.00	12.70	20.70
	Minimum	0.00	0.00	0.00	7.30
System Height (in)	Mean	29.40	29.20	35.10	29.90
	Standard Deviation	2.70	3.10	5.60	2.80
	Maximum	37.20	37.20	55.20	42.00
	Minimum	18.00	16.80	20.40	25.20
Post-Spacing (ft)	Mean	6.00	6.40	-	16.20
	Standard Deviation	3.60	1.30	-	0.20
	Maximum	22.10	12.60	-	16.80
	Minimum	1.20	2.50	-	16.00
Front-Slope Ratio (V:H)	Mean	0.05	0.05	0.05	0.05
	Maximum	0.00	0.04	0.05	0.05
	Minimum	0.06	0.06	0.06	0.08
Back-Slope Ratio (V:H)	Mean	0.05	0.05	0.05	-
	Maximum	0.04	0.05	0.05	-
	Minimum	0.06	0.05	0.06	-
Speed Limit (mph)	Mean	72.00	73.00	66.00	72.00
	Standard Deviation	5.50	6.40	6.60	6.40
	Maximum	75.00	75.00	75.00	75.00
	Minimum	50.00	50.00	50.00	50.00
AADT (vehicles/day)	Mean	6303.00	6002.00	8135.00	4465.00
	Standard Deviation	2567.00	1676.00	1913.00	950.00
	Maximum	12,903.00	12,903.00	10,995.00	7208.00
	Minimum	749.00	1624.00	3388.00	3432.00
AADTT (trucks/day)	Mean	2037.00	2488.00	1892.00	898.00
	Standard Deviation	1110.00	803.00	860.00	133.00
	Maximum	3689.00	3413.00	3689.00	1690.00
	Minimum	131.00	284.00	235.00	590.00

This Table was published in Journal of Safety Research; Volume 71; Investigating the Effect of Geometric Dimensions of Median Traffic Barriers on Crashes: Crash Analysis of Interstate Roads in Wyoming using Actual Crash Datasets; Molan, M., Moomen, M., and Ksaibati, K.; pages 163–171; Copyright Elsevier (2019) [58].

shows geometric features collected during the field survey. Figure 2 is a graphical representation of the cross-section for some of the variables mentioned in Table 1. The crash, traffic, and geometric datasets were combined to form a comprehensive database on which the analysis was undertaken. Several of the variables from the combined dataset served as input parameters based on previous safety barrier studies from the literature and publications by the authors [20,21,22]. Other variables not explored in previous studies were also included in the input dataset to gain additional insights on their impacts on traffic barrier safety.

Injury-severity categories were analyzed using the KABCO injury scale, which comprises five injury categories as a reference [59]. The severity categories are fatal injury, incapacitating injury, non-incapacitating injury, possible injury, and no injury (property damage only). However, given that there was a limited number of fatal and incapacitating injury crashes in comparison with non-injury crash category, it was not statistically feasible to estimate all five injury-severity categories. As a result, three injury categories resulted from the grouping; no injury, minor injury (non-incapacitating injury and possible injury), and severe injury (fatal and incapacitating injuries).

Table 3. Summary of target crash data collected.

Variable		Box Beam Barriers		W-Beam Barriers		Concrete Barriers		Cable Barriers		Overall
		No.	%	No.	%	No.	%	No.	%	No.	%
Crash Severity	Severe injury	31	3	14	4	10	3	2	2	57	3
	Minor injury	152	15	46	12	95	26	9	10	302	16
	No injury	863	82	315	84	265	71	77	88	1520	81
Surface Condition	Dry	296	28	118	31	91	25	20	23	525	28
	Wet	120	12	45	13	59	16	14	16	238	13
	Snowy/Icy	630	60	212	56	220	59	54	61	1116	59
Vehicle Type	Passenger Car	405	38	135	36	153	41	34	39	727	39
	SUV and Van	196	19	59	16	105	28	18	20	378	20
	Pickup	350	33	124	33	87	24	28	32	589	31
	Heavy Vehicle (>10,000 lbs)	89	9	54	14	23	6	8	9	174	9
	Motorcycle	6	1	3	1	2	1	0	0	11	1
Rollover Involved	Yes	65	6	33	9	20	5	4	5	122	6
	No	981	94	342	91	350	95	84	95	1757	94
Total		1046	100	375	100	370	100	88	100	1879	100

[This Table was published in Journal of Safety Research; Volume 71; Investigating the Effect of Geometric Dimensions of Median Traffic Barriers on Crashes: Crash Analysis of Interstate Roads in Wyoming using Actual Crash Datasets; Molan, M., Moomen, M., and Ksaibati, K.; pages 163–171; Copyright Elsevier (2019) [58]].

shows a summary of the target crash statistics for this study.

Due to the presence of three possible injury-severity outcomes in the model, it is important to note that the model cannot be directly interpreted in terms of the sign of the estimated coefficients. Also, the addition of random parameters complicates interpreting the effect of the variable directly in terms of only the estimated parameters. This implies that a positive parameter may in fact be associated with a reduction in probability [60]. Due to the possible ambiguity of the direction of variable impact on crash severity, it is useful to consider the marginal effects of the variables on the probabilities of the injury severity. In this study, all variables considered for estimating the models are 0 and 1 indicator variables. For indicator variables, marginal effects are the effect of the independent variables as they change from zero to one in terms of the injury-severity outcome probabilities [51].

4. Results and Discussion

The NLOGIT 6.0 software was used in estimating the mixed logit model.

Table 4. Mixed logit model results of injury severity of median barrier crashes (distribution estimates for random parameters in italics).

Variable	Parameter Estimate	t-Stat	Marginal Effects
			No Injury	Minor Injury	Severe Injury
Constant [NI]	−2.978	−11.27
Median Barrier Characteristics
Concrete barrier on front side-slope indicator (1 if concrete barrier is located on a front slope; 0 otherwise) [SI]	0.5027	2.86	−0.0014	−0.01	0.0114
Box beam barriers with lateral offset less than 2 ft indicator (1 if box beam barrier has less than 2 ft offset; 0 otherwise) [SI]	0.458	2.96	−0.0022	−0.0122	0.0144
Roadway Attributes
Rural interstate indicator (1 if crash occurred on a rural interstate; 0 otherwise) [SI]	1.46	11.85	−0.0162	−0.0808	0.0969
Downgrade indicator (1 if crash occurred on a downgrade; 0 otherwise) [N1]	1.137	7.16	0.0042	−0.0009	−0.0032
Vehicle Characteristics
Sports utility vehicle indicator (1 if vehicle is a sports utility vehicle; 0 otherwise) [MI]	−0.894	−2.02	0.0008	−0.0015	0.0007
Standard deviation of “sports utility vehicle” [MI]	1.588	2.24
Pickup vehicle indicator (1 if vehicle is a pickup; 0 otherwise) [MI]	−1.736	−3.46	0.0023	−0.0141	0.1183
Standard deviation of “pickup vehicle” [MI]	1.63	2.53
Heavy vehicle indicator (1 if vehicle is a heavy vehicle; 0 otherwise) [MI]	−1.424	−4.32	0.0012	−0.0094	−0.0082
Crash Characteristics
Rollover crash indicator (1 if crash is a rollover; 0 otherwise) [MI]	2.848	8.37	−0.0084	0.031	−0.0226
Rollover crash indicator (1 if crash is a rollover; 0 otherwise) [NI]	2.591	7.16	0.0162	−0.0076	0.0086
Other Variables
Dry road surface indicator (1 if crash occurred on a dry road surface; 0 otherwise) [NI]	1.426	4.75	0.0223	−0.0065	−0.0158
Snowy weather indicator (1 if crash occurred during snowy weather; 0 otherwise) [SI]	1.037	6.10	−0.0022	−0.0288	0.0316
Improper restraint indicator (1 if driver was improperly restrained; 0 otherwise) [MI]	1.107	3.02	−0.0096	0.0146	−0.0136
Standard deviation of “improperly restrained” [MI]	2.001	1.89

shows the estimation results of the mixed logit model for median traffic barriers, while

Table 5. Model statistics.

Number of observations	1880
Restricted Log-likelihood	−2065.39
Log-likelihood at convergence-fixed parameters	−965.24
Log-likelihood at convergence-random parameters	−960.38
McFadden Pseudo R-Square	0.53
F-Measure	0.72
G-Mean	0.58

shows the model statistics. All parameters included in the model were tested and found statistically significant at a 0.1 level of significance or higher. Overall, sixteen parameters were found to be significant. Three parameters were found to be random at a 0.1 level or higher. These were the indicator variables for sports utility vehicle, pickup vehicle, and improper restraint. Different distributions were analyzed for the random parameters but the normal distribution was found to produce the best statistical fit for all the random parameters.

It is important to note that multinomial logit models are estimated based on the consideration that variables defining the choice probability functions are classified into two groups [51]. These are variables that differ across alternatives and those that do not. Due to this, estimated parameters related to variables that do not vary across outcome alternatives are estimated in i − 1 terms (i is the number of severity categories). Constant terms are considered variables that do not vary across the severity categories and were estimated for two out of the three severity categories. In other words, the category whose constant has been set to zero is considered as the base or reference category. The choice of a constant term to set as the reference category is arbitrary [51]. For this analysis, the medium-injury category was set as the reference category.

4.1. Median Barrier Characteristics

Turning to the specific results, median concrete barriers on front side-slopes had a higher probability to result in fatal and severe injury crashes. The average marginal effects show that the probability of severe injuries increase by 0.0114 with minor and no-injuries decreasing by 0.0100 and 0.0014 for concrete median barriers installed on front side-slopes This is an expected finding as rigid (concrete) barriers should not be installed on side-slopes steeper than 1:10 (vertical: horizontal), based on general design policies [9]. Another reason for this finding is that past studies identified that the face-slope is one of the most important design elements in concrete barriers and could cause rollovers from vehicles hitting concrete barriers [9,61]. Therefore, the higher risk of severe crashes may be associated with the fact that median concrete barriers on front side-slopes had an inappropriate face-slope (not based on design policies).

Median box beam barriers with a lateral offset less than 2 ft tended to experience a higher probability of severe injury crashes. This may be because of the shorter perception–reaction time for drivers in crashes involving median box beam barriers with short lateral offsets. In other words, drivers might be unable to reduce speed and change the collision angle before the crash compared to when a wider lateral offset is available on roadsides.

4.2. Roadway Attributes

The rural interstate indicator variable was found to be a significant fixed parameter in the severe injury function. Table 4 shows that the probability of severe injury increases by 0.0969, while minor and no-injury severities decrease by −0.0808 and −0.0162, respectively, for median barrier crashes. This finding is consistent with previous studies that have found that severe injury crashes are higher on rural interstates such as those in Wyoming due to higher speed limits on these sections [62,63].

Traffic barrier crashes that occur on downgrades were found to have a lower probability of resulting in severe injury crashes compared to level sections. According to the estimated average marginal effects in Table 4, the predicted probability specific to severe and minor injury crashes decrease by 0.0032 and 0.0009, respectively, while the predicted probability for no-injury crashes increases by 0.0042. While these results may look counterintuitive at first, an explanation in the context of Wyoming interstate highways justifies the findings. Truck traffic on the states interstates is high because of mining and drilling activities. On downgrades, heavy trucks and recreational vehicles usually travel at reduced speeds leading to platooning [64]. This leads to a reduced travel speed and lower injury severities.

4.3. Vehicle Characteristics

The sports utility vehicle indicator variable in the minor injury function was normally distributed with mean −0.984 and standard deviation 1.875. With these estimates, 28.67% of this distribution is above zero and 71.33% is below zero. This implies that compared with other vehicle types, almost 30% of traffic barrier crashes involving sports utility vehicle will increase the probability of minor injury crashes and decreases the probability of minor injuries for about 70% of median barrier crashes. The average marginal effects indicate an increase in the probability of no-injury and severe injury crashes by 0.0008 and 0.0007, respectively, while probability of minor injuries decreases by 0.0015. The effects of sports utility vehicles on crash injury severity in the literature has been mixed. Khattak and Rocha suggest that sport utility vehicles have a high propensity to rollover [65]. Compared to passenger vehicles, sports utility vehicles have eight times the risk of rollovers in barrier crashes [66]. Median barrier crashes involving rollovers are associated with higher injury severities [67]. However, sports utility vehicles also provide more protection to its occupants compared to passenger vehicles. On the other hand, Ulfarsson and Mannering found that rollover crashes increase the probability of minor injuries for females drivers in sports utility vehicles while increasing injury severity for male drivers [68]. These findings suggest there is some unobserved heterogeneity with regard to the impact of sport utility vehicles on crash injury severity. The random parameter for this variable may account for some of the unobserved heterogeneity.

The parameter for pickup vehicles defined in the minor injury function resulted in a parameter that is normally distributed with a mean of −1.736 and standard deviation 1.630 (Table 4). This shows that 14.34% of the distribution is above zero and 85.66% is below zero. The implication is that most drivers in pickups involved in traffic barrier crashes (84.36%) are less likely to sustain minor injuries, while 15.64% of drivers in pickups in traffic barrier crashes have a high probability of being involved in minor injury crashes. The average marginal effect shows that the risk of severe injury crashes increases by 0.1183 and minor injury decreases by 0.0414 for pickup vehicles. Similar to sport utility vehicles, pickups are predisposed to rollover crashes compared to passenger vehicles [66]. This is likely accounting for the increased probability of severe injury crashes for pickup vehicles involved in barrier crashes.

Traffic barrier crashes involving heavy vehicles decrease the likelihood of minor and severe injury crashes, which is intuitive. This finding may result from the added protection afforded to occupants of heavy vehicles compared to other vehicle classes. The average marginal effects (Table 4) shows that the probabilities of minor and severe injury crashes decrease by 0.0094 and 0.0082 for heavy vehicle-traffic barrier collisions. This result is similar to findings for side barriers in a previous study [69].

Powered-two-wheeler (PTW) crashes (including motorcycles) into median barriers is a safety issue of major concern to stakeholders in highway safety. Gabler pointed out that, in the U.S., motorcycles account for over 42% of all fatalities as a result of collisions with safety barriers and roadside objects though they represent only 2% of the vehicle fleet [70]. Savolainen and Russo found that motorcycle crashes into median barriers resulted in severe injury outcomes in comparison to other vehicle types [23]. This finding was attributed to the lack of protection associated with motorcyclists. This was supported by another study which went on to suggest that the severe crash outcomes observed in median barrier motorcycle crashes can be attributed to the speeding behavior of motorcyclists [71]. Though other studies have found motorcycles to be associated with higher barrier crash severity, the motorcycle variable was not statistically significant for this analysis. This may be attributed to the relatively lower frequency of motorcycle crashes into median barriers observed for this study. More data may have to be collected on other highway functional classes that have median barriers installed to understand the actual impact median barriers on motorcycle crashes.

4.4. Crash Characteristics

The rollover crash indicator was a significant fixed parameter in the minor and no-injury outcomes. Rollover crashes were found to increase crash severity. The average marginal effects suggest rollover crashes are associated with increased probabilities of minor and severe injuries in comparison to other crash types (Table 4). This finding is consistent with other studies that found that traffic barrier crashes that result in rollovers result in higher injury severities [67,72]. Rollover crashes are complex and mostly result in severe injuries because occupants move independently of the vehicle during the rollovers [73]. This may lead to impacts with the inside of the vehicle or ejections in worse scenarios.

4.5. Other Variables

Three variables were found significant for this category. These were dry road surface, snowy weather, and improper restraint indicator variables. The dry road surface indicator variable was found to be significant in the no-injury-severity function. The average marginal effects suggest that the probability of no-injury crashes increase by 0.0233 and decrease by 0.0065 and 0.0158 for minor and severe injury crashes, respectively (Table 4). This is intuitive as vehicles are much more under control and less likely to skid on dry road conditions compared to other road surface conditions. A similar result was found by other studies for median barrier crashes [58,67].

Traffic barrier crashes in snowy weather were found to be significant in the severe injury category. Table 4 shows that the probability of no-injury and minor injury crashes decrease by 0.0022 and 0.0288, respectively, while the probability of sustaining severe injuries increases by 0.0316. Snowfall may increase the probability of injury crashes due to the risks of vehicles going off the road. However, the effect of snowy weather on crash severity in the literature is still an ongoing discussion [74]. Previous research suggests that severe injury crashes increase in the first few months of snowy weather but decrease subsequently [75]. Other research has found that snowy weather decreases crash injury severity [76,77,78]. Further research is needed to assess the impact of snowy weather on traffic barrier injury severity.

The effect of improper restraint was found to be significant in the minor injury category. This parameter was normally distributed with a mean of 1.107 and a standard deviation of 2.001. With the estimated parameters, 29.01% of the distribution is less than zero and 70.99% of the distribution is greater than zero. This implies that about 30% of improperly restrained vehicle occupants involved in traffic barrier crashes decrease the probability of minor crash injuries and approximately 70% of vehicle occupants increase the probability of minor injuries. The average marginal effects indicate a decrease in probability of 0.0096 and 0.0136 for no-injury and severe injury crashes, and a probability increase of 0.0146 for minor injury crashes. This result is generally intuitive but the decrease in probability for fatal injuries attributable to improper restraint requires a further look. In this regard, it is important to note that the impact of safety restraints on injury severity may be moderated by several unobserved variables. As an example, the effectiveness of a safety restraint in preventing injury severity may be affected by physical frame, precise sitting posture of individual, or driving style [47]. Most of these factors are not usually available in crash data. Using a random parameter captures the unobserved heterogeneity associated with this variable.

5. Conclusions

Median barriers are installed to prevent or reduce the impact of potential severe cross-median crashes. Previous analyses of traffic barrier-related injury crashes have generally not taken into consideration the geometric characteristics of the barriers. This study explored the heterogeneous impact of factors including roadway attributes, vehicle characteristics, crash characteristics, and most importantly, median barrier characteristics on injury crashes in Wyoming. A field survey was undertaken to record barrier characteristics and geometric variables (including height, post-spacing, and lateral) of over 110 miles of median traffic barriers. The mixed logit model was utilized to account for unobserved factors that are conventionally not present in a crash database. The impacts of some of the variables on driver-injury severity in the model were found to vary across observations. The results showed that three variables have a normally distributed random distribution. These were the sports utility vehicle indicator, pickup vehicle indicator, and improper restraint indicator variables. Some important findings with respect to the random parameters were that sport utility and pickup vehicles reduce the probability of minor injury crashes but are associated with higher probabilities of severe injury crashes. This was attributed to the higher likelihood of rollovers associated with sport utility vehicles and pickups. The improper restraint indicator variable was found to vary across observations and increased the probability of minor injury crashes in median traffic barrier crashes by approximately 70%.

With respect to median barrier characteristics, the concrete barrier located on front side-slope and box beam barriers with offset less than 2 ft indicator variables were found to be fixed statistically significant indicator variables in the severe injury category. Both variables were found to be associated with a high probability of severe injury crashes. The average marginal effects indicated concrete barrier located on front side-slope and box beam barriers with offset less than 2 ft indicator variables increased the likelihood of severe injury crashes by 0.0114 and 0.0144, respectively.

Other variables found to be statistically significant in the model were rural interstate, downgrade, rollover, dry road surface, heavy vehicle, and snowy weather indicator variables. These variables were found to be fixed across observations. Some important findings from these variables were that rural interstates are associated with severe injury crashes while downgrades were found to decrease the probability of minor and severe injury crashes for median barrier crashes. Similar to other studies, heavy vehicles were associated with a decrease in injury severity while rollover crashes were found to increase severe injuries.

The results of this study indicated that the effect of sports utility vehicles, pickups, and improper restraints on injury severity varies across observations. Future studies should further analyze the heterogeneous effects of these variables (in means and variances) on median barrier injury severity to reveal further insights.

From the results obtained, there is a need to increase safety campaigns on the use of restraints and seatbelts on highways [79]. These campaigns should also be designed to encourage drivers to exercise caution when driving in inclement weather including snowstorm events, which are common in Wyoming. Additionally, signs should be installed on mountainous highways warning drivers of the need to reduce their speeds on steep downgrades to reduce crash risks.

The results from this study may be used as a critical reference to policy makers to understand the factors impacting barrier safety in Wyoming and to develop appropriate countermeasures and strategies to mitigate such barrier crashes.

This study is limited by the lack of data related to barrier installation method, working width, angle of vehicle impact, dynamic deflection, vehicle intrusion, and speed at impact. This information should be collected and analyzed for future studies. Also, future studies should perform crash tests on median barriers based on barrier type. This will provide information on factors such as barrier deflection, extent of vehicle intrusion and deflection, among other relevant information for the use of highway practitioners and policy makers.