Traffic barriers have been identified as a measure to mitigate the severity of run of the road (ROR) crashes. However, while roadside crashes account for only 16% of all crashes in the U.S., a substantial proportion of those crashes are high-severity crashes [1]. In order to mitigate the severity of those crashes, extensive efforts have been made to identify contributory factors to those crashes in accurate ways.

One of the main shortcomings of the standard mixed model is that the method allocates a single random effect estimate for all barriers crash populations. While modeling crash datasets, accounting for that shortcoming is especially important due to a high amount of heterogeneity in the crashes due to much seen and unseen variations across observations.

Statistical methods have been undertaken to model factors to crashes while accounting for the data heterogeneity. For instance, mixed models have often been used to describe the behavior of individual observations [2].

While a driver characteristic such as gender might be random, that impact might be varied across observation based on some other observed covariates. For instance, female drivers might be more likely to drive under special road conditions, and thus the impact on the response might vary based on those factors. The impacts have been often called heterogeneity in means or coefficients [3] or heterogeneity in taste [4]. Not accounting for that impact might result in biased estimations of error terms and possibly biased parameters estimations.

Taste heterogeneity (TH) can be defined as variations in models’ random coefficients based on other explanatory variables across the population. It is compared to the standard random effects model, where there is a single random coefficient across all observations. Those models are especially popular in choice modeling or travel behavior.

For instance, a traveler's choice of choosing a specific type of travel mode, e.g., train or bus, might be influenced by random parameters of time of travel. On the other hand, the random parameter of time of travel itself might be impacted by other characteristics such as travelers’ income [5]. In this study, the heterogeneity in taste is used to possibly account for endogeneity bias resulting from various driver or environmental characteristics. For instance, taste heterogeneity as an alternative form of endogeneity bias was used for the evaluation of various factors on car ownership [6]. The importance of accounting for TH has been highlighted by a large body of studies in the literature [7]. It should be noted that the TH could be implemented in scenarios where the coefficients are fixed e.g [8] or random e.g [9].

The rest of this manuscript is organized as follows: the literature review presents some studies that applied methods to account for the heterogeneity in dataset observations. The methodology section discusses the mathematical formulation of the mixed model and mixed model with heterogeneity in taste. The data section outlines the dataset. Then, the results section and discussion will outline and discuss the results.

The mixed model can be viewed as an extension of the standard logit model, where one of the main shortcomings of the standard logit model is that the model does not account for unobserved heterogeneity that might exist across observations. To address this issue, the mixed logit model is employed to account for heterogeneity across crashes by letting the impacts of predictors vary across observations. The standard mixed logit model could be achieved by assuming continuous heterogeneity (e.g., normal distribution), allowing parameters to vary across crashes [13]. However, the limitation of the standard mixed model is that the model only accounts for unobserved heterogeneity in the coefficients without clarifying how the heterogeneity in means might vary based on various coefficients or accounting for observed heterogeneity.

However, often assuming the independent means of random parameters might result in biased estimates as they ignore the underlying relationships between those factors (endogeneity). Modifications of the mixed model could be made to address possible interrelations across variables. One of the approaches is by allowing the random coefficient means to be dependent on observed characteristics of individuals [14].

The random parameter model could be written as:

(1)

Where y_i* is a realization of crash severity of observed individual crash i being independent, x_i is the observed vector of covariates, and here ε_i is the error terms following logit distribution. The probability density function of y_i* based on binary logit, could be written as:

(2)

Where p is the cumulative distribution function (CDF), which could be written as Pi = exp(Y)/1+ exp(Y), γ is the sum of various coefficients to be estimated and related covariates. The log-likelihood, on the other hand, could be written as:

(3)

Where i is related to observation crash and n is the number of coefficients to be estimated. The above discussion is similar to the standard logit and the mixed logit model. The difference is that we have R number of attributes/columns to the number of draws and they will be summed up to the fixed parameters process, and the probability in 2 will be estimated for R number of columns.

So, the probability is 3 over the value of β_i, for instance, could be written as:

(4)

And with approximation through a random draw, the above could be written as:

(5)

Where the above is used in Equation 3 for creating the log likelihood.

For random parameters of β_i, the parameter varies based on each individual crash or individual-specific predictor, where the changes are based on the distribution function of g(β_i|0). The random parameter in this study follows a normal distribution and we have:

(6)

Where ω_i ~ N (0,1), and L is a diagonal matrix containing the standard deviation (δ) of the random parameters, and in case of incorporating the correlated random parameters, L includes the correlation on its off-diagonal elements. It is also intuitive that in case of having no random parameter, the (δ) of the model, or L, will be zero and the model turn into the standard logit model.

Also, it should be noted that here we accounted for observed heterogeneity to change β_i based on some observed attributes so we have:

(7)

Where is a matrix accommodating the parameters to be estimated, and s_i is the vector of related covariates.

The findings are presented in Table 2 in three subsections. The first section elaborates on the results of the fixed effects, followed by random effects in the second subsection. To compare the performances of different considered models, various goodness of fit measures were considered and included in Table 2.

3.3. Comparison Across the Two Models

The standard logit model was expanded to the mixed logit model and then to the mixed model with heterogeneity in taste. The model with heterogeneity in taste assumes that unseen heterogeneity being due to gender and AADT vary across individual barrier crashes according to some observable variables. The results are intuitive, indicating that the mean of the random parameter of gender, for instance, decreases based on higher traffic count and shoulder width as follows:

(8)

Here π corresponds to the matrix that accommodates the vectors of shoulder width and AADT. ω_ir accommodates random draws being normally distributed, N(0,I)and L is the lower-triangular Cholesky factor [18].

The goodness of the two models was compared in terms of Alkaike information criterion (AIC). The goodness of fit is a valid measure for comparison as the method penalizes for the number of included predictors. As can be seen, there is a significant improvement from the standard logit model (BIC=1,644) to the standard mixed (1,564) and mixed model with heterogeneity in taste (1,555). In addition to the improvement in the model fits, significant changes could be observed in the magnitudes of the points estimated and their standard errors.

Especially significant changes could be observed across the two mixed models across those parameters in which heterogeneity in tastes was considered. In terms of fixed parameters, differences could be observed across the two mixed models and standard logit models. For instance, consider rollover crashes where the point estimates of mixed models are very close, they are more varied when compared with the standard logit model. Similar explanations could be made regarding posted speed limit across the mixed model and the standard logit model. The variations are expected and reflected in the goodness of fit of models.

Run of roadway crashes account for a significant proportion of severe traffic crashes. Traffic barriers have been installed to mitigate the severity of those crashes. Despite efforts in traffic barriers designs improvement, the severity of those crashes still persists. The standard logit model is a common method that has been implemented in the literature review for modeling crash severity.

In this study, the application of the standard logit model has been extended to the mixed model for analyzing crashes to account for the randomness of individual crashes. However, a standard mixed model with random parameters might not capture the real impact of random parameters and the dependence of those factors on other variables. Researchers have given more flexibility to the random parameters by setting constrain off those parameters for giving them more flexibility to vary across observations. It has been achieved by changing the random parameters’ means based on other predictors.

In general, improvements were observed moving from the standard logit model to the standard mixed and mixed model with heterogeneity in taste. In the analysis, it was found that the means of the random parameter of gender is dependent on the shoulder width and AADT, indicating that an increase in ADTT and shoulder width decrease the likelihood of female drivers being involved in higher severity of barriers crashes. Also, the means of AADT was found to be dependent on the means of ADTT. The impacts were considered after no multicollinearity was observed between those two predictors.

Besides comparison based on goodness of fits, the models could also be compared in terms of error terms and model coefficients’ estimate. In terms of models’ estimates of coefficients, the models’ coefficients and standard errors vary especially across standard logit models and the other two mixed models. Still, significant differences could be observed across the two mixed models, especially across parameter estimations for those variables in which heterogeneity in taste was considered. In general, higher error terms could be observed for the mixed model with heterogeneity in taste. The result is consistent with the literature review that not accounting for taste heterogeneity should not be seen as a deficiency of the model but as a sign that the error term is decreased [19].

The improvement in a random model with heterogeneity in taste might be due to the fact that a complemented model could better account for crash randomness by taking into consideration the endogeneity of gender, shoulder width, traffic, and also AADT. In general, the results of the analysis were in line with the literature review showing higher posted speed limit, shoulder width, presence of concrete barriers, motorcycle riders, dry road conditions and having past citation records will increase the likelihood of higher severity crashes for barriers.

The main goal of this study was to test the presence of taste heterogeneity in the population of traffic barrier crashes. The results lead to an important insight into the importance of accounting for taste heterogeneity. That was especially important as it allows us to account for heterogeneity in taste resulting from possible endogeneity across various predictors. Future studies should take into consideration the residual taste heterogeneity to see if a further improvement could be achieved. Also, they should take into consideration the unobserved inter-alternative correlation.

On the other hand, it has been argued that although the mixed model could model various data well, the results would depend on setting a correct mixing distribution [20], often set arbitrary as normal by practitioners. The shortcoming could be accounted for in future study using a discrete distribution in the latent class model.