## 5 Modeling Results and Discussion
### Full Bayesian Estimation
Among the crash models calibrated in this study, all parameters of the MN-logit model are fixed value (Table 4) because it cannot capture heterogeneity. In the RI-logit model, the coefficients of risk variable are fixed value, but the coefficient of intercept is random (Table 4). The heterogeneity in crash data is captured by the random variation of the intercept. Further, in the collision model established by the RP-logit function, both random coefficients and fixed coefficients are included (Table 5). Meanwhile, the number of random coefficients and fixed coefficients varies with vehicle types. In the SRP-logit model, three types of coefficients are accommodated, including the fixed coefficients, random coefficients, and coefficients of spatial variation (Table 5). Hence, the collision models established in this study accommodate the coefficients that can demonstrate the respective characteristics. The comparison of the fitting performance of these models is shown below. From Table 3, some substantial differences in \(DIC\) and \(CA\) values across different logit approaches were found. First, passenger car crash function constructed by the MN-logit model had a higher \(DIC\) value (8859) than other models and the differences in \(DIC\) is greater than 5, 8827 for the RP-logit model, 8839 for the RI-logit model, and 8821 for the SRP-logit model. Similar phenomena were exhibited in the crash severity models of motorcycle, pickup, and truck. Meanwhile, the \(CA\) values associated with the MN-logit model were lower than other models, which remained stable across motor vehicle types. These findings indicated that the RI-logit model, the RP-logit model, and the SRP-logit model perform better than the MN-logit model and highlighted that the fitting performance and classification accuracy of the model can be improved by accommodating unobserved heterogeneity in crash severity analysis. [PERSON] et al. [40] reached the same conclusion and gave a detailed discussion. Second, among the three candidate heterogeneity models (RP-logit model, RI-logit model, SRP-logit model), the SRP-logit model was a superior approach for fitting rural SV crash severity, followed by the RP-logit model and the RI-logit model, and these findings were supported by all the vehicle types in this research. More specifically, in the SRP-logit model, the RP-logit model, and the RI-logit model, the \(DIC\) values of passenger cars were 8821, 8827, and 8839, respectively, and of motorcycles were 8610, 8615, and 8631, respectively, and of trucks were 4184, 4205, and 4217, respectively. According to the statistical results, the \(DIC\) values of the three candidate models can be expressed in descending order, as follows--RI-logit model, RP-logit model, and SRP-logit model--and this finding remained stable across the three vehicle types. Furthermore, the differences in \(DIC\) were greater than 5. Hence, for passenger car, motorcycle, and truck crashes, the SRP-logit model exhibited the best fit performance, followed by the RP-logit model and the RI-logit model. Similar findings can be obtained by checking the \(CA\) values of such three vehicle types. For pickup crashes, the \(DIC\) values of the SRP-logit model, RP-logit model, and RI-logit model were 2126, 2128, and 2137, respectively. It was found that the RP-logit model outperformed the RI-logit model. Note that the differences in \(DIC\) values between the SRP-logit model and the RP-logit model was less than 5; thus, \(DIC\) cannot be used for model diagnosis and \(CA\) values were adopted to determine the fitting performance. The values of \(CA_{1}\), \(CA_{2}\), \(CA_{3}\), and \(CA_{whole}\) in the SRP-logit model were 79.1%, 27.6%, 15.2%, and 75.8%, respectively, and in the RP-logit model were 73.6%, 20.8%, 8.9%, and 66.3%, respectively. It can be easily found that the prediction accuracy of the SRP-logit model outperforms that of the RP-logit model.