# Two-Component Bayesian Hierarchical Models for Cost-Benefit Analysis of Traffic Barrier Crash [PERSON]
## 11 Best Fit Model  
Three models were compared to select the best model for conducting the optimization process. Since Poisson distribution was used as a starting technique for modeling the count response, Bayesian hierarchical two-component model with the Poisson distribution was considered in addition to NB distribution. Besides those two models, a one-layer NB Bayesian hierarchical technique was considered. Similar predictors were considered for the three models so that a fair comparison could be made. Truck traffic, instead of AADT, was used for the first layer, logistic regression, of the two-component models as it was found that ADTT to be significant for the logistic regression part. Since the one-layer NB could accommodate all the predictors in one layer and there was a correlation between ADTT and AADT, ADTT was not considered in that model. However, as discussed in the method section, DIC could penalize for the number of included predictors. The results are presented in Tables 3-5. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Mean** & **SD** & **2.50\%** & **97.50\%** \\ \hline alpha1 & 0.278 & 0.087 & 0.108 & 0.45 \\ \hline alpha2 & 0.429 & 0.153 & 0.148 & 0.756 \\ \hline alpha3 & 0.446 & 0.085 & 0.28 & 0.613 \\ \hline b1 & 0.001 & 0.0002 & 0.0009 & 0.01 \\ \hline b10 & 2.213 & 0.156 & 1.909 & 2.537 \\ \hline b12 & 0.403 & 0.098 & 0.216 & 0.6 \\ \hline b2 & 1.991 & 0.19 & 1.614 & 2.373 \\ \hline b3 & 0.199 & 0.087 & 0.028 & 0.369 \\ \hline b4 & 0.138 & 0.084 & \(-\)0.028 & 0.302 \\ \hline b5 & \(-\)0.328 & 0.086 & \(-\)0.497 & \(-\)0.16 \\ \hline b6 & 0.703 & 0.086 & 0.537 & 0.875 \\ \hline b7 & 0.438 & 0.102 & 0.241 & 0.642 \\ \hline b8 & 0.005 & 0.0001 & 0.004 & 0.006 \\ \hline b9 & \(-\)0.42 & 0.15 & \(-\)0.716 & \(-\)0.12 \\ \hline \multicolumn{5}{c}{DIC = 12,679.8} \\ \hline \end{tabular}
\end{table} 
Table 4: Estimation results of single-layer model, with NB distribution. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Mean** & **SD** & **2.50\%** & **97.50\%** \\ \hline \multicolumn{5}{c}{**2 nd layer, Poisson model**} \\ \hline alpha1 & 1.739 & 0.025 & 1.69 & 1.788 \\ \hline alpha2 & 2.521 & 0.038 & 2.447 & 2.595 \\ \hline alpha3 & 2.346 & 0.022 & 2.302 & 2.39 \\ \hline b1 & 0.001 & 0.0002 & 0.001 & 0.001 \\ \hline b10 & 1.351 & 0.025 & 1.301 & 1.4 \\ \hline b12 & \(-\)0.361 & 0.021 & \(-\)0.404 & \(-\)0.32 \\ \hline b2 & 0.917 & 0.034 & 0.849 & 0.981 \\ \hline b3 & \(-\)0.066 & 0.021 & \(-\)0.107 & \(-\)0.026 \\ \hline b4 & \(-\)0.334 & 0.022 & \(-\)0.377 & \(-\)0.291 \\ \hline b5 & \(-\)0.667 & 0.02 & \(-\)0.707 & \(-\)0.627 \\ \hline b6 & 0.299 & 0.022 & 0.258 & 0.343 \\ \hline b7 & \(-\)0.335 & 0.023 & \(-\)0.379 & \(-\)0.29 \\ \hline b8 & 0.001 & 0.0001 & 0.001 & 0.001 \\ \hline b9 & 0.331 & 0.036 & 0.26 & 0.399 \\ \hline \multicolumn{5}{c}{**1 st layer, logistic regression**} \\ \hline alpha & \(-\)0.88 & 0.074 & \(-\)1.027 & \(-\)0.734 \\ \hline b17 & \(-\)3.034 & 0.19 & \(-\)3.415 & \(-\)2.668 \\ \hline B16 & 0.001 & 0 & 0.001 & 0.002 \\ \hline \multicolumn{5}{c}{DIC = 54,658.1} \\ \hline \end{tabular}
\end{table} 
Table 3: Estimation results of two-layer model, with Poisson distribution for the second layer. Although the signs/directions are consistent across one layer and two-layer models with NB distribution, important variations could be observed between those models and two-layer Poisson distribution. The differences are especially important across b12 (Lighting conditions), b7 (gender), and b9 (interaction term). This is expected as Poisson cannot accommodate over-dispersion and an excess number of zeroes. This model also resulted in a worst fit (DIC = 54,685) compared with two-layer NB (DIC = 12,559) and one-layer NB (DIC = 12,680). Based on the discussed DIC values, the two-layer NB provides the best fit model. The results indicated that alcohol involvement, clear weather conditions, being a female driver, and having an improper restrain increased the EPDO of barrier crashes. On the other hand, driving in light conditions decreased the EPDO crashes.