## 4 Results
### Best Fit Model  
Three models were considered in this study based on various intuitive variables as hierarchies and the objective hierarchy assigned by the FMM. Those models were compared in terms of goodness of fit based on the DIC. The assigned objective hierarchies resulted in a lower DIC of 11,057, compared with the state highway system as a hierarchy with DIC = 11,063, and barrier types as a hierarchy with DIC of 11,060. Thus, due to lowest DIC value, an objective hierarchy was used for the analysis. The sum of the predicted EPDO and real EPDO along with their variances are presented in Table 2. The results from using subjective hierarchies are not presented in Table 2 to conserve space.  
Although the range between the sum of the predicted and real EPDO is not too wide, the variance difference of these two variables are rather wide. This is likely due to the randomness of crashes. For instance, the maximum of the predicted EPDO is 1733. This high value is related to the fact that this observation has the worst criteria based on the statistical results: for instance, the majority of drivers hitting this barrier were male, occurring on the interstate system with a long barrier length.  
Since the objective of this study is not just about conducting statistical analysis, but rather the application of machine learning techniques, this study does not detail the analytical results. For instance, the results highlight that being unrestrained, being female drivers, being a non-Wyoming resident,  
\begin{table}
\begin{tabular}{l l c c c c} \hline \hline
**Variable** & **Variable Description** & **Mean** & **SD** & **Min** & **Max** \\ \hline \(n\) = 2432 & & & & & **All barriers** \\ Response (EPDO) & & 9.245 & 1355 & 1 & 555 \\ \# Cluster I & & & & 67 & & \\ \# Cluster II & & & 35 & & \\ \# Cluster III & & & & 1496 & & \\ \# Cluster IV & & & & 250 & & \\ b1 & Restrain condition, having some form of restraint as reference (0), & & & 0.116 & 0.076 & 0 & 1 \\ b2 & Gender, males as reference (0),1 otherwise & & 0.335 & 0.157 & 0 & 1 \\ b3 & Residency, Wyoming residence as reference (0), others as 1 & & 0.467 & 0.188 & 0 & 1 \\ b4 & Weather condition, clear weather condition as reference (0), & & 0.50 & 0.181 & 0 & 1 \\ b5 & Lighting conditions, light condition as reference (0), 1 otherwise & & 0.387 & 0.172 & 0 & 1 \\ b6 & AADT & & 2438 & 2,510,230 & 27 & 8854 \\ b7 & Length, continuous, ft & & 740 & 2,144,682 & 14 & 35,470 \\ b8 & Barrier types & Box beam as 1 & & 1.626 & 0.861 & 1 & 3 \\ b9 & Barrier height, continuous, inch & & 2.458 & 0.057 &
\begin{tabular}{c} \(<\)12 \\ in \\ b10 \\ B12 \\ \end{tabular} & Shoulder width, less than or equal 1.5 ft as 0, otherwise 1 & & 0.925 & 0.070 & 0 & 1 \\ B12 & Highway system, state highway as reference (0), interstate as 1 & & 0.548 & 0.248 & 0 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Descriptive statistics of the trained dataset.  
and driving in adverse weather conditions reduce the likelihood of the number of the EPDO crashes. Although for some predictors some minor uncertainty about the significance of the predictors is observed, the important predictors of interaction of shoulder width and barrier heights were found to be significant. It should be noted that barrier types, shoulder width, AADT, and barrier length were considered especially to account for exposure. Alpha [1] to alpha [4] are the clusters assigned to the model hierarchy while B0 is an intercept. It should be noted the Sum of real EPDO is 28,945, while the sum of predicted EPDO is 17,084. On the other hand, variance of real EPDO is 30,942, and the variance of predicted EPDO is 1355. Again the findings' comparison across models, more consistent results for predictions values, are in line with the previous study [3].