## 4 Methodology
### Model Diagnosis  
The fitting performance of a crash model cannot be illustrated by regression coefficients. To effectively diagnose the usefulness of statistical techniques for fitting rural SV crashes, deviance information criterion (_DIC_) and classification accuracy (_CA_) values were calculated to determine the goodness-of-fit and prediction accuracy of the crash model,  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Models** & **Types** & **Characteristics** \\ \hline MN-logit model & Fixed effects model &
\begin{tabular}{c}  
respectively. Meanwhile, these methods have been widely used to test full Bayesian model and the detailed explanations can be found in [PERSON] et al. [63] and [PERSON] et al. [43].  
The _DIC_ is considered as a Bayesian generalization of [PERSON]'s information criterion, which comprehensively accommodates model complexity and goodness-of-fit. According to [PERSON] et al. [63], the _DIC_ can be defined as:  
\[DIC_{value}=\overline{D}+pD,=pD=\overline{D}-\hat{D}, \tag{16}\]  
where \(D\) denotes the unstandardized deviance; \(\overline{D}\) denotes the posterior mean deviance of \(D\) which can be employed to evaluate model fitness; \(pD\) denotes the number of effective parameters which can be used to test model complexity; \(\hat{D}\) denotes the point estimation of model parameters. A smaller _DIC_ value is associated with a better goodness-of-fit. Typically, differences of more than 10 can exclude models with higher _DIC_; differences between 5 and 10 are substantial; if the difference in _DIC_ is less than 5 and the model inferences are substantially different, then it is misleading to determine the fit performance by the _DCI_ value.  
In addition, _CA_ value of crash severity \(k\) was calculated as a supplement to the model evaluation criterion. The _CA_ value in this research was divided into two parts, including the _CA_ value of the whole dataset and the _CA_ value of a specific severity \(k\). The larger the _CA_ value, the better the prediction accuracy of the model.  
The _CA_ value of the whole dataset was defined as the proportion of accurately predicted samples in the whole dataset. The specific calculation method was shown below.  
\[CA_{whole}=\frac{\sum_{\hat{Y}_{i}=Y_{i}}\hat{Y}_{i}}{\sum_{\hat{Y}_{i}}Y_{i}} \times 100\%,\ \forall i\in\left\{1,2,\ldots,\,I\right\}, \tag{17}\]  
where \(\hat{Y}_{i}\) and \(Y_{i}\) respectively represent the predicted and the real outcome of severity level for crash \(i\).  
The _CA_ value of a specific severity \(k\) was defined as the proportion of accurately predicted samples to all observations associated with severity \(k\), which can be expressed as:  
\[CA_{k}=\frac{\sum_{\hat{Y}_{i}=Y_{i}=k}\hat{Y}_{i}}{\sum_{Y_{i}=k}Y_{i}}\times 1 00\%,\ \forall k\in\left\{1,2,3\right\}. \tag{18}\]  
In this research, no injury, slight injury, and FS injuries were denoted by \(CA_{1}\), \(CA_{2}\), and \(CA_{3}\), respectively.  
During the process of diagnosing the fit performance, the _DIC_ value will be checked first and models with larger _DIC_ will be excluded if the difference in _DIC_ is greater than 5; the _CA_ value is then used to calibrate the decision. If the difference in _DIC_ is less than 5, then the model performance will be directly determined by the CA value.