# Temporal Instability Analysis of Injury Severities for Angle and Non-Angle Crashes at Roundabouts
## 5 Temporal Instability and Transferability Tests
\begin{table}
\begin{tabular}{c c c c c} \hline
**Year Group** & **2012–2013** & **2014–2015** & **2016–2017** & **2018–2019** \\ \hline \(LL\big{(}\mathbf{\beta}_{t_{i}t_{i+1}}\big{)}\) & \(-\)447.59 & \(-\)687.43 & \(-\)759.74 & \(-\)344.63 \\ Sub-group 1 & 2012 & 2014 & 2016 & 2018 \\ \(LL\big{(}\mathbf{\beta}_{t_{i}}\big{)}\) & \(-\)215.38 & \(-\)315.24 & \(-\)378.12 & \(-\)257.97 \\ Sub-group 2 & 2013 & 2015 & 2017 & 2019 \\ \(LL\big{(}\mathbf{\beta}_{t_{i+1}}\big{)}\) & \(-\)225.54 & \(-\)362.71 & \(-\)373.87 & \(-\)81.57 \\ \(\chi^{2}\) value & 13.34 & 18.96 & 15.50 & 10.18 \\ Degrees of freedom & 10 & 13 & 16 & 11 \\ Confidence level & 79.47\% & 87.57\% & 51.16\% & 48.57\% \\ \hline \end{tabular}
\end{table} 
Table 4: Likelihood ratio test results between consecutive years (Non-angle crash). \begin{table}
\begin{tabular}{c c c c c} \hline
**Year Group** & **2012–2013** & **2014–2015** & **2016–2017** & **2018–2019** \\ \hline \(LL\big{(}\mathbf{\beta}_{t_{i}t_{i+1}}\big{)}\) & \(-\)174.39 & \(-\)293.34 & \(-\)259.37 & \(-\)187.49 \\ Sub-group 1 & 2012 & 2014 & 2016 & 2018 \\ \(LL\big{(}\mathbf{\beta}_{t_{i}}\big{)}\) & \(-\)79.97 & \(-\)130.95 & \(-\)127.31 & \(-\)144.67 \\ Sub-group 2 & 2013 & 2015 & 2017 & 2019 \\ \(LL\big{(}\mathbf{\beta}_{t_{i+1}}\big{)}\) & \(-\)85.45 & \(-\)152.97 & \(-\)125.98 & \(-\)37.94 \\ \(\chi^{2}\) value & 17.94 & 18.84 & 12.16 & 9.76 \\ Degrees of freedom & 12 & 13 & 14 & 12 \\ Confidence level & 88.25\% & 87.19\% & 40.65\% & 36.30\% \\ \hline \end{tabular}
\end{table} 
Table 3: Likelihood ratio test results between consecutive years (Angle crash). The formula for the difference analysis of any two periods is as follows:  
\[X^{2}=-2[LL(\beta_{1g1_{1}})-LL(\beta_{t_{1}})] \tag{10}\]  
where \(LL(\beta_{t_{2}t_{1}})\) is the log-likelihood of the model when using the significant variables of period \(t_{2}\) and using the data of period \(t_{1}\), \(LL(\beta_{t_{1}})\) is the log-likelihood of the model taking the data from period \(t_{1}\), using the same variables but not limited to the convergence parameters of period \(t_{2}\). To ensure the completeness of the test, the periods \(t_{1}\) and \(t_{2}\) are exchanged, and the same calculation is performed again. The temporal stability test results for the pairwise periods are shown in Tables 5 and 6 according to angle and non-angle crashes, respectively. Subsequently, the temporal instabilities of injury severity in angle and non-angle crashes were clarified using the likelihood ratio test. \[X_{g}^{2}=-2\big{[}LL\big{(}\beta_{2012-2019g}\big{)}-LL\big{(}\beta_{2012-201 3g}\big{)}-LL\big{(}\beta_{2014-2015g}\big{)}-LL\big{(}\beta_{2016-2017g}\big{)} -LL\big{(}\beta_{2018-2019g}\big{)}\big{]} \tag{11}\]  
where \(LL\big{(}\beta_{2012-19g}\big{)}\) is the log-likelihood at the convergence of the corresponding model using the data \(g\) for all angle crashes or all non-angle crashes for the period 2012-2019, \(LL\big{(}\beta_{2012-13g}\big{)}\) is the log-likelihood when the model converges using the crash data \(g\) in 2012-2013, \(LL\big{(}\beta_{2014-15g}\big{)}\) is the log-likelihood when the model converges using the crash data \(g\) in 2014-2015, \(LL\big{(}\beta_{2016-17g}\big{)}\) is the log-likelihood when the model converges using the crash data \(g\) in 2016-2017, \(LL\big{(}\beta_{2018-19g}\big{)}\) is the log-likelihood when the model converges using the crash data \(g\) in 2018-2019. For angle crashes, the \(X^{2}\) estimate is 90.58, corresponding to the \(X^{2}\) distribution with 33 degrees of freedom. This result provides 99.99% confidence that the null hypothesis that the parameters are equal across four periods (2012-2013, 2014-2015, 2016-2017, and 2018-2019) can be rejected. For non-angle collision crashes, the \(X^{2}\) estimate is 83.76, corresponding to an \(X^{2}\) distribution with 26 degrees of freedom. This result also provides 99.99% confidence that the null hypothesis that the parameters are equal across four periods can be rejected.