# Temporal Instability Analysis of Injury Severities for Angle and Non-Angle Crashes at Roundabouts
## 5 Temporal Instability and Transferability Tests  
Among the temporal instability studies [11; 12], the likelihood ratio test is one of the most commonly used methods. This study conducted two sets of likelihood ratio tests. One group addressed the differences between angle and non-angle crashes. The other aimed to explore the changes in influencing factors over time. Each period (2012-2013, 2014-2015, 2016-2017, and 2018-2019) was tested to compare the injury severity of angle and non-angle crashes. The equation of the likelihood ratio test is:  
\[X_{t}^{2}=-2\Big{[}LL(\mathbf{\beta}_{\mathrm{all},t})-LL\Big{(}\mathbf{\beta}_{ \mathrm{non-angle},t}\Big{)}-LL\Big{(}\mathbf{\beta}_{\mathrm{angle},t}\Big{)} \Big{]} \tag{8}\]  
where \(LL(\mathbf{\beta}_{\mathrm{all},t})\) is the log-likelihood when the model converges using all angle crashes and non-angle crashes data in period \(t\), \(LL\Big{(}\mathbf{\beta}_{\mathrm{non-angle},t}\Big{)}\) is the log-likelihood when the model converges using the non-angle crash data in period \(t\), and \(LL\Big{(}\mathbf{\beta}_{\mathrm{angle},t}\Big{)}\) is the log-likelihoodwhen the model converges using the angle crash data in period \(t\). The \(X^{2}\) estimates for the four periods 2012-2013, 2014-2015, 2016-2017, and 2018-2019 are 54.18, 22.05, 30.61, and 23.56, respectively. These estimates are distributed with 10, 12, 9, and 10 degrees of freedom, indicating a confidence level above 90%. Then, the assumption that the parameters of angle and non-angle crashes are equal can be rejected. To validate the two-year combinations, the likelihood test is further applied:  
\[\chi^{2}_{c}=-2\big{[}LL\big{(}\mathbf{\beta}_{t_{i}t_{i+1}}\big{)}-LL\big{(}\mathbf{ \beta}_{t_{i}}\big{)}-LL\big{(}\mathbf{\beta}_{t_{i+1}}\big{)}\big{]} \tag{9}\]  
where \(LL\big{(}\mathbf{\beta}_{t_{i}t_{i+1}}\big{)}\) represents the log-likelihood at convergence when using both year \(i\) and year \(i+1\) data, \(LL\big{(}\mathbf{\beta}_{t_{i}}\big{)}\) and \(LL\big{(}\mathbf{\beta}_{t_{i+1}}\big{)}\) represent the log-likelihood at convergence when using only year \(i\) and year \(i+1\) data, respectively. The results in Tables 3 and 4 show that the test confidence level is much lower than 90%. That is, for angle and non-angle collisions, the null hypothesis that the parameters are equal between two test years cannot be rejected. The results support the rationality of combining the data from two years in this study. (Due to convergence problems caused by a small number of observations, the likelihood ratio tests for one-year groups were inconclusive and unable to validate one-year groups. Therefore, this study considered two-year or multi-year groupings. In detail, two-year groupings from even to odd years (such as 2012-2013) in this study were based on likelihood ratio tests for potential temporal stability (Tables A1 and A2 in Appendix A). If the confidence level obtained by the test is above 90%, the tested models can be considered to be statistically different; otherwise, they are considered to be the same. Starting with 2012 and 2013, the chronological assessment tests were performed year by year for consecutive years as follows: if the two periods were different, they would be modeled separately; if there was no significant difference between the two periods, they would be combined into one group; and then conduct likelihood ratio tests to validate the combination of the previously combined group and its subsequent consecutive years. Following these steps, we tested the yearly data sequentially, searching for a next-year combination with the current group until the end of the data were reached.