## 5 Likelihood Ratio Tests  
To examine the differences in conflict severity outcomes between different ETC and MTC vehicle-following patterns and the spatial dependence in sub-segments, four groups of likelihood ratio tests were conducted.  
To compare the models developed for four different vehicle-following patterns between MTC and ETC vehicles, the first series of likelihood ratio tests were conducted, which can be expressed as follows [19,29]:  
\[\begin{array}{c}\chi_{1}^{2}=-2[\text{LL}(\beta_{\text{All}})-\text{LL}(\beta _{\text{ETC-ETC}})-\text{LL}(\beta_{\text{ETC-MTC}})-\text{LL}(\beta_{\text{MTC -ETC}})\\ -\text{LL}(\beta_{\text{MTC-MTC}})]\end{array} \tag{11}\]  
where LL(\(\beta_{\text{All}}\)) is the log-likelihood at the convergence of the model estimated with all data, LL(\(\beta_{\text{ETC-ETC}}\)), LL(\(\beta_{\text{ETC-MTC}}\)), LL(\(\beta_{\text{MTC-ETC}}\)), and LL(\(\beta_{\text{MTC-MTC}}\)) represent the log-likelihood values at the convergence of the estimated models using different ETC and MTC vehicle-following patterns.  
Moreover, to investigate the spatial stability of variables between different sub-segments of the diverging area, three sub-models were developed for three parts of the diverging area (denoted as D\({}_{1}\), D\({}_{2}\), and D\({}_{3}\); D\({}_{1}\) represents the downstream of the diverging area, including sub-segment 1-4; D\({}_{2}\) represents the midstream of the diverging area, including sub-segment 5-8; D\({}_{3}\) represents the upstream of the diverging area, including sub-segment 9-12, as shown in Figure 4). The second series of likelihood ratio tests were utilized [11]:  
\[\chi_{2}^{2}=-2\Big{[}\text{LL}\Big{(}\beta_{y_{1}y_{2}}\Big{)}-\text{LL}\Big{(} \beta_{y_{1}}\Big{)}\Big{]} \tag{12}\]  
where LL\(\Big{(}\beta_{y_{1}y_{2}}\Big{)}\) is the log-likelihood at convergence of a model containing parameters from \(y_{2}\) while using data subset \(y_{1}\), and LL\(\Big{(}\beta_{y_{1}}\Big{)}\) is the log-likelihood at convergence of the model using data subset \(y_{1}\).  
Tables 2-5 present whether the null hypothesis that the parameters are stable in the three parts of the diverging area can be rejected based on the results. For example, as shown in Table 2, using the converged parameters of the D\({}_{2}\) model (y\({}_{2}\)) as the initial values and applying them to the D\({}_{1}\) data (y\({}_{1}\)) yields \(\chi^{2}=27.428\) with 9 degrees of freedom, indicating that the null hypothesis that the two parts of the diverging area are the same can be rejected at a 99.88% confidence level. Likewise, using the converged parameters of the D\({}_{1}\) model as the starting values and applying them to the D\({}_{2}\) data gave \(\chi^{2}=22.918\) with 8 degrees of freedom, demonstrating that the null hypothesis that the parts of the diverging area are the same can be rejected at a 99.65% confidence level. Other subsections have similar results that the null hypotheses can be rejected at a high confidence level.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline \multirow{2}{*}{\(\mathbf{y_{1}}\)} & \multicolumn{2}{c}{\(\mathbf{y_{2}}\)} \\ \cline{2-4}  & \multicolumn{1}{c}{**D\({}_{1}\) (0–120 m)**} & \multicolumn{1}{c}{**D\({}_{2}\) (120–240 m)**} & \multicolumn{1}{c}{**D\({}_{3}\) (240–360 m)**} \\ \hline D\({}_{1}\) (0–120 m) & – & 22.918 (8) & 27.420 (14) \\  & & [99.65\%] & [98.30\%] \\ D\({}_{2}\) (120–240 m) & 27.428 (9) & – & 28.678 (14) \\  & [99.88\%] & & [98.85\%] \\ D\({}_{3}\) (240–360 m) & 23.748 (9) & 25.475 (8) & – \\  & [99.53\%] & [99.87\%] & \\ \hline \hline \end{tabular} Notes: the numbers in parentheses represent the degrees of freedom for the Likelihood Ratio Test.  
\end{table}
Table 2: Likelihood ratio test results between different diverging areas (ETC-ETC model).