## 3 Methodology
### The Random Parameter Ordered Logit (RPOL) Model  
As mentioned above, the random parameters ordered logit model with heterogeneity in means and variances is widely applied to explore the accident mechanism as it can flexibly capture the unobserved heterogeneity. Thus, the model is employed in this study to investigate factors contributing to conflict risks. The typical form of the utility function is defined as follows:  
\[\mathrm{U_{mn}}=\beta_{\mathbf{m}}\mathbf{X_{mn}}+\varepsilon_{mn} \tag{3}\]  
where \(\mathrm{U_{mn}}\) is the function determining the probability of conflict severity m in sample n, \(\beta_{\mathbf{m}}\) represents a vector of the estimated parameters, \(\mathbf{X_{mn}}\) is the n th vector of explanatory variables affecting conflict-severity m. \(\varepsilon_{mn}\) is a stochastic error term accounting for unobserved effects and assumed to follow the generalized extreme value distribution yielding a standard multinomial logit model:  
\[\mathrm{P_{mn}}=\int\frac{\exp(\beta_{\mathbf{mn}}\mathbf{X_{mn}})}{\sum\exp( \beta_{\mathbf{mn}}\mathbf{X_{mn}})}\mathrm{f}(\beta|\varphi)\mathrm{d}\beta \tag{4}\]  
where \(\mathrm{P_{mn}}\) denotes the probability of risk level m in sample n, \(\mathrm{f}(\beta|\varphi)\) is the density function of vector \(\beta\) and \(\varphi\) is the predefined parameter vector of the density function. \(\beta_{\mathbf{mn}}\) is a vector of estimable parameters to capture the heterogeneity in the mean and variance, which is defined as:  
\[\beta_{\mathbf{mn}}=\beta_{\mathbf{m}}+\mathrm{v_{mn}}\mathbf{Z_{mn}}+\sigma_ {mn}\exp(\boldsymbol{\omega_{mn}}\mathbf{W_{mn}})\mathrm{v_{mn}} \tag{5}\]  
where \(\beta_{\mathbf{m}}\) represents the mean parameters, \(\mathrm{v_{mn}}\) is the estimable parameters, \(\mathbf{Z_{mn}}\) is the vector of collision-specific variables with the heterogeneity in the mean, \(\mathbf{W_{mn}}\) is another vector of explanatory variables that explain the standard heterogeneity in the standard deviation \(\sigma_{mn}\), with the corresponding parameter vector \(\boldsymbol{\omega_{mn}}\), and \(\mathrm{v_{mn}}\) is an error term. The potential conflict risk propensity function for i th vehicle-following pair is determined by Equation (6):  
\[\mathrm{Y_{m}^{*}}=\beta_{\mathbf{m}}^{{}^{\prime}}\mathbf{X_{m}}+\varepsilon _{m} \tag{6}\]  
\[\mathrm{Y_{m}}=1,\text{if }\mathrm{Y_{m}^{*}}\leq\mathrm{u_{m0}}\] \[\mathrm{Y_{m}}=2,\text{if }\mathrm{u_{m0}}\leq\mathrm{Y_{m}^{*}} \leq\mathrm{u_{m1}} \tag{7}\]  
\[\mathrm{Y_{m}}=3,\text{if }\mathrm{Y_{m}^{*}}\geq\mathrm{u_{m1}}\]  
where \(\mathrm{Y_{m}}=1,2,3\) in Equation (7) denotes the most severe, medium, and lowest collision risk level, respectively. \(\mathrm{u_{m1}}\) and \(\mathrm{u_{m0}}\) are the estimated thresholds. The probability of vehicle group i to be j th collision risk level is shown in Equation (8):  
\[\mathrm{P(y}=1)=\Phi(-\beta_{\mathbf{m}}\mathbf{X_{m}})\] \[\mathrm{P(y}=2)=\Phi(\mathrm{u_{1}}-\beta_{\mathbf{m}}\mathbf{X_ {m}})-\Phi(-\beta_{\mathbf{m}}\mathbf{X_{m}}) \tag{8}\] \[\mathrm{P(y}=3)=1-\Phi(\mathrm{u_{1}}-\beta_{\mathbf{m}}\mathbf{X_ {m}})\]  
where \(\Phi()\) denotes the standard normal cumulative distribution function, \(\mathrm{y}=0\) indicates the lowest conflict level, \(\mathrm{y}=1\) indicates the medium conflict risk level, and \(\mathrm{y}=2\) indicates the highest conflict risk level. The random parameter models are estimated with 500 Halton draws as past researchers [19; 41] have found that 500 Halton draws could provide stable parameters. The stepwise process is also conducted for the parameter selection.