# Temporal Instability Analysis of Injury Severities for Angle and Non-Angle Crashes at Roundabouts
## 4 Methodology  
To study crash injury severity, various discrete models have been adopted, including ordered logit models, multinomial logit models, nested logit models, latent-class logit models, mixed logit models, and others [31, 44, 52]. In current studies, possible unobserved heterogeneity and temporal instabilities are often explained by random parameter approaches. This study uses random parameter logit models with possible heterogeneity in means and variances, which aims to discover potential heterogeneity and temporal instabilities in angle and non-angle crashes at roundabouts. Experiments were carried out in the statistical software package NLOGIT V5.0. The framework of the methodology section is shown in Figure 2.  
First, a function that determines the injury severity of crashes is defined as,  
\[S_{kn}=\mathbf{\beta}_{k}\mathbf{X}_{kn}+\varepsilon_{kn} \tag{1}\]  
where \(S_{kn}\) is an injury severity function determining the probability of injury severity \(k\) in crash \(n\) at roundabouts. \(\mathbf{\beta}_{k}\) is the vector of estimated parameters, \(\mathbf{X}_{kn}\) is a vector of explanatory variables affecting the injury severity \(k\) in angle and non-angle crashes, and \(\varepsilon_{kn}\) is a disturbance term assumed to be a generalized extreme value distribution. Then, the multinomial logit model is expressed as follows [52]:  
\[\mathrm{P}_{n}(k)=\frac{\mathrm{EXP}[\mathbf{\beta}_{k}\mathbf{X}_{kn}]}{\sum_{ \forall\text{K}}\mathrm{EXP}(\mathbf{\beta}_{k}\mathbf{X}_{kn})} \tag{2}\]  
Figure 2: Framework of the methodology. where \(\mathrm{P_{n}}(k)\) represents the probability of injury severity \(k\) in an angle or non-angle crash \(n\) at a roundabout, \(k\) is a set of three possible injury severities. Considering possible parameter fluctuations, the injury severity probabilities for angle and non-angle collisions are rewritten as [31]:  
\[\mathrm{P_{n}}(k)=\int\frac{\mathrm{EXP}(\beta_{k}\mathbf{X}_{kn})}{\sum_{\forall k }\mathrm{EXP}(\beta_{k}\mathbf{X}_{kn})}f(\beta_{k}\mid\mathbf{\varphi}_{k})d\beta_{k} \tag{3}\]  
where \(f(\beta_{k}\mid\mathbf{\varphi}_{k})\) is the density function of a vector \(\mathbf{\beta}_{k}\), \(\mathbf{\varphi}_{k}\) is a vector of parameters characterizing the density function, other terms are consistent with previous definitions. Given the possibility of unobserved heterogeneity in the means and variances of the parameters, \(\mathbf{\beta}_{kn}\) is allowed to be a vector of estimated parameters that vary across crashes. The vector is defined as [31]:  
\[\mathbf{\beta}_{kn}=\mathbf{\beta}_{k}+\mathbf{\Theta}_{kn}\mathbf{Z}_{kn}+\sigma_{kn}EXP( \mathbf{\
abla}_{kn}\mathbf{W}_{kn})v_{kn} \tag{4}\]  
where \(\mathbf{\beta}_{k}\) is the mean parameter estimate for all angle and non-angle crashes at the roundabout, \(\mathbf{\Theta}_{kn}\) is a vector related to estimated parameters, and \(\mathbf{Z}_{kn}\) is a vector of explanatory variables that discovers heterogeneity in the mean that determines the injury severity \(k\) for angle and non-angle crashes at roundabouts. The heterogeneity in the standard deviation \(\sigma_{kn}\) of random parameters is captured by \(\mathbf{W}_{kn}\), with estimable parameters in vector \(\mathbf{\
abla}_{kn}\), and \(v_{kn}\) is the error term. We used the simulated maximum likelihood method with 1000 Halton draws to estimate model parameters [31]. As for the possible distribution of parameters, the normal distribution, logarithmic distribution, and uniform distribution are tested, and the results show that the normal distribution gives the best statistical fit.