# Traffic Collision Severity Modeling Using Multi-Level Multinomial Logistic Regression Model
## 4 Modelling
### Multi-Level Multinomial Logistic Regression  
The Multi-Level Multinomial Logistic Regression model, an extension of the Multinomial Logistic Regression, presents an effective framework for handling hierarchical or nested data structures. The context of this study, aimed at comprehending the impact of behavioural aspects on traffic collision severities, outlines the application of this methodology to analyze crash severity while appropriately addressing nested data structures [39]. The Multi-Level Multinomial Logistic Regression model is valid for modeling crash severity because it accounts for the hierarchical structure of the data and handles the categorical nature of the dependent variable. In addition, crash severity is of a multinomial outcome nature. Several previous studies utilized the Multi-Level Multinomial Logistic Regression for modeling crash severity [40; 41].  
The diversity in crash severity across various locations revealed an intriguing phenomenon. Within specific groups, the variation was smaller than that observed across these groups. This observation became pronounced when considering geographic divisions, distinctions between rural and urban areas, land use patterns, climate zones, and functional areas. Spatial disparities contributed significantly to the severity of injuries resulting from crashes. Therefore, a meticulous analysis of crash data necessitated the consideration of all levels of clustering. Neglecting to account for cluster-specific effects could lead to statistical inaccuracies, manifesting as biased parameter estimates, underestimated standard errors, and an exaggerated perception of statistical significance, as highlighted by [PERSON] et al. [42].  
The model was meticulously tailored to the unique structure of the dataset, accommodating the inherent variability in crash severity. During this process, the study categorized the dataset into distinct levels of crash severity, each compared against a reference point. This categorization aided the model in discerning the factors that differentiate, for example, a minor collision from a more severe one. Such granularity enriched the analysis, enabling us to capture the subtleties associated with varying crash outcomes [43].  
Furthermore, the model transcended individual cases and acknowledged that diverse groups may exhibit distinct patterns. Here, the concept of random effects assumed significance. These effects were incorporated to account for the variations between groups, which might experience crashes differently due to unique circumstances. The general equation for the Multi-Level Multinomial Logistic Regression model can be expressed as follows discussed by [33; 44]:  
\[\log P\big{(}Y_{ij}=j\mid X_{ij}\big{)}=\beta_{oj}+\beta_{1j}X_{1 ij}+\beta_{2j}X_{2 ij}+\cdots+\beta_{kj}X_{kij}+u_{oj}+u_{1j}W_{1j} \tag{2}\] \[+u_{2j}W_{2j}+\cdots+u_{kj}W_{kj}\]  
The probability of crash severity category \(j\) for individual \(i\) within group \(j\) given predictor variables \(X_{ij}\) is represented by \(P\big{(}Y_{ij}=j\mid X_{ij}\big{)}\). The fixed effects coefficients for individual-level predictors are denoted by \(\beta_{oj},\ \beta_{1j},\ \beta_{2j},\ \ldots,\ \beta_{kj}\). The individual-level predictor variables influencing crash severity are represented by \(X_{1 ij},\ X_{2 ij},\ \ldots,\ X_{kij}\). The random effects coefficients for group-level predictors are represented by \(u_{oj},\ u_{1j},\ u_{2j},\ \ldots,\ u_{kj}\). The group-level predictor variables influencing crash severity are denoted by \(W_{1j},\ W_{2j},\ \ldots,\ W_{kj}\).