A multivariate count model is developed by introducing a simple and practical formula. The formulation begins with a modification of the standard ordered response model to adopt the count outcomes nature. This modification is accomplished by introducing a non-linear asymmetric interdependence structure among the error terms using the copula-based model. To avoid simulation maximum-likelihood for evaluating the multi-outcome density, we utilize the composite marginal likelihood (CML) approach. The proposed copula-based model with the CML approach allows for asymmetric (tail) dependency without a need for a simulation mechanism. Non-parametric graphical techniques with the empirical copula as well as conventional goodness-of-fit statistics are utilized to guide copula selection. In addition, unobserved heterogeneity across observations is also addressed through a heterogeneous dispersion parameter in the proposed model. The heterogeneous dispersion parameter model is a suitable alternative to random parameter count models in that captures heterogeneity in variance, while allowing for closed form while the latter needs numerical integration or simulation. We apply these techniques to study the interdependence structure among four types of traffic crashes using three years (2005–2007) of cross-sectional crash data record for 274 multilane freeway segments in the State of Washington, USA. These four categories of crash types are the rear end; sideswipe; fixed objects and other crash types. The empirical results show a significant presence of unobserved heterogeneous dependency across these types of crashes. The results indicate the important role of unobserved heterogeneity in variance and covariance structure estimation. An important outcome of this result is that it can affect inference on the relative impact of roadway geometrics on crash occurrence. For example, we find that horizontal curve related parameters on freeway segments substantially increase the joint likelihood of rear-end, sideswipe, fixed objects and other crash types, when compared to the characteristics of vertical curves.
- Composite marginal likelihood
- Crash type correlations
- Heterogeneous dispersion
- Multivariate count data
- Variance-covariance structure