Application of different negative binomial parameterizations to develop safety performance functions for non-federal aid system roads

•The model outperforms the NB model., regardless of the variance/dispersion structure.•The dispersion and variance structures can significantly affect the model performance.•Each family of models developed in this study (NB-L and NB) favors a specific dispersion structure.•The choice of the variance structure does not only depend on the data, but also on the NB formulation.•A NB-L model with a reasonably chosen dispersion/variance structure yield better estimates.