Application of the Poisson-Tweedie distribution in analyzing crash frequency data

This paper describes a study that applies the Poisson-Tweedie distribution in developing crash frequency models. The Poisson-Tweedie distribution offers a unified framework to model overdispersed, underdispersed, zero-inflated, spatial, and longitudinal count data, as well as multiple response variables of similar or mixed types. The form of its variance function is simple, and can be specified as the mean added to the product of dispersion and mean raised to the power P. The flexibility of the Poisson-Tweedie distribution lies in the domain of P, which includes positive real number values. Special cases of the Poisson-Tweedie distribution models include the linear form of the negative binomial (NB1) model with P equal to 1.0, the geometric Poisson (GeoP) model with P equal to 1.5, the quadratic form of the negative binomial (NB2) model with P equal to 2.0, and the Poisson Inverse Gaussian (PIG) model with P equal to 3.0. A series of models were developed in this study using the Poisson-Tweedie distribution without any restrictions on the value of the power parameter as well as with specific values of the power parameter representing NB1, GeoP, NB2, and PIG models. The effects of fixed and varying dispersion parameters (i.e., dispersion as a function of covariates) on the variance and expected crash frequency estimates were also examined. Three years (2012-2014) of crash data from urban three-leg stop-controlled intersections and urban four-leg signalized intersections in the state of Florida were used to develop the models. The Poisson-Tweedie models or the GeoP models were found to perform better when the dispersion parameter was constant or fixed. With the varying dispersion parameter, the NB2 and PIG models were found to perform better, with both performing equally well. Also, the fixed dispersion parameter values were found to be smaller in the models with a higher value of the power parameter. The variation across the models in their estimates of weight factor, expected crash frequency, and potential for safety improvement of hazardous sites based on the empirical Bayes method was also discussed.