A framework for modelling overdispersed count data, including the Poisson-shifted generalized inverse Gaussian distribution

A variety of methods of modelling overdispersed count data are compared. The methods are classified into three main categories. The first category are ad hoc   methods (i.e. pseudo-likelihood, (extended) quasi-likelihood, double exponential family distributions). The second category are discretized continuous distributions and the third category are observational level random effects models (i.e. mixture models comprising explicit and non-explicit continuous mixture models and finite mixture models). The main focus of the paper is a family of mixed Poisson distributions defined so that its mean μμ is an explicit parameter of the distribution. This allows easier interpretation when μμ is modelled using explanatory variables and provides a more orthogonal parameterization to ease model fitting. Specific three parameter distributions considered are the Sichel and Delaporte distributions. A new four parameter distribution, the Poisson-shifted generalized inverse Gaussian distribution is introduced, which includes the Sichel and Delaporte distributions as a special and a limiting case respectively. A general formula for the derivative of the likelihood with respect to μμ, applicable to the whole family of mixed Poisson distributions considered, is given. Within the framework introduced here all parameters of the distributions are modelled as parametric and/or nonparametric (smooth) functions of explanatory variables. This provides a very flexible way of modelling count data. Maximum (penalized) likelihood estimation is used to fit the (non)parametric models.