# Over/Under-dispersion, zero-infation and Poisson-Tweedie models

Abstract: The standard distributions for the analysis of count and proportion data are the Poisson and binomial distributions. Frequently, in practice they are too restrictive in that the variability in the data is either signicantly greater (overdispersed) or less (underdispersed) than that implied by the models variance function. For the analysis of count data, Nelder and McCullagh (1989) says that overdispersion is the norm and not the exception and this has been well studied, see Hinde and Demétrio (1999) and many subsequent articles presenting a wide range of distributions. Although less common, underdispersion can arise, typically from dependent responses. For instance, when there is competition between plants and animals this can induce negative correlation in temporal and spatial counting processes. Here we will also consider how underdispersion can occur as a result of features of the underlying counting, or data collection, process. The range of distributions for modelling underdispersed count data is relatively limited, although models can be derived in specic situations.A class of general models is presented based on Poisson-Tweedie factorial dispersion models with variance phi mu^p, where mu is the mean, phi and p are the dispersion and Tweedie power parameters, respectively. This class of models provides a flexible and comprehensive family including many standard discrete models. The family provides for modelling of overdispersed count data, including Neyman Type A, Polya-Aeppli, negative binomial, Poisson-inverse Gaussian and Hermite distributions, and can also accommodate zero-infation and underdispersion. For a general approach we consider an extended version of the Poisson-Tweedie model and discuss estimation of regression, dispersion and Tweedie power (variance function) parameters. A full description of the approach is given in Bonat et al. (2017).ReferencesBonat, W., J rgensen, B, Kokonendji, C., Hinde, J. and Demétrio, C.G.B. (2017)Extended Poisson-Tweedie: properties and regression models for count data. Statistical Modelling(accepted)Hinde, J. and Demétrio, C.G.B. (1998) Overdispersion: Models and estimation. Computational Statistics and Data Analysis. 27, 151{170.McCullagh, P. and Nelder, J.A. (1989). Generalized Linear Models. Chapman andHall.