Modelling Count Responses with Overdispersion

Abstract

We frequently encounter outcomes of count that have extra variation. This paper considers several alternative models for overdispersed count responses such as a quasi-Poisson model, zero-inflated Poisson model and a negative binomial model with a special focus on a generalized linear mixed model. We also explain various goodness-of-fit criteria by discussing their appropriateness of applicability and cautions on misuses according to the patterns of response categories. The overdispersion models for counts data have been explained through two examples with different response patterns. Poisson distribution is the standard model for count responses, for which the variance of responses is expected to equal mean. But violations in the assumptions characterizing a Poisson distribution may lead to the presence of extra variation in the analysis of counts and rates in longitudinal studies. There may exist a dependence between the elemental units in a study where experimental units are clusters. When responses are observed in clusters they frequently exhibit extra variation than the permitted variance of the assumed model. We often encounter this extra variation in a real data of count or binomial responses. Overdispersion may sometimes be observed when the residual variation obtained is greater than which can be attributed to the sampling variation assumed by the model. McCullagh and Nelder (1989) showed that overdispersion is not uncommon in practice. Overdis- persion should be considered deliberately in modelling count responses. The ordinary Poisson general linear model(GLM) cannot be fitted well in the presence of overdispersion. In the analysis of clus- tered or longitudinal data the generalized linear mixed model(GLMM) and the generalized estimating equations(GEE) approaches are most popular. The GLMM incorporates random subject effects into a GLM by allowing subjects variability; however, the GEE method solves equations that include the correlation structure of repeated responses to estimate regression coefficients. Thall and Vail (1990) discussed covariance models for longitudinal count data with overdisper- sion, and Jowaheer and Sutradhar (2002) applied a GEE method to analyze longitudinal count data; however, Sutradhar et al. (2007) suggested three kinds of mixture models to account for the overdis- persion in binomial data. The beta-binomial, the finite mixture, and the zero-inflated binomial model all belong to the same class of mixture models. They suggested a chi-squared type goodness-of-fit statistic to test the assumed null model against the alternative three kinds of models without con- sidering covariate variables. Recently, Morel and Neerchal (2012) have extensively studied various overdispersion models using SAS.