A zero-inflated overdispersed hierarchical Poisson model

Abstract

Count data are most commonly modeled using the Poisson model, or by one of its many extensions. Such extensions are needed for a variety of reasons: (1) a hierarchical structure in the data, e.g., due to clustering, the collection of repeated measurements of the outcome, etc.; (2) the occurrence of overdispersion (or underdispersion), meaning that the variability encountered in the data is not equal to the mean, as prescribed by the Poisson distribution; and (3) the occurrence of extra zeros beyond what a Poisson model allows. The first issue is often accommodated through the inclusion of random subject-specific effects. Though not always, one conventionally assumes such random effects to be normally distributed. Overdispersion is often dealt with through a model developed for this purpose, such as, for example, the negative-binomial model for count data. This can be conceived through a random Poisson parameter. Excess zeros are regularly accounted for using so-called zero-inflated models, which combine either a Poisson or negative-binomial model with an atom at zero. The novelty of this article is that it combines all these features. The work builds upon the modelling framework defined by Molenberghs et al. ( 2010 ) in which clustering and overdispersion are accommodated for through two separate sets of random effects in a generalized linear model.