Score Tests for Zero-inflation and Over-dispersion in Generalized Linear Models

Discrete data in the form of counts often exhibit extra variation that cannot be explained by a simple model, such as the binomial or the Poisson. Also, these data sometimes show more zero counts than what can be predicted by a simple model. Therefore, a discrete generalized linear model (Poisson or binomial) may fail to fit a set of discrete data either because of zero-inflation, because of overdispersion, or because there is zero-inflation as well as over-dispersion in the data. Previous published work deals with goodness of fit tests of the generalized linear model against zero-inflation and against over-dispersion separately. In this paper we deal with the class of zero-inflated over-dispersed generalized linear models and propose procedures based on score tests for selecting a model that fits such data. For over-dispersion we consider a general over-dispersion model and specific overdispersion models. We show that in certain cases and under certain conditions, the score tests derived using the general over-dispersion model and those developed under specific over-dispersion models are identical. Empirical level and power properties of the tests are examined by a limited simulation study. Simulations show that the score tests, in general, hold nominal levels well and have good power properties. Two illustrative examples and a discussion are presented.