A goodness-of-fit test for overdispersed binomial (or multinomial) models

Overdispersion or extra variation is a common phenomenon that occurs when binomial (multinomial) data exhibit larger variances than that permitted by the binomial (multinomial) model. This arises when the data are clustered or when the assumption of independence is violated. Goodness-of-fit (GOF) tests available in the overdispersion literature have focused on testing for the presence of overdispersion in the data and hence they are not applicable for choosing between the several competing overdispersion models. In this paper, we consider a GOF test proposed by Neerchal and Morel [1998. Large cluster results for two parametric multinomial extra variation models. J. Amer. Statist. Assoc. 93(443), 1078–1087], and study its distributional properties and performance characteristics. This statistic is a direct analogue of the usual Pearson chi-squared statistic, but is also applicable when the clusters are not necessarily of the same size. As this test statistic is for testing model adequacy against the alternative that the model is not adequate, it is applicable in testing two competing overdispersion models.