Testing for Overdispersion in Poisson and Binomial Regression Models

Abstract

In this article a method for obtaining tests for overdispersion with respect to a natural exponential family is derived. The tests are designed to be powerful against arbitrary alternative mixture models where only the first two moments of the mixed distribution are specified. Various tests for extra-Poisson and extra-binomial variation are obtained as special cases; the use of a particular test may be motivated by a consideration of the mechanism through which the overdispersion may arise. The common occurrence of extra-Poisson and extra-binomial variation has been noted by several authors. However, the Poisson and binomial models remain valid in many instances and, because of their simplicity and appeal, it is of real interest to ascertain when they apply. This paper develops a unifying theory for testing for overdispersion and generalizes tests previously derived, including those by Fisher (1950), Collings and Margolin (1985), and Prentice (1986). It also shows the Pearson statistic to be a score test for overdispersion in a certain situation.

Key Words:

• Count data
• Exponential family
• Extra-binomial variation
• Extra-Poisson variation
• Mixture model
• Proportion