A comparison of tests for overdispersion in generalized linear models

Abstract

This paper compares performances of several statistics in tests of overdispersion; in particular, tests based on the score statistics proposed by Smith and Heitjan (1993) and Dean (1992) for testing overdispersion in binomial and Poisson generalized linear models and a test based on the Pearson χ² statistic. The latter has been used by several authors as a measure of overdispersion (e.g., McCullagh and Nelder, 1989). We find that, when nominal type I error rates are acceptable, the power of the score tests can be significantly less than that with the Pearson statistic, especially for binomial data. The Pearson test, however, proves to be liberal for Poisson and binomial data containing a large number of zeros and other small counts. In this latter situation, the score tests prove to have better power compared to the Pearson statistic although they do have somewhat conservative type I errors. As well, inclusion of the small sample correction term suggested by Dean (1992) produced a test which either was unacceptably liberal in performance, provided little improvement over the uncorrected version or proved inferior to the Pearson statistic. Our results also indicate that substitution of the observed information matrices in place of the expected information matrices in the score tests results in tests which are extremely liberal. In addition, we extend the score tests to multinomial generalized linear models, but discover the computing effort required to explore power to be impractical for data sets of the type that we are interested in analyzing. Finally, one serious disadvantage of the Smith and Heitjan score tests compared to the Pearson χ² test and Dean's score test noted is that the former are not invariant under covariate recodings.

Keywords:

• Extra-binomial variation
• extra-poisson variation
• the pearson statistic
• score tests