Poisson Overdispersion Estimates Based on the Method of Asymmetric Maximum Likelihood

Abstract

A common difficulty in regression problems with Poisson (or binomial or other exponential family) response variables is over-dispersion: the scatter around the fitted regression is too large by the standards of Poisson variability. This article concerns the description, estimation, and testing of various patterns of overdispersion, with particular emphasis on the Poisson case. Asymmetric maximum likelihood (AML) is a method of fitting regressions for the conditional percentiles of the response variable as a function of the predictors (e.g., the conditional 90th percentile of y given x). Distances between the various regression percentiles give a direct assessment of overdispersion. The discussion is carried through in terms of an archaeological data set, where we see that the counts are overdispersed by a factor of 1.35 in one part of the covariate space, but not at all in another. Moreover, the overdispersion is about 40% larger in the positive response direction than in the negative. The AML estimates are easy to compute and relate nicely to the usual maximum likelihood estimates for generalized linear regression.

Key Words:

• Deviance
• Regression percentiles
• Tilt statistic