Diagnostic tools for overdispersion in generalized linear models

Graduation date: 1990Data in the form of counts or proportions often exhibit more\ud variability than that predicted by a Poisson or binomial\ud distribution. Many different models have been proposed to account\ud for extra-Poisson or extra-binomial variation. A simple model\ud includes a single heterogeneity factor (dispersion parameter) in the\ud variance. Other models that allow the dispersion parameter to vary\ud between groups or according to a continuous covariate also exist but\ud require a more complicated analysis. This thesis is concerned with\ud (1) understanding the consequences of using an oversimplified model\ud for overdispersion, (2) presenting diagnostic tools for detecting the\ud dependence of overdispersion on covariates in regression settings for\ud counts and proportions and (3) presenting diagnostic tools for\ud distinguishing between some commonly used models for overdispersed\ud data.\ud The double exponential family of distributions is used as a\ud foundation for this work. A double binomial or double Poisson\ud density is constructed from a binomial or Poisson density and an\ud additional dispersion parameter. This provides a completely\ud parametric framework for modeling overdispersed counts and\ud proportions.\ud The first issue above is addressed by exploring the properties\ud of maximum likelihood estimates obtained from incorrectly specified\ud likelihoods. The diagnostic tools are based on a score test in the\ud double exponential family. An attractive feature of this test is\ud that it can be computed from the components of the deviance in the\ud standard generalized linear model fit. A graphical display is\ud suggested by the score test. For the normal linear model, which is a\ud special case of the double exponential family, the diagnostics reduce\ud to those for heteroscedasticity presented by Cook and Weisberg\ud (1983)