Modeling the effects of overdispersion in generalized linear models

Distribution free models are developed for analyzing overdispersion where both the mean and dispersion parameters are modeled. Some of these models may require some specification of the variance-mean relationship of the response variable and as such belong to the class of moment methods.;The quasi-likelihood and extended quasi-likelihood methods belong to this class of moment methods. The quasi-likelihood method allows for modeling the mean parameters without any structural form for the dispersion, which may be regarded as nuisance parameters. The extended quasi-likelihood method allows for structured dispersion models where the dispersion parameters can be modeled by including a set of explanatory variables in the same manner as in the modeling of the mean parameters. However, in the modeling of the dispersion parameters, the extended quasi-likelihood function implicitly assumes that the distribution of the dispersion sub-model is that of a gamma random variable.;The structured modeling of the dispersion parameters with the mean parameters is a method for accounting for overdispersion. The generalized extended quasi-likelihood function provides techniques of modeling overdispersion through the joint modeling of the mean and dispersion parameters. The function does not assume any distribution for the dispersion sub-model. The variance function in the dispersion sub-model is presented as a power function form where the parameter in its exponent can be estimated using the information contained in the data.;A majority of the techniques for modeling overdispersion developed here and elsewhere are applied to the Ames Salmonella data (Margolin, Kaplan, and Zeiger, 1981) and the "Seed" data (Crowder, 1978). The analyses suggest that the extensions of the quasi-likelihood present reliable and useful techniques. A simulation study is performed to examine the efficiency of proposed and existing methods of overdispersion modeling.