Adjusted Likelihood Methods for Modelling Dispersion in Generalized Linear Models∗

This paper considers double generalized linear models, which allow the mean and dispersion to be modelled simultaneously in a generalized linear model context. Estimation of the dispersion parameters is based on a χ1 approximation to the unit deviances, and the accuracy of the saddle-point approximation which underlies this is discussed. Approximate REML methods are developed for estimation of the dispersion, and these are related to the likelihood adjustment methods of McCullagh and Tibshirani (1990) and Cox and Reid (1987). The approximate REML methods can be implemented with very little added complication in a generalized linear model setting by adjusting the working vector and working weights. S-Plus functions for double generalized linear models are described. Through two data examples it is shown that the approximate REML methods are more robust than maximum likelihood, in the sense of being less sensitive to perturbations in the mean model.