HIERARCHICAL GENERALIZED LINEAR-MODELS

We consider hierarchical generalized linear models which allow extra e rror components in the linear predictors of generalized linear models. The distribution of these components is not restricted to be normal; this allows a broader class of models, which includes generalized line ar mixed models. We use a generalization of Henderson's joint likeliho od, called a hierarchical or h-likelihood, for inferences from hierarc hical generalized linear models, This avoids the integration that is n ecessary when marginal likelihood is used. Under appropriate condition s maximizing the h-likelihood gives fixed effect estimators that are a symptotically equivalent to those obtained from the use of marginal li kelihood; at the same time we obtain the random effect estimates that are asymptotically best unbiased predictors. An adjusted profile h-lik elihood is shown to give the required generalization of restricted max imum likelihood for the estimation of dispersion components, A scaled deviance test for the goodness of fit, a model selection criterion for choosing between various dispersion models and a graphical method for checking the distributional assumption of random effects are proposed . The ideas of quasi-likelihood and extended quasi-likelihood are gene ralized to the new class. We give examples of the Poisson-gamma, binom ial-beta and gamma-inverse gamma hierarchical generalized linear model s. A resolution is proposed for the apparent difference between popula tion-averaged and subject-specific models. A unified framework is prov ided for viewing and extending many existing methods.