Hierarchical Generalized Linear Models

Abstract

SUMMARY We consider hierarchical generalized linear models which allow extra error 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 linear mixed models. We use a generalization of Henderson's joint likelihood, called a hierarchical or h-likelihood, for inferences from hierarchical generalized linear models. This avoids the integration that is necessary when marginal likelihood is used. Under appropriate conditions maximizing the h-likelihood gives fixed effect estimators that are asymptotically equivalent to those obtained from the use of marginal likelihood; at the same time we obtain the random effect estimates that are asymptotically best unbiased predictors. An adjusted profile h-likelihood is shown to give the required generalization of restricted maximum 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 generalized to the new class. We give examples of the Poisson–gamma, binomial–beta and gamma–inverse gamma hierarchical generalized linear models. A resolution is proposed for the apparent difference between population-averaged and subject-specific models. A unified framework is provided for viewing and extending many existing methods.