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Hierarchical generalized linear models. (With discussion). (English) Zbl 0880.62076
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 C. R. Henderson’s joint likelihood [Biometrics 31, 423-447 (1975; Zbl 0335.62048)], 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.

##### MSC:
 62J12 Generalized linear models (logistic models)