Hierarchical generalised linear models: A synthesis of generalised linear models, random-effect models and structured dispersions

Hierarchical generalised linear models are developed as a synthesis of generalised linear models, mixed linear models and structured dispersions. We generalise the restricted maximum likelihood method for the estimation of dispersion to the wider class and show how the joint fitting of models for mean and dispersion can be expressed by two interconnected generalised linear models. The method allows models with (i) any combination of a generalised linear model distribution for the response with any conjugate distribution for the random effects, (ii) structured dispersion components, (iii) different link and variance functions for the fixed and random effects, and (iv) the use of quasilikelihoods in place of likelihoods for either or both of the mean and dispersion models. Inferences can be made by applying standard procedures, in particular those for model checking, to components of either generalised linear model. We also show by numerical studies that the new method gives an efficient estimation procedure for substantial class of models of practical importance. Likelihood‐type inference is extended to this wide class of models in a unified way.