Random-effect models are becoming increasingly popular in the analysis of data. Lee and Nelder (2006) introduced double hierarchical generalized linear models (DHGLMs) in which not only the mean but also the residual variance (overdispersion) can be further modelled as random-effect models. In this article, we introduce DHGLMs that allow random-effect models for both the variances of random effects and the residual variance. We show how to use this general model class for the analysis of data and discuss how to select the best fitting model using the likelihood and various model-checking plots.
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