ABSTRACT
This chapter discusses how HGLMs can be fitted using interconnected and augmented GLMs. It presents a few examples illustrating how the dispersion parameters can be modeled in HGLMs. It aims to present extensions of the IWLS algorithm for GLMs to HGLMs, in light of the h-likelihood. HGLMs extend GLMs by allowing random effects in the linear predictor. It is well known that the PQL estimator is seriously biased especially for binary data. A linear mixed model can be written as a weighted least squares problem by augmenting the response vector. The weighted least squares estimator for this model is identical to Henderson's mixed model equations. The same h-likelihood framework can be used to extend the fitting algorithm and inference for Double HGLMs, where random effects are allowed in dispersion models, and these can be further extended for outcomes from different DHGLMs with different distributions, i.e., multivariate DHGLM analysis.
