Random effects in generalized linear models and the em algorithm

Nelder and Wedderburn (1972) gave a practical fitting procedure that encompassed a more general family of data distributions than the Gaussian distribution and provided an easily understood conceptual framework. In extending the framework to more than one error structure the technical difficulties of the fitting procedure have tended to cloud the concepts, Here we show that a simple extension to the fitting procedure is possible and thus pave the way for a fuller examination of mixed effects models in generalized linear model distributions. It is clear that we should not, and do not have to, confine ourselves to fitting random effects using the Gaussian distribution. In addition, in some quite general mixing distribution problems the application of the EM algorithm to the complete data likelihood leads to iterative schemes that maximize the marginal likelihood of the observed data variable.