ABSTRACT
Generalized linear mixed models (GLMMs) extend normal linear mixed models (NLMMs) in the same way that generalized linear models (GLMs) extend normal linear models (NLMs), that is, they permit a wide range of non-Normal responses to be modeled using random effects as ¯exibly as in the Normal case. GLMMs still consist of three components as discussed in Chapter 4 (linear predictor, link function, and error distribution), but the linear predictor contains random effect terms as discussed in Chapter 5. Most modern books explaining mixed models discuss both NLMMs and GLMMs. So the same books referenced at the beginning of Chapter 5 provide useful coverage of the models discussed in this chapter: Brown and Prescott (2006) or McCulloch and Searle (2001) from a classical perspective, and Gelman et al. (2004) or Ntzoufras (2009) from a Bayesian one. In the Bayesian literature, both NLMMs and GLMMs are often referred to as hierarchical models due to the hierarchy in the model parameters that the random effects assumption imposes. The term “hierarchical linear model” is usually reserved for the NLMM because of the non-linearity caused by the link functions typically used in GLMMs as discussed in Chapter 4. When random effects are included in the model, it is important to be aware of how a non-linear link function affects how the parameters can be interpreted. Care is needed because, unlike in linear models, averaging over the random effects on the link scale and back-transforming is not equivalent to averaging on the original scale. So, for example, when random effects are used to model differences between subjects in a clinical trial, the parameters representing the effects of treatments are subject speci˜c, that is, they represent the expected difference that would occur for an individual who was given one treatment instead of another. These should not be interpreted as population effects. The differences between these two types of inferences are often covered in more detail in books discussing longitudinal data, where marginal models are typically recommended when population effects are of interest; see Fitzmaurice et al. (2004) or Diggle et al. (2002), for example. Marginal models cannot be ˜tted using BugsXLA.
