Generalized Linear Mixed Models

Abstract

For analyzing repeated measures data, the necessity of considering the relationships between outcome variables as well as between outcome variables and explanatory variable are of concern. We have discussed about such models in previous chapters. All the models proposed in various chapters are fixed effect models. However, in some cases, the dependence between outcomes from repeated observations for each cluster or group as well as explanatory variables may not be adequate if a population-averaged marginal model based on a fixed effect model is considered. As the joint dependence model is ignored in modeling for different groups or clusters in a population-averaged fixed effect model, an alternative approach is to consider random variation in groups or clusters in addition to fixed marginal effects. In Chap. 12, GEE is introduced as an extension of GLM based on quasi-likelihood methods. In GEE, we have considered repeated observations in groups for each subject and a fixed effect population-averaged model is shown which is represented by the link function $g(\mu_{ij} ) = X_{ij} \beta$ where $i = 1,\ldots,n\,{\text{and}}\,j = 1,\ldots,J_{i}$. In this chapter, an extension to generalized mixed model is discussed by introducing random effect for clusters. Examples are displayed for identity, logit, and log link functions. The mixed models are also shown in case of multinomial data for both nominal or ordinal categories. Several examples are included in this chapter to highlight the estimation and test procedures.