ABSTRACT
In Chapters 7 to 9, we showed how to analyse data arising from designed experiments using multi-stratum ANOVA to take proper account of structure in the experimental units and that this approach led to appropriate estimates of parameter standard errors. However, multi-stratum ANOVA does not apply to unbalanced structures, and so its use is limited. In Chapter 11, we saw that combining the explanatory and structural components of the model – the so-called intra-block analysis – gives good results only for certain types of structure. We therefore need a more general approach to account for structure, and in this chapter, we introduce the class of linear mixed models. This class extends multi-stratum ANOVA to the cases of unbalanced and non-orthogonal structures, and extends regression models to include a structural component. We start with a short discussion of the need to include structure in models (Section 16.1) and then give a more formal denition of the linear mixed models that we use to achieve this (Section 16.2). We then describe methods for investigating the explanatory component of the model (Section 16.3) and aspects of the structural component (Sections 16.4 and 16.5) before considering prediction (Section 16.6) and model checking (Section 16.7). We analyse a data set in some detail to illustrate the concepts discussed in the previous sections (Section 16.8), and we explain some of the difculties that can be encountered with this more general form of model (Section 16.9). Finally, we give a general overview of extensions to this class of models (Section 16.10).
