In this chapter we discuss in more detail the mixed model with normally distributed errors. We will refer to this as the 'normal mixed model'. Of course, this does not imply that values of the response variables follow normal distributions, because they are, in fact, mixtures of effects with different means. In practice, though, if a variable appears to have a normal distribution, the assumption of normal residuals and random effects is often reasonable.
In the examples introduced in Sections 1.1-1.4 we defined several mixed models using a notation chosen to suit each situation. In Section 2.1 we define the mixed model using a general matrix notation which can be used for all types of mixed model. Matrix notation may at first be unfamiliar to some readers and it is outwith the scope of this book to teach matrix algebra. A good introductory guide is Matrices for Statistics by Healy (1986). Once grasped, though, matrix notation can make the overall theory underlying mixed models easier to comprehend. Mixed models methods based on classical statistical techniques are described in Section 2.2, and in Section 2.3 the Bayesian approach to fitting mixed models will be introduced. These two sections can be omitted by readers who do not desire a detailed understanding of the more theoretical aspects of mixed models. In Section 2.4 some practical issues related to the use and interpretation of mixed models are considered, and a worked example illustrating several of the points made in Section 2.4 is described in Section 2.5. For those who wish a more in-depth understanding of the theory underlying mixed models the textbook Mixed Models: Theory and Applications byDemidenko (2004) is recommended.
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