Mixed Models and Variance Components

Abstract

Traditionally, linear models have been divided into three categories: fixed effects models, random effects models, and mixed models. The categorization depends on whether the β vector in Y = Xβ + e is fixed, random, or has both fixed and random elements. Random effects models always assume that there is a fixed overall mean for observations, so random effects models are actually mixed. Variance components are the variances of the random elements of β. Sections 1 through 3 discuss mixed models in general and prediction for mixed models. Sections 4 through 9 present methods of estimation for variance components. Section 10 examines exact tests for variance components. Section 11 uses the ideas of the chapter to develop the interblock analysis for balanced incomplete block designs. Searle, Casella, and McCulloch (1992) give an extensive discussion of variance component estimation. Khuri, Mathew, and Sinha (1998) give an extensive discussion of testing in mixed models. The methods considered in this chapter are presented in terms of fitting general linear models. In many special cases, considerable simplification results. For example , the RCB models of (11.1.5) and Exercise 11.4, the split plot model (11.3.1), and the subsampling model (11.4.2) are all mixed models with very special structures. 12.1 Mixed Models The mixed model is a linear model in which some of the parameters, instead of being fixed effects, are random. The model can be written Y = Xβ + Zγ + e, (1) where X and Z are known matrices, β is an unobservable vector of fixed effects, and γ is an unobservable vector of random effects with E(γ) = 0, Cov(γ) = D, and Cov(γ, e) = 0. Let Cov(e) = R.