A Review of Linear Mixed Models

Abstract

In this chapter we will introduce the basic concepts and matrix algebra methods used to perform linear mixed models analysis. For readers who are more familiar with traditional analysis of variance (ANOVA) based on ordinary least squares methods, we first will review the ANOVA and compare ANOVA to mixed models analysis to help introduce this topic. We will show that under certain conditions, results from ANOVA and mixed models analysis are largely equivalent, but that when data are unbalanced or when we want to relax certain assumptions of the ANOVA, the mixed model analysis has properties that make it preferable. In this section, we focus on hypothesis testing and estimation using an empirical data set to show how these analyses are conducted for different methods and for different software packages. Only a few details of the mathematical machinery involved in the mixed models analysis will be covered here. A more detailed description of mixed model theory will be covered in later sections of the book. For readers interested in a more formal treatment of the argument details, they can be found in Sorensen and Gianola (Likelihood, Bayesian, and MCMC Methods in Quantitative Genetics. Springer Science & Business Media, 2007). Details of ANOVA for balanced and unbalanced data can be found in Rawlings et al. (Applied regression analysis: A research tool. New York: Springer, 2001) and Milliken and Johnson (Analysis of messy data, Designed Experiments (Vol. 1). Boca Raton: Chapman & Hall/CRC, 2004).