An Example-Based Tour in Linear Mixed Models

Abstract

Linear models provide an appropriate framework for studying the relation between a continuous dependent variable (response) and one or more independent variables (factors or covariates). For analysis of variance models an independent variable typically takes as value a level of some factor, e. g., the independent variable can be a specific treatment from a given number of possible treatments. We therefore could think about the independent variable as a discrete variable. For regression models we typically interpret the covariate as a continuous variable. In this chapter the focus is on ANOVA models, except for Example 2.4 where we model a linear trend. If interest is restricted to the factor levels included in the study we call the ANOVA model a fixed-effects model. In many situations however the factor levels in the study can be considered as randomly selected from a population of all possible factor levels. If all factors in the model have this interpretation we call the ANOVA model a random-effects model. If some factors are fixed and other factors are random we say that the ANOVA model is a mixed-effects model. For fixed-, random-, and mixed-effects models we assume that the relation between the dependent variable and the factors is linear in the parameters. This collection of models is referred to as the class of general linear mixed models. Such models are extremely relevant in applied statistics. Applications in clinical trials, agriculture, economy,… are numerous. Before we give a formal description of general linear mixed models we discuss in more detail the difference between fixed and random effects. Our discussion will be example-based. In statistical analysis fixed and random effects are handled in a different way, therefore a clear understanding of differences and similarities is needed for good modeling practice.