Linear Mixed-Effects Models: Basic Concepts and Examples

Abstract

Many common statistical models can be expressed as linear models that incorporate both fixed effects, which are parameters associated with an entire population or with certain repeatable levels of experimental factors, and random effects, which are associated with individual experimental units drawn at random from a population. A model with both fixed effects and random effects is called a mixed-effects model. Mixed-effects models are primarily used to describe relationships between a response variable and some covariates in data that are grouped according to one or more classification factors. Examples of such grouped data include longitudinal data, repeated measures data, multilevel data, and block designs. By associating common random effects to observations sharing the same level of a classification factor, mixed-effects models flexibly represent the covariance structure induced by the grouping of the data. In this chapter we present an overview of linear mixed-effects (LME) models, introducing their basic concepts through the analysis of several real-data examples, starting from simple models and gradually moving to more complex models. Although the S code to fit these models is shown, the purpose here is to present the motivation for using LME models to analyze grouped data and not to concentrate on the software for fitting and displaying the models. This chapter serves as an appetizer for the material covered in later chapters: the theoretical and computational methods for LME models described in Chapter 2 and the linear mixed-effects modeling facilities available in the nlme library, covered in detail in Chapter 4. The examples described in this chapter also serve to illustrate the breadth of applications of linear mixed-effects models. 4 1. Linear Mixed-Effects Models