Predictors can be fixed or random, and their classification affects how we fit and interpret statistical models. A mismatch between their treatment in the model and their interpretation is a common problem. This chapter focuses on categorical predictors and introduces nested or hierarchical designs that combine fixed and random effects. For these designs, we distinguish between those where the random effects correspond to replicate experimental and sampling units and those that also include multiple observational units within each replicate. We also consider factorial mixed models and introduce hybrid designs that combine factorial and nested components. We describe the fitting of these models using traditional “ANOVA” approaches using OLS and present an alternative approach used in the following chapters – linear mixed models or multilevel models. These modeling approaches are illustrated for multilevel nested designs and factorial designs with and without replication.

The analysis of variance is introduced as a method of testing differences among means of more than two groups of observations. We outline the basic assumptions of ANOVA models, focusing on the expected homogeneity of variances across the compared groups, which is assessed by the Bartlett test. The decomposition of variability in the response variable (its total sum of squares) into among-group and within-group (residual) variation leads to the definition of the F-ratio, which is the central test statistic in ANOVA models. We also introduce the distinction between fixed and random effects and discuss the F test power as well as the robustness of the test to violations of ANOVA model assumptions. The first part of the chapter, dealing with one-way ANOVA, concludes with a description of the multiple comparisons procedure. We focus on two types - Tukey's test and Dunnett's test. This chapter concludes by presenting a nonparametric counterpart of one-way ANOVA, the Kruskal-Wallis test. The methods described in this chapter are accompanied by a carefully-explained guide to the R code needed for their use, including the multcomp package.