Linear Mixed-Effects Models in Cluster-Randomized Studies

Abstract

Mixed-effects models are used in behavioral research to analyze hierarchical data. Hierarchical data can be repeated measures data (e.g., repeated assessments nested within study participants), longitudinal data (e.g., longitudinal data collected at staggered time intervals, nested within individual participants), or multilevel data (e.g., group-randomized clinical trials in which participants are nested within psychotherapy groups; individuals within couple dyads in couples therapy; students nested within classes; and classes nested within schools). An important statistical issue is that data units that belong to the same clusters are likely to show correlated outcome endpoints. This Intra-Class Correlation (ICC) must be taken into account in the data analysis. This chapter contains an example analysis from Flay et al. (1995) in testing the efficacy of behavioral interventions in a group-randomized trial. This chapter is organized as follows. A brief summary of the Flay et al. (1995) study design and data structure is presented first. Next we present data visualization and exploratory analyses. The intervention effects are tested in a linear mixed-effects model. The same model is also explained in a multilevel modeling approach as described in Bryk and Raudenbush (2002). Data analyses primarily rely on the lme4 package by Douglas Bates, Martin Maechler, and Ben Bolker; additional analyses are done using the nlme package by Jose Pinheiro, Douglas Bates, Saikat DebRoy, Deepayan Sarkar, and the R Core team. The ICCs in the study are calculated. A matrix representation of the linear mixed model is described in Sect. 11.6, in part because the matrix notation is the standard modeling notation in the literature. We conclude this chapter with an example on how to estimate, using simulations, the statistical power of a study design with clusters.