Power Calculations for Binary Moderator in Cluster Randomized Trials

Cluster randomized trials (CRTs), or studies in which intact groups of individuals are randomly assigned to a condition, are becoming more common in the evaluation of educational programs, policies, and practices. The website for the National Center for Education Evaluation and Regional Assistance (NCEE) reveals they have launched over 30 evaluation studies in the past decade, the majority of them utilizing a randomized trial. Clearly there are a large number of randomized trials of educational programs, policies, and practices either complete or currently in the field. The overarching goal of these randomized trials is generate rigorous evidence of whether or not a program works. In order to do this, studies must also be designed to detect moderator effects. The power to detect moderator effects at the student, cluster, or site level in CRTs has received much less attention in the literature than the power for the main effect of treatment. The purpose of this paper is to extend the work on power calculations for moderator effects to include moderator effects at any level for the following 4 types of CRTs: 2-level CRT, 3-level CRT, 3-level multi-site cluster randomized trial (MSCRT), and 4-level MSCRT. In addition to providing the calculations and R code to do the calculations, we start to develop intuition around the minimum detectable effect size for moderator effects using sample sizes from CRTs in the field of education. This paper represents the next step towards building the capacity of researchers to design CRTs that move beyond the main effect of treatment. Designing studies to detect not only whether or not an intervention works, but for whom or under what circumstances is critical. The results from this study suggest that in many cases, if a study is powered to detect a reasonable main effect of treatment and it has a reasonable number of individuals per cluster, then it will also be powered to detect an individual level moderator (although not shown in this proposal).