Sample size requirements to detect an intervention by time interaction in longitudinal cluster randomized clinical trials

In designing a longitudinal cluster randomized clinical trial (cluster-RCT), the interventions are randomly assigned to clusters such as clinics. Subjects within the same clinic will receive the identical intervention. Each will be assessed repeatedly over the course of the study. A mixed-effects linear regression model can be applied in a cluster-RCT with three-level data to test the hypothesis that the intervention groups differ in the course of outcome over time. Using a test statistic based on maximum likelihood estimates, we derived closed-form formulae for statistical power to detect the intervention by time interaction and the sample size requirements for each level. Importantly, the sample size does not depend on correlations among second-level data units and the statistical power function depends on the number of second- and third-level data units through their product. A simulation study confirmed that theoretical power estimates based on the derived formulae are nearly identical to empirical estimates.