Statistical power and sample size requirements to detect an intervention by time interaction in four-level longitudinal cluster randomized trials.

Cluster/group randomized controlled trials (CRTs) have a long history in the study of health sciences. CRT is a special type of intervention trial in which a complete group is randomly assigned to a study condition (or intervention). It is typically performed when individual randomization is difficult/impossible without substantial risk of contamination across study arms or prohibitive from the cost or group dynamics point of view. In this article, the aim is to design and analyze four-level longitudinal cluster randomized trials. The main interest here is to study the difference between treatment groups over time for such a four-level hierarchical data structure. This work is motivated by a real-life study for education based HIV prevention. Such trials are not only popular for administrative convenience, ethical considerations, subject compliance, but also help to reduce contamination bias. A random intercept mixed effects linear regression including a time by intervention interaction is used for modeling. Closed form expression of the power function to detect the interaction effect is determined. Sample size equations depend on correlation among schools but not on correlations among classes or students while, the power function depends on the product of number of units at different levels. Optimal allocation of units under a fixed cost by minimizing the expected standardized variance is also determined and are shown to be independent of correlations among units in any level. Results of detailed simulation studies find the theoretical power estimates based on the derived formulae close to the empirical estimates.