Optimal Sampling of Units in Three-Level Cluster Randomized Designs: An ANCOVA Framework

Field experiments with nested structures assign entire groups such as schools to treatment and control conditions. Key aspects of such cluster randomized experiments include knowledge of the intraclass correlation structure and the sample sizes necessary to achieve adequate power to detect the treatment effect. The units at each level of the hierarchy have a cost associated with them, however, and thus, researchers need to take budget and costs into account when designing their studies. This article uses analysis of covariance and provides methods for computing power within an optimal design framework that incorporates costs of units at different levels and covariate effects for three-level cluster randomized balanced designs. The optimal sample sizes are a function of the variances at each level and the cost of each unit. Overall, the results suggest that when units at higher levels become more expensive, the researcher should sample units at lower levels. The covariates affect the sampling of units and the power estimates. Fewer units need to be sampled at levels where covariates explain considerable proportions of the variance.