Incorporating cost in power analysis for three-level cluster-randomized designs

In experimental designs with nested structures, entire groups (such as schools) are often assigned to treatment conditions. Key aspects of the design in these cluster-randomized experiments involve knowledge of the intraclass correlation structure, the effect size, and the sample sizes necessary to achieve adequate power to detect the treatment effect. However, the units at each level of the hierarchy have a cost associated with them and thus researchers need to decide on sample sizes given a certain budget, when designing their studies. This article provides methods for computing power within an optimal design framework that incorporates costs of units in all three levels for three-level cluster-randomized balanced designs with two levels of nesting at the second and third level. The optimal sample sizes are a function of the variances at each level and the cost of each unit. Overall, larger effect sizes, smaller intraclass correlations at the second and third level, and lower cost of Level 3 and Level 2 units result in higher estimates of power.