Power for t-test comparisons of unbalanced cluster exposure studies

Studies of individuals sampled in unbalanced clusters have become common in health services and epidemiological research, but available tools for power/sample size estimation and optimal design are currently limited. This paper presents and illustrates power estimation formulas for t-test comparisons of effect of an exposure at the cluster level on continuous outcomes in unbalanced studies with unequal numbers of clusters and/or unequal numbers of subjects per cluster in each exposure arm. Iterative application of these power formulas obtains minimal sample size needed andlor minimal detectable difference. SAS subroutines to implement these algorithms are given in the Appendices. When feasible, power is optimized by having the same number of clusters in each arm k(A) = k(B) and (irrespective of numbers of clusters in each arm) the same total number of subjects in each arm n(A)k(A) = n(B)k(B). Cost beneficial upper limits for numbers of subjects per cluster may be approximately (5/rho) -5 or less where rho is the intraclass correlation. The methods presented here for simple cluster designs may be extended to some settings involving complex hierarchical weighted cluster samples.