Statistical power in complex experimental designs

We show that power and sample size tables developed by Cohen (1988, pp. 289-354, 381-389) produce incorrect estimates for factorial designs: power is underestimated, and sample size is overestimated. The source of this bias is shrinkage in the implied value of the noncentrality parameter, lambda, caused by using Cohen's adjustment to n for factorial designs Cpp. 365 and 396). The adjustment was intended to compensate for differences in the actual versus presumed (by the tables) error degrees of freedom; however, more accurate estimates are obtained if the tables are used without adjustment. The problems with Cohen's procedure were discovered while testing subroutines in DATASIM 1.2 for computing power and sample size in completely randomized, randomized-blocks, and split-plot factorial designs. The subroutines give the user the ability to generate power and sample size tables that are as easy to use as Cohen's, but that eliminate the conservative bias of his tables. We also implemented several improvements relative to ''manual'' use of Cohen's tables: (1) Since the user can control the specific values of 1-beta, n, and f used on the rows and columns of the table, interpolation is never required; (2) exact as opposed to approximate solutions for the noncentral F distribution are employed; (3) solutions for factorial designs, including those with repeated measures factors, take into account the actual error degrees of freedom for the effect being tested; and (4) provision is made for the computation of power for applications involving the doubly noncentral F distribution.