Approximate Power for Repeated-Measures ANOVA Lacking Sphericity

Abstract

Violation of sphericity of covariance across repeated measures inflates Type I error rates in univariate repeated-measures analysis of variance. Hence the use of Geisser—Greenhouse or Huynh—Feldt is now common (to provide improved Type I error rate). With nonsphericity, no method has been available to compute power. A convenient method is suggested for approximating power and test size under nonsphericity. New approximations are suggested for (a) a weighted sum of independent noncentral χ²,s, (b) the trace of a noncentral Wishart (or pseudo-Wishart) matrix, (c) the expected values of and , and (d) the noncentral test statistic, whether corrected or uncorrected. The new approximations are extensions of the work of Box (1954a,b) and Satterthwaite (1941). The method performed well when evaluated against published and new simulations.

Key Words:

• Equicorrelation
• Error rates
• Noncentral chi square
• Noncentral pseudo-Wishart
• Noncentral Wishart
• Sample size