Analysis of Repeated Measures Under Second-Stage Sphericity: An Empirical Bayes Approach

The conventional multivariate analysis of repeated measures is applicable in a wide variety of circumstances, in part, because assumptions regarding the pattern of covariances among the repeated measures are not required. If sample sizes are small, however, then the estimators of the covariance parameters lack precision and, as a result, the power of the multivariate analysis is low. If the covariance matrix associated with estimators of orthogonal contrasts is spherical, then the conventional univariate analysis of repeated measures is applicable and has greater power than the multivariate analysis. If sphericity is not satisfied, an adjusted univariate analysis can be conducted, and this adjusted analysis may still be more powerful than the multivariate analysis. As sample size increases, the power advantage of the adjusted univariate test decreases, and, for moderate sample sizes, the multivariate test can be more powerful. This article proposes a hybrid analysis that takes advantage of the strengths of each of the two procedures. The proposed analysis employs an empirical Bayes estimator of the covariance matrix. Existing software for conventional multivariate analyses can, with minor modifications, be used to perform the proposed analysis. The new analysis behaves like the univariate analysis when samples size is small or sphericity is nearly satisfied. When sample size is large or sphericity is strongly violated, then the proposed analysis behaves like the multivariate analysis. Simulation results suggest that the proposed analysis controls test size adequately and can be more powerful than either of the other two analyses under a wide range of non-null conditions.