Understanding, testing, and relaxing sphericity of repeated measures ANOVA with manifest and latent variables using SEM

This article demonstrates how to perform univariate repeated measures ANOVA (U-RM-ANOVA) as a special case of structural equation models (SEMs). In the literature, sphericity is usually defined in terms of variances of pairwise differences of within-subject conditions. This article illustrates the original definition by Huynh and Feldt (1970) in terms of (co)variances of contrasts using SEM and demonstrates how to impose, test, and relax sphericity, and how to test main/interaction effects with and without the assumption of sphericity in SEM. We perform two simulation studies. The first study compares Mauchly’s sphericity test with an SEM based test and shows that the two approaches have a very similar Type 1 error and power. The second study compares U-RM-ANOVA with SEM for different degrees of departure from sphericity and shows that U-RM-ANOVA and SEM have similar statistical properties in terms of Type 1 error, power, as well as similar bias and efficiency of effect size estimates of main and interaction effects. We furthermore show how to implement sphericity in latent variable models and provide software to perform the proposed tests and analyses.