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

Methodology, vol. 19, pp. 60–95

Abstract

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.

Authors 4

1. Benedikt Langenberg

   Bielefeld University

   Affiliation as printed

   Psychological Methods, Faculty of Psychology and Sports Science, Bielefeld University, Bielefeld, Germany

2. Jonathan Lee Helm

   San Diego State University

   Affiliation as printed

   Department of Psychology, San Diego State University, San Diego, CA, USA

3. Thomas Günther Aachen Department of Child and Adolescent Psychiatry Child Neuropsychology Section Institute of Psychology Psychological Diagnostics and Intervention

   RWTH Aachen University

   Affiliation as printed

   Child Neuropsychology Section, Department of Child and Adolescent Psychiatry, Psychotherapy and Psychosomatics, RWTH Aachen University, Aachen, Germany; Psychological Diagnostics and Intervention, Institute of Psychology, RWTH Aachen University, Aachen, Germany

   Child Neuropsychology Section, Department of Child and Adolescent Psychiatry, Psychotherapy and Psychosomatics, RWTH Aachen University, Aachen, Germany

   Psychological Diagnostics and Intervention, Institute of Psychology, RWTH Aachen University, Aachen, Germany

4. Axel Mayer

   Bielefeld University

   Affiliation as printed

   Psychological Methods, Faculty of Psychology and Sports Science, Bielefeld University, Bielefeld, Germany