Repeated measures ANOVA and adjusted F-tests when sphericity is violated: which procedure is best?

María J Blanca; Jaume Arnau; F Javier García-Castro; Rafael Alarcón; Roser Bono

doi:10.3389/fpsyg.2023.1192453

Repeated measures ANOVA and adjusted F-tests when sphericity is violated: which procedure is best?

Front Psychol. 2023 Aug 30:14:1192453. doi: 10.3389/fpsyg.2023.1192453. eCollection 2023.

Authors

María J Blanca¹, Jaume Arnau², F Javier García-Castro³, Rafael Alarcón¹, Roser Bono^{2

4}

Affiliations

¹ Department of Psychobiology and Behavioral Sciences Methodology, University of Malaga, Málaga, Spain.
² Department of Social Psychology and Quantitative Psychology, University of Barcelona, Barcelona, Spain.
³ Department of Psychology, Universidad Loyola Andalucia, Seville, Spain.
⁴ Institute of Neurosciences, University of Barcelona, Barcelona, Spain.

Abstract

Introduction: One-way repeated measures ANOVA requires sphericity. Research indicates that violation of this assumption has an important impact on Type I error. Although more advanced alternative procedures exist, most classical texts recommend the use of adjusted F-tests, which are frequently employed because they are intuitive, easy to apply, and available in most statistical software. Adjusted F-tests differ in the procedure used to estimate the corrective factor ε, the most common being the Greenhouse-Geisser (F-GG) and Huynh-Feldt (F-HF) adjustments. Although numerous studies have analyzed the robustness of these procedures, the results are inconsistent, thus highlighting the need for further research.

Methods: The aim of this simulation study was to analyze the performance of the F-statistic, F-GG, and F-HF in terms of Type I error and power in one-way designs with normal data under a variety of conditions that may be encountered in real research practice. Values of ε were fixed according to the Greenhouse-Geisser procedure ( $\hat{ε}$ ). We manipulated the number of repeated measures (3, 4, and 6) and sample size (from 10 to 300), with $\hat{ε}$ values ranging from the lower to its upper limit.

Results: Overall, the results showed that the F-statistic becomes more liberal as sphericity violation increases, whereas both F-HF and F-GG control Type I error; of the two, F-GG is more conservative, especially with large values of $\hat{ε}$ and small samples.

Discussion: If different statistical conclusions follow from application of the two tests, we recommend using F-GG for $\hat{ε}$ values below 0.60, and F-HF for $\hat{ε}$ values equal to or above 0.60.

Keywords: Greenhouse-Geisser adjustment; Huynh-Feldt adjustment; Mauchly test; power; robustness.