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
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 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo>̂</mml:mo></mml:mover></mml:mrow></mml:math> ). We manipulated the number of repeated measures (3, 4, and 6) and sample size (from 10 to 300), with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo>̂</mml:mo></mml:mover></mml:mrow></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo>̂</mml:mo></mml:mover></mml:mrow></mml:math> and small samples. Discussion If different statistical conclusions follow from application of the two tests, we recommend using F-GG for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo>̂</mml:mo></mml:mover></mml:mrow></mml:math> values below 0.60, and F-HF for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo>̂</mml:mo></mml:mover></mml:mrow></mml:math> values equal to or above 0.60.