AUTHOR=Blanca María J. , Arnau Jaume , García-Castro F. Javier , Alarcón Rafael , Bono Roser 

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

JOURNAL=Frontiers in Psychology

VOLUME=14

YEAR=2023

URL=https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2023.1192453

DOI=10.3389/fpsyg.2023.1192453

ISSN=1664-1078

ABSTRACT=<sec id="sec1001"><title>Introduction</title><p>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 <italic>F</italic>-tests, which are frequently employed because they are intuitive, easy to apply, and available in most statistical software. Adjusted <italic>F</italic>-tests differ in the procedure used to estimate the corrective factor <italic>ε</italic>, the most common being the Greenhouse-Geisser (<italic>F-GG</italic>) and Huynh-Feldt (<italic>F-HF</italic>) adjustments. Although numerous studies have analyzed the robustness of these procedures, the results are inconsistent, thus highlighting the need for further research.</p></sec><sec id="sec2001"><title>Methods</title><p>The aim of this simulation study was to analyze the performance of the <italic>F</italic>-statistic, <italic>F-GG</italic>, and <italic>F-HF</italic> 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 <italic>ε</italic> were fixed according to the Greenhouse–Geisser procedure (<inline-formula><mml:math id="M1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo>̂</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>). We manipulated the number of repeated measures (3, 4, and 6) and sample size (from 10 to 300), with <inline-formula><mml:math id="M2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo>̂</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula> values ranging from the lower to its upper limit.</p></sec><sec id="sec3001"><title>Results</title><p>Overall, the results showed that the <italic>F</italic>-statistic becomes more liberal as sphericity violation increases, whereas both <italic>F-HF</italic> and <italic>F-GG</italic> control Type I error; of the two, <italic>F-GG</italic> is more conservative, especially with large values of <inline-formula><mml:math id="M3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo>̂</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula> and small samples.</p></sec><sec id="sec4001"><title>Discussion</title><p>If different statistical conclusions follow from application of the two tests, we recommend using <italic>F-GG</italic> for <inline-formula><mml:math id="M4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo>̂</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula> values below 0.60, and <italic>F-HF</italic> for <inline-formula><mml:math id="M5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo>̂</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula> values equal to or above 0.60.</p></sec>