How to proceed when both normality and sphericity are violated in repeated measures ANOVA

Abstract: Adjusted F-tests have typically been proposed as an alternative to the F-statistic in repeated measures ANOVA. Despite considerable research, it remains unclear how these statistics perform under simultaneous violation of normality and sphericity. Accordingly, our aim here was to conduct a detailed examination of Type I error and power of the F-statistic and the Greenhouse-Geisser (F-GG) and Huynh-Feldt (F-HF) adjustments, manipulating the number of repeated measures (3-6), sample size (10-300), sphericity (Greenhouse-Geisser epsilon estimator, , from its lower to upper limit), and distribution shape (slight to extreme deviations from normality). The findings show that the behavior of F-GG and F-HF depends on the degree of violation of both normality, sphericity, and sample size. Overall, we suggest using F-GG under violation of sphericity and slight or moderate deviations from normality in all sample size; with severe deviations from both normality and sphericity F-GG may be used with a sample size larger than 10; and with extreme deviation from both normality and sphericity this statistic may be used with a sample size larger than 30. In the event of discrepant results between F-GG and F-HF, the choice depends on the value.