The study presents a series of alternative ANOVA-based methods which offered a remedy to the traditional F test to accommodate violations of normality and sphericity assumptions. Specially Robust Means Modeling (RMM), developed from structured means modeling (SMM), a branch of structural equation modeling (SEM), is introduced to circumvent the sphericity assumption while alleviating the violation of normality assumption. Maximum likelihood, Satorra-Bentler scaled chi-square, asymptotic distribution-free (ADF) methods and its corrections, as well as residual-based ADF methods (RES) and its corrections, are included in this RMM category.

A Monte Carlo simulation is designed to evaluate Type I error robustness and power of the ANOVA-based methods and the proposed RMM methods under the fully crossed conditions including degree of non-sphericity, degree of non-normality, sample size, and the number of levels of the repeated measures. The study gains strong ground for RMM methods to be recommended over ANOVA-based methods under almost all conditions except when the model was complex (i.e. 8 levels) and sample size was small (15, 30).