partitioning out the between-subjects variability, the F-value will increase because the sum of squares error term will be smaller resulting in a smaller MSError. It is noteworthy that partitioning variability reduces degrees of freedom from the F-test, therefore the between-subjects variability must be significant enough to offset the loss in degrees of freedom. If between-subjects variability is small this process may actually reduce the F-value. Assumptions As with all statistical analyses, specific assumptions should be met to justify the use of this test. Violations can moderately to severely affect results and often lead to an inflation of type 1 error. With the rANOVA, standard univariate and multivariate assumptions apply. The univariate assumptions are: Normality—For each level of the within-subjects factor, the dependent variable must have a normal distribution. Sphericity—Difference scores computed between two levels of a within-subjects factor must have the same variance for the comparison of any two levels. (This assumption only applies if there are more than 2 levels of the independent variable.) Randomness—Cases should be derived from a random sample, and scores from different participants should be independent of each other. The rANOVA also requires that certain multivariate assumptions be met, because a multivariate test is conducted on difference scores. These assumptions include: Multivariate normality—The difference scores are multivariately normally distributed in the population. Randomness—Individual cases should be derived from a random sample, and the difference scores for each participant are independent from those of another participant. F test As with other analysis of variance tests, the rANOVA makes use of an F statistic to determine significance. Depending on the number of within-subjects factors and assumption violations, it is necessary to select the most appropriate of three tests: Standard Univariate ANOVA F test—This test is commonly used given only two levels of the within-subjects factor (i.e. time point 1 and time point 2). This test is not recommended given more than 2 levels of the within-subjects factor because the assumption of sphericity is commonly violated in such cases. Alternative Univariate test—These tests account for violations to the assumption of sphericity, and can be used when the within-subjects factor exceeds 2 levels. The F statistic is the same as in the Standard Univariate ANOVA F test, but is associated with a more accurate p-value. This correction is done by adjusting the degrees of freedom downward for determining the critical F value. Two corrections are commonly used: the Greenhouse–Geisser