Repeated measures ANOVA calculator with sphericity checks.

What is a repeated measures ANOVA?

A repeated measures ANOVA tests whether the mean of a numeric outcome changes across three or more measurements on the same people, such as a pre-test, post-test and follow-up. Because each person acts as their own control, differences between people are removed from the error term, which makes the test more powerful.

Formula

SS error is what remains after removing the variation between conditions and between participants. Partial η² = SS conditions ÷ (SS conditions + SS error).

Sphericity and corrections

Sphericity means the variances of the differences between all pairs of conditions are equal. Mauchly's test checks it. If p < .05, multiply both degrees of freedom by the Greenhouse-Geisser ε (or Huynh-Feldt ε when ε is above about .75) and report the corrected p value.

Worked example: three time points

Ten participants were measured at pre-test (M = 4.50), post-test (M = 5.60) and follow-up (M = 7.50).

1. Mauchly's test: W = .869, χ²(2) = 1.12, p = .571, so sphericity holds.
2. F(2, 18) = 62.82, p < .001.
3. Partial η² = .87, a very large effect.
4. Bonferroni post hoc tests then show which time points differ.

How to report it (APA style)

“Mauchly's test indicated that the assumption of sphericity was met, χ²(2) = 1.12, p = .571. A one-way repeated measures ANOVA showed that scores differed significantly across time, F(2, 18) = 62.82, p < .001, partial η² = .87.”

References

• Greenhouse, S. W., & Geisser, S. (1959). On methods in the analysis of profile data. Psychometrika, 24(2), 95–112.
• Huynh, H., & Feldt, L. S. (1976). Estimation of the Box correction for degrees of freedom from sample data in randomized block and split-plot designs. Journal of Educational Statistics, 1(1), 69–82.
• Mauchly, J. W. (1940). Significance test for sphericity of a normal n-variate distribution. Annals of Mathematical Statistics, 11(2), 204–209.