Several-samples repeated measures tests

In repeated measures ANOVA, values in each row are observations on the same “subject”. Repeated measures ANOVA is the extension of the paired t test to several samples. Each column (sample) must contain the same number of values.

Missing values are not supported.

In addition to the parametric F test, a permutation test with 9999 random permutations is also given.

For mathematical details, see the Past manual.

Sphericity estimates and corrections

An assumption of repeated measures ANOVA is sphericity, meaning equal variances of the differences between all combinations of groups. A statistic called epsilon approaches 1 for data meeting the sphericity assumption. For smaller values of epsilon, a correction can be applied to the degrees of freedom of the F test, providing a corrected p value for the ANOVA. PAST provides two versions of this procedure, Greenhouse-Geisser (Greenhouse & Geisser 1959) and Huynh-Feldt (Huynh & Feldt 1976).

Tukey’s pairwise post-hoc tests

The "post-hoc" pairwise comparisons are based on the Tukey test. The Studentized Range Statistic Q is given in the lower left triangle of the array, and the probabilities p(equal) in the upper right.

Friedman test

The Friedman test is a non-parametric test for equality of medians in several repeated-measures univariate groups. It can be regarded as the non-parametric version of repeated-measures ANOVA, or the repeated-measures version of the Kruskal-Wallis test (Bortz et al. 2000).

The post hoc tests are by simple pairwise Wilcoxon, exact for n<20, asymptotic for n>=20. These tests have higher power than the Friedman test.


Bortz, J., Lienert, G.A. & Boehnke, K. 2000. Verteilungsfreie Methoden in der Biostatistik. 2nd ed. Springer.

Greenhouse, S.W. & Geisser, S. 1959. On methods in the analysis of profile data. Psychometrika 24:95-112.

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 1:69-82.

Published Aug. 31, 2020 9:11 PM - Last modified Aug. 22, 2021 4:22 PM