# 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.

#### References

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.