# Power Analysis for repeated measures ANOVA

I am trying to run a post-hoc power analysis on a previously published report. The intent is to calculate the average obtained power for (the equivalent of) Cohen's effects of d = .20, .50, .80) for all main effects and contrasts.

The study design is as follows: N = 240, repeated measures mixed (split-plot) 2x4(x4) ANOVA. To simplify, the first two factors are between-subjects. Factor A (gender) has 2 levels (male, female obviously), Factor B (ethnicity) has 4 levels totalling 8 experimental groups. Groups are balanced with n = 30 at the smallest unit of observation. Each of the 8 groups performs the same task under 4 different conditions. No information about the degree to which the responses are correlated is given. Pretty straightforward. Hopefully I haven't left anything out.

I can use STATA, GPower, or other online sources to calculate power for one and two way repeated measures ANOVA, but I haven't found a way to do this particular design.

Has anyone had any luck configuring either R, STATA syntax, or done some other voodoo to calculate power for this design?

• How do you plan to set one d for all main effects and contrasts simultaneously? – John Nov 19 '13 at 1:35
• Oh, I don't. I'm finding effect sizes for each separately: a different one for each main effect and interaction. Then, these effect sizes are weighted and averaged so that all tests (that test main hypotheses) that appear in any one article have a single average power for each of the three cohens effect sizes. Cohen actually came up with this procedure, and it's been used multiple times in various disciplines as a meta-analysis of sorts. – CircusMaximus Nov 19 '13 at 10:45

To calculate the effect size for a 2-way repeated ANOVA on both factors, you can use two formulas:

$$\eta^2_{partial} =\frac{SS} {SS+SS_{Error}}$$ where $SS$ is the sum of squares.

The number you'll get, must then be multiplied with 100, so you'll have the percentage of explained influence of the factor on your dependent variable.

The other way to do this, is with the formulas:

$f^2 = F \cdot \frac{ df}{df_{error}} \rightarrow \eta^2_{partial} = \frac{f^2} {1+f^2}$

Likewise you'll have to multiple your result with 100, so you'll get the explained variance.

Both these formulas tell you about the effect size for your sample, but not for the population.

• I formatted your formulae using MathJax. Please check that I translated the second one and third one correctly, and edit if I did not. – Marquis de Carabas May 4 '16 at 20:37