Suppose we have a three way interaction of three factors called age, sex and type. Each factor has two levels, age and sex are between subject and type is within subject.

Pilot data has suggested an effect size, partial eta-squared, of 0.10 as worthy of interest. We wish to do a power calculation to see how many people we will need to detect an eta-squared of at least 0.10 with a power of 0.80 and a Type I error of 5%.

There are two between subjects factors (age, sex) with 2 levels each do $$b_{1}=b_{2}$$ = 2. There is one within subjects factor (type) with 4 levels so $$w_{1}$$ = 4. We can now use these to work out our inputs.

num = numerator df = (2-1)(2-1)(2-1)=1

d1 = (2-1) + (2-1) + (2-1)*(2-1) = 3

d2 = (2-1)(2-1) = 1

prod = 4

Putting in these together, assuming conservatively no correlation between types, with an alpha=0.05, partial eta-squared of 0.10 and a power of 0.80 gives a total sample size of 38 required.

If we assume an average correlation of 0.25 amongst the types the partial eta-squared rises to 0.14 and we only need a total sample size of 27 with a power of 0.80.

Reference

Faul, F. & Erdfelder, E. (1992) GPOWER: A priori, post-hoc, and compromise power analyses for MS-DOS [Computer program]. Bonn, Germany: Bonn University, Dep. of Psychology.