Suppose we have a three way interaction of three factors called age, sex and type. Age and sex have two levels and are between subject and type has four levels and 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 the power to detect an eta-squared of at least 0.10 with a total sample size of 20 and a Type I error of 5%.

There are two between subjects factors (age, sex) each with 2 levels so b1 = b2 = 2. Their interaction (which makes up the term of interest) has (2-1)*(2-1)=1 df. There is one within subject factor (type) with 4 levels so w1 = 4. We can now use these to work out our inputs.

num = numerator df = (2-1)(2-1)(4-1)=3

bsum = (2-1) + (2-1) + (2-1)*(2-1) = 3 (sum of dfs for age, sex and age*sex interaction)

wdf = (4-1) = 3 (for type)

Now putting these together and taking the conservative assumption that the types are uncorrelated with an alpha=0.05, partial eta-squared of 0.10 and a total sample size of 20 gives a power of 0.43.

If we assume the types have an average of correlation of 0.25 then a partial eta-squared of 0.10 and a total sample size of 20 has a power of 0.56.

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.