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 the df for both age and sex equals (2-1)=1. Their interaction (which makes up the term of interest) has a df of (2-1)*(2-1)=1. There is one within subject factor (type) with 4 levels so the df for type is (4-1)=3. We can now use these to work out our inputs.
num = numerator df of age by sex by type interaction = (2-1)(2-1)(4-1)= 3
bsum = (2-1) + (2-1) + (2-1)*(2-1) = 3 (sum of dfs for between subject factor terms: age, sex and age*sex interaction)
wdf = (4-1) = 3 (df for the within subject factor type)
Now putting these together, taking the conservative assumption that the types are uncorrelated, an alpha of 0.05, partial eta-squared of 0.10 and a total sample size of 20 gives a power of 0.43.
If we assume that the average correlation between a pair of types is 0.25 then an alpha of 0.05, 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.