Suppose we have a three way interaction of three factors called age, sex and type. Age and sex are between subject and have two levels and type is within subject and comprises four levels.
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. Their interaction (which comprises the term of interest) has (2-1)*(2-1)=1 df. 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 (sum of age, sex and age*sex interaction dfs).
d2 = (4-1) = 3
prod = 4
Putting 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 19 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 14 for 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.