Suppose we have a three way interaction of three factors called age, sex and type. Age and sex are between subject and each have two levels whilst type is a within subject factor and has 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 and, so, both have dfs of (2-1)=1. 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 type has a df of (4-1)=3. We can now use these to work out our inputs.

num = numerator df = df for age*sex*type = (2-1)(2-1)(4-1)=3

bsum = (2-1) + (2-1) + (2-1)*(2-1) = 3
(sum of dfs for the between subject factor terms: age, sex and age*sex)

wdf = (4-1) = 3 (df for the within subject factor, type)

Putting these together, assuming conservatively the types are uncorrelated, for a power of 0.80 we require 39 people assuming an alpha of 0.05 and a partial eta-squared of 0.10. 

If we assume an average correlation of 0.25 amongst pairs of types with a partial eta-squared of 0.10 we only need a total sample size of 30 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.