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 is one between subjects factor with 2 levels do $$b_{1}$$ = 2. There are two within subject factors each with 2 levels so $$w_{1}$$ = $$w_{2}$$ = 2. 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
d2 = (2-1)(2-1) = 1
prod = 2
Putting in these together 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.