I need to know about the power of a simple effect that follows from a 2x2 ANOVA interaction.
Let's call the factors in this design GROUP(Patient, Control) and MANIPULATION(Expectation, Random), giving me 4 cells. Our very specific hypothesis is the following: controls will adapt their behavior according to the experimental manipulation, but patients won't.
So, formally, the dependent variable changes only in the Control group when the manipulation is 'Expectation'. Thus, the means of three cells do not differ significantly, whereas the fourth mean differs significantly from the rest. If I were testing data, I would simply look for the interaction and then the corresponding simple effect.
However I need to a priori determine the sample size needed to test this hypothesis, and thus need to know the power of this test. This is where I need help. So far, I've done the power analysis for a GROUP X MANIPULATION interaction. But here's the problem: I do not know whether this power analysis also covers the subsequent simple effects tests.
I'd like to ask you the following:
Does a power analysis for a 2x2 ANOVA interaction also 'apply' to the subsequent testing of the simple effect?
If not, is there a way how I can formalize my hypothesis 'in a single test' whose power can be determined?
If not, how else could I estimate power and sample size for this hypothesis?