I'm designing a study where I have a control group and a treatment group and measure rates of completion (0 if complete, 1 if not complete) of the different groups (between subjects design). I consider the treatment group successful if a N-1 two proportions test shows a statistically significant difference.
However, I also want to know if there is a statistically significant difference between two different recruitment methods. My thought was to take the difference between the rates of completion in the treatment group and the control group and compare that between the two different recruitment methods (e.g., method #1 produces a boost of +3 percentage points in the treatment group and method #2 produces a boost of +6 percentage points in the treatment group) with a N-1 two proportions test. Is that correct?
Additionally, what equation could I use to calculate the necessary sample size for achieving statistical significance with various assumptions about the difference in percentage point increases (assuming a desired power of 80% and an alpha of 0.05)?
Update/conclusion: we ended up deciding to go with a factorial logistic regression.