The physicians' health study was randomized clinical trial, one goal of which was to assess the effect of aspirin in preventing myocardial infraction (MI). Participants were 22,000 male physicians ages 40-84 and free of cardiovascular disease in 1982. The physicians were randomized to either active aspirin (one white pill containing 325 mg of aspirin taken every other day) or aspirin placebo (one white placebo pill taken every other day). As the study progressed, it was estimated from self-report that 10% of the participants in the aspirin group were not complying (that is, were not taking their study [aspirin] capsules). Thus the dropout rate was 10%. Also, it was estimated from the self-report that 5% of the participants in the placebo group were taking aspirin regularly on their own outside the study protocol. Thus the drop-in rate was 5%. The issue is: How does this lack of compliance affect the sample size and power estimates for the study? Suppose we assume that the incidence of MI is .005 per year among participants who actually take placebo and that aspirin prevents 20% of MI (i.e., relative risk = p_1/p_2= 0.8). We also assume that the duration of the study is 5 years and that the dropout rate in the aspirin group = 10% and the drop-in rate in the placebo group = 5%. How many participants need to be enrolled in each group to achieve 80% power using a two-sided test with significance level =.05?
Question: How many participants need to be enrolled in each group to have a 90% chance of detecting a significant difference a two-sided test with α=0.05 if compliance is perfect?