The p-values of the hypothesis testing about relative risks

Logistic regression analysis can provide an estimate of the odds ratio (OR), adjusted for a number of confounders. It is known that it is approximately equal to the associated adjusted relative risk (RR) if the incidence of an outcome of interest is rare in a cohort study or a clinical trial study. In this paper, we consider the incidence of an outcome that is relatively common in a study population, and investigate the relationship between the two null hypotheses that an OR and the associated RR are equal to unity. It is shown that the p-values associated with the two null hypotheses are asymptotically equivalent. This complements the existing formula for converting an adjusted estimate of the OR and the associated confidence interval to their counterparts of the RR.