Sensitivity Analysis on Odds Ratios

The classical Cornfield inequalities state that if a third confounding variable is fully responsible for an observed association between the exposure and the outcome variables, then the association between both the exposure and the confounder, and the confounder and the outcome, must be at least as strong as the association between the exposure and the outcome, as measured by the risk ratio. The work of Ding and VanderWeele on assumption-free sensitivity analysis sharpens this bound to a bivariate function of the 2 risk ratios involving the confounder. Analogous results are nonexistent for the odds ratio, even though the conversion from odds ratios to risk ratios can sometimes be problematic. We present a version of the classical Cornfield inequalities for the odds ratio. The proof is based on the mediant inequality, dating back to ancient Alexandria. We also develop several sharp bivariate bounds of the observed association, where the 2 variables are either risk ratios or odds ratios involving the confounder.