Interval identification of natural effects in the presence of outcome‐related unmeasured confounding

With reference to a binary outcome and a binary mediator, we derive identification bounds for natural effects under a reduced set of assumptions. Specifically, no assumptions about confounding are made that involve the outcome; we only assume no unobserved exposure‐mediator confounding as well as a pair of conditions termed Partially Constant Cross‐World Dependence (PC‐CWD) and Logit Constancy (LC). These assumptions pose fewer constraints on the counterfactual probabilities than the set of assumptions they replace. The proposed strategy permits to achieve interval identification of the total effect, which is no longer point identified under the considered set of assumptions. Our derivations are based on postulating a logistic regression model for the mediator as well as for the outcome. However, in both cases the functional form governing the dependence on the explanatory variables is allowed to be arbitrary, thereby resulting in a semi‐parametric approach. To account for sampling variability, we provide delta‐method approximations of standard errors to build uncertainty intervals from identification bounds. The method is compared to some alternative ones in a simulation study and then applied to a dataset gathered from a Spanish prospective cohort study, with the aim to evaluate whether the effect of smoking on lung cancer risk is mediated by the onset of pulmonary emphysema.