  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 condition termed partially constant cross-world dependence (PC-CWD), which
poses fewer constraints on the counterfactual probabilities than the usual
cross-world independence assumption. The proposed strategy can be used also 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 in order to build uncertainty
intervals from identification bounds. The proposed method is applied to a
dataset gathered from a Spanish prospective cohort study. The aim is to
evaluate whether the effect of smoking on lung cancer risk is mediated by the
onset of pulmonary emphysema.
