Natural direct and indirect effects on the exposed: effect decomposition under weaker assumptions

We define natural direct and indirect effects on the exposed. We show that these allow for effect decomposition under weaker identification conditions than population natural direct and indirect effects. When no confounders of the mediator-outcome association are affected by the exposure, identification is possible under essentially the same conditions as for controlled direct effects. Otherwise, identification is still possible with additional knowledge on a nonidentifiable selection-bias function which measures the dependence of the mediator effect on the observed exposure within confounder levels, and which evaluates to zero in a large class of realistic data-generating mechanisms. We argue that natural direct and indirect effects on the exposed are of intrinsic interest in various applications. We moreover show that they coincide with the corresponding population natural direct and indirect effects when the exposure is randomly assigned. In such settings, our results are thus also of relevance for assessing population natural direct and indirect effects in the presence of exposure-induced mediator-outcome confounding, which existing methodology has not been able to address.