Natural direct and indirect effects formalize traditional notions of mediation analysis into a rigorous causal framework and have recently received considerable attention in epidemiology and in the social sciences. Sufficient conditions for identification of natural direct effects were formulated by Judea Pearl under a nonparametric structural equations model, which assumes certain independencies between potential outcomes. A common situation in epidemiology is that a confounder of the mediator is affected by the exposure, in which case, natural direct effects fail to be nonparametrically identified without additional assumptions, even under Pearl's nonparametric structural equations model. In this paper, the authors show that when a single binary confounder of the mediator is affected by the exposure; the natural direct effect is nonparametrically identified under a monotonicity assumption about the effect of the exposure on the confounder. A similar result is shown to hold for a vector of binary confounders of the mediator under a certain independence assumption about the confounders. Finally, the authors show that natural direct effects are more generally identified if there is no-additive mean interaction between the mediator and confounders of the mediator affected by exposure. When correct, this latter assumption is particularly appealing because it does not require monotonicity of effects of the exposure, additionally, it places no restriction on the nature of the confounders of the mediator which can be continuous or polytomous.
Tchetgen Tchetgen, Eric J. and VanderWeele, Tyler J., "On Identification of Natural Direct Effects when a Confounder of the Mediator is Directly Affected by Exposure" (June 2012). COBRA Preprint Series. Working Paper 93.