for marginal effects by considering conditional natural direct and indirect effects, given a subset of pre-exposure variables [Tchetgen Tchetgen and Shpitser (2011)]. These models are particularly important in making inferences about so-called moderated mediation effects, a topic of growing interest, particularly in the field of psychology [Preacher, Rucker and Hayes (2007)]. In related work, we have recently extended our results to a survival analysis setting [Tchetgen Tchetgen (2011)].
A major limitation of the current paper is that it assumes that the mediator is measured without error, an assumption that may be unrealistic in practice and, if incorrect, may result in biased inferences about mediated effects. We note that much of the recent literature on causal mediation analysis makes a similar assumption. In future work, it will be important to build on the results derived in the current paper to appropriately account for a mis-measured mediator [Tchetgen Tchetgen and Lin (2012)].
APPENDIX
PROOF OF THEOREM 1. Let $F_{O;t} = F_{Y|M,X,E;t} F_{M|E,X;t} F_{E|X;t} F_{X;t}$ denote a one-dimensional regular parametric submodel of $M_{\text{nonpar}}$, with $F_{O,0} = F_O$, and let
The efficient influence function $S_{\theta_0}^{\text{eff,nonpar}}(\theta_0)$ is the unique random variable to satisfy the following equation:
for $U$ the score of $F_{O;t}$ at $t=0$, and $\nabla_t = 0$ denoting differentiation w.r.t. $t$ at $t=0$. We observe that