Then, under the consistency, sequential ignorability and positivity assumptions, Imai, Keele and Tingley (2010) showed that
so that $\mathbb{E}(Y_{1,M_0})$ and $\mathbb{E}(Y_e)$, $e=0,1$, are identified from the observed data, and so is the mean natural direct effect $\mathbb{E}(Y_{1,M_0}) - \mathbb{E}(Y_0) = \theta_0 - \delta_0$ and the mean natural indirect effect $\mathbb{E}(Y_1) - \mathbb{E}(Y_{1,M_0}) = \delta_1 - \theta_0$. For binary $Y$, one might alternatively consider the natural direct effect on the risk ratio scale $\mathbb{E}(Y_{1,M_0})/\mathbb{E}(Y_0) = \theta_0/\delta_0$ or on the odds ratio scale ${\mathbb{E}(Y_{1,M_0})\mathbb{E}(1-Y_0)}/{\mathbb{E}(1-Y_{1,M_0})\mathbb{E}(Y_0)} = {\theta_0(1-\delta_0)}/{\delta_0(1-\theta_0)}$ and similarly defined natural indirect effects on the risk ratio and odds ratio scales. It is instructive to contrast the expression (2) for $\mathbb{E}(Y_{1,M_0})$ with the expression (3) for $e=1$ corresponding to $\mathbb{E}(Y_1)$, and to note that the two expressions bare a striking resemblance except the density of the mediator in the first expression conditions on the unexposed (with $E=0$), whereas in the sec- ond expression, the mediator density is conditional on the exposed (with $E=1$). As we demonstrate below, this subtle difference has remarkable implications for inference.
Pearl (2001) was the first to derive the M-functional $\theta_0 = \mathbb{E}(Y_{1,M_0})$ under a different set of assumptions. Others have since contributed alternative sets of identifying assumptions. In this paper, we have chosen to work under the sequential ignorability assumption of Imai, Keele and Yamamoto (2010), Imai, Keele and Tingley (2010), but note that alternative related assumptions exist in the literature [Robins and Greenland (1992), Pearl (2001), van der Laan and Petersen (2005), Hafeman and Vanderweele (2011)]; however, we note that Robins and Richardson (2012) disagree with the label “sequential ignorability” because its terminology has previously carried a different interpretation in the literature. Nonetheless, the assumption entails two ignorability-like assumptions that are made sequentially. First, given the observed pre-exposure confounders, the exposure assignment is assumed to be ignorable, that is, statistically independent of potential outcomes and potential mediators. The second part of the assumption states that the mediator is ignorable given the observed exposure and pre-exposure confounders. Specifically, the second part of the sequential ignorability assumption is conditional on the observed value of the ignorable treatment and the observed pretreatment confounders. We note that the second part of the sequential ignorability assumption is