Targeted Maximum Likelihood Estimation of Natural Direct Effect

In many causal inference problems, one is interested in the direct causal effect of an exposure on an outcome of interest that is not mediated by certain intermediate variables. Robins and Greenland (1992) and Pearl (2000) formalized the definition of two types of direct effects (natural and controlled) under the counterfactual framework. Since then, identifiability conditions for these effects have been studied extensively. By contrast, considerably fewer efforts have been invested in the estimation problem of the natural direct effect. In this article, we propose a semiparametric efficient, multiply robust estimator for the natural direct effect of a binary treatment using the targeted maximum likelihood framework of van der Laan and Rubin (2006) and van der Laan and Rose (2011). The proposed estimator is asymptotically unbiased if either one of the following holds: i) the conditional outcome expectation given exposure, mediator, and confounders, and the mediated mean outcome difference are consistently estimated; (ii) the exposure mechanism given confounders, and the conditional outcome expectation are consistently estimated; or (iii) the exposure mechanism given confounders, and a ratio of conditional mediator densities are consistently estimated. Moreover, case (iii) implies in particular that estimation of the conditional mediator density may be replaced by consistent estimation of the exposure mechanism and the conditional distribution of exposure given confounders and mediator. If all three conditions hold, then the effect estimate is asymptotically efficient.