Effect Decomposition in the Presence of Treatment-induced Confounding: A Regression-with-residuals Approach

Analyses of causal mediation are often complicated by treatment-induced confounders of the mediator-outcome relationship. In the presence of such confounders, the natural direct and indirect effects of treatment on the outcome, into which the total effect can be additively decomposed, are not identified. An alternative but similar set of effects, known as randomized intervention analogues to the natural direct effect (R-NDE) and the natural indirect effect (R-NIE), can still be identified in this situation, but existing estimators for these effects require a complicated weighting procedure that is difficult to use in practice. In this paper, we introduce a new method for estimating the R-NDE and R-NIE that involves only a minor adaption of the comparatively simple regression methods used to perform effect decomposition in the absence of treatment-induced confounding. It involves fitting linear models for (a) the conditional mean of the mediator given treatment and a set of baseline confounders and (b) the conditional mean of the outcome given the treatment, mediator, baseline confounders, and the treatment-induced confounders after first residualizing them with respect to the observed past. The R-NDE and R-NIE are simple functions of the parameters in these models when they are correctly specified and when there are no unobserved variables that confound the treatment-outcome, treatment-mediator, or mediator-outcome relationships. We illustrate the method by decomposing the effect of education on depression symptoms at midlife into components operating through income versus alternative factors. R and Stata packages are available for implementing the proposed method.