Causal Mediation Analysis: Frameworks and Robustness

• Causal mediation analysis is a statistical framework that decomposes total effects into direct and mediator-driven indirect paths.
• It employs semiparametric methods and robust estimators to achieve efficient inference even under model misspecification.
• Sensitivity analysis is incorporated to quantify violations of mediator ignorability and ensure reliability in observational studies.

Causal mediation analysis is a rigorous statistical framework that decomposes the total effect of an exposure (treatment, intervention) on an outcome into direct and indirect pathways, the latter operating through a post-exposure mediator. It formalizes counterfactual contrasts quantifying how much of the causal effect is carried along specific mechanisms. The development of semiparametric theory for mediation functionals has enabled precise, robust inference even in high-dimensional and observational paper settings, directly connecting efficiency theory and multiple robustness to mediation estimands.

1. Semiparametric Framework and Identification

The semiparametric framework establishes identification of marginal natural direct and indirect effects under standard assumptions: consistency, positivity, and sequential ignorability. For i.i.d. observed data $O = (Y, E, M, X)$, with $E$ binary (exposure), $M$ (mediator), $Y$ (outcome), and $X$ (pre-exposure confounders), the core mediation ("M-functional") is

$$\theta_0 = \iint \mathbb{E}[Y | E = 1, M = m, X = x] \, f_{M|E,X}(m | E=0, X=x) \, f_X(x) \, d\mu(m, x).$$

This target encodes a counterfactual logic: holding the mediator distribution at what it would be without exposure while assessing outcomes under exposure. Such “mixed model” integration is central to natural (in)direct effect identification.

The necessary sequential ignorability assumptions are:

• $\{ Y_{e'}, M_e \} \perp E \mid X$
• $\{ Y_{e'}, M_e \} \perp M \mid E = e, X$

These guarantee nonparametric identification of both mediation and direct effect estimands as functionals of the observed data law, permitting modeling flexibility for nuisance components:

• Outcome regression: $\mathbb{E}[Y|E, M, X]$
• Mediator density: $f_{M|E,X}(m|E,X)$
• Propensity score: $f_{E|X}(e|X)$

2. Efficiency and Robustness

The semiparametric efficiency bound for mediation functionals is derived, providing the minimum achievable variance (the Cramér–Rao lower bound under nonparametric models). The efficient influence function (EIF), which characterizes all regular asymptotically linear estimators, is given explicitly for the mediation functional $\theta_0$: $$S^{\mathrm{eff}}_{\theta_0} = \frac{I\{E=1\} f_{M|E,X}(M|E=0,X)}{f_{E|X}(1|X) f_{M|E,X}(M|E=1,X)} [ Y - \mathbb{E}(Y|X,M,E=1)] \ + \frac{I\{E=0\}}{f_{E|X}(0|X)} [ \mathbb{E}(Y|X,M,E=1) - \eta(1,0,X) ] + \eta(1,0,X) - \theta_0,$$ where $$\eta(1,0,X) = \int \mathbb{E}(Y|X, M=m, E=1) f_{M|E,X}(m|E=0,X) d\mu(m)$$.

Multiple robustness is established via submodels: consistency is ensured if at least one of the combinations (outcome+mediator, outcome+exposure, mediator+exposure) is correct. The union model covering these ensures "triply robust" inference.

3. Multiply Robust Locally Efficient Estimators

The construction of estimators proceeds by plugging estimates into empirical EIFs. The proposed triply-robust estimator for $\theta_0$ is: $$\widehat{\theta}_0^{\mathrm{triply}} = \mathbb{P}_n \Bigg[ \frac{I\{E=1\} \widehat{f}_{M|E,X}(M|E=0, X)}{\widehat{f}_{E|X}(1|X) \widehat{f}_{M|E,X}(M|E=1, X)} \{ Y - \widehat{\mathbb{E}}(Y|E=1, M, X)\}$$

$$+ \frac{I(E=0)}{\widehat{f}_{E|X}(0|X)} \{ \widehat{\mathbb{E}}(Y|E=1, M, X) - \widehat{\eta}(1,0,X) \} + \widehat{\eta}(1,0,X) \Bigg],$$

with

$$\widehat{\eta}(1,0,X) = \int \widehat{\mathbb{E}}(Y|E=1, M=m, X) \widehat{f}_{M|E,X}(m|E=0,X) d\mu(m).$$

The estimator is consistent and asymptotically normal if any of the three model pairs is correctly specified, and achieves efficiency bound when all are.

Extensions to direct and indirect effects (via $E[Y_e]$ and contrasts) parallel this approach.

4. Sensitivity Analysis for Mediator Ignorability

Acknowledging that mediator-outcome ignorability is stringent and rarely fully credible, the framework introduces a sensitivity parameterization. Define a selection bias function $t(e, m, x)$ summarizing unmeasured confounding: $$t(e, m, x) = \mathbb{E}[ Y_{1,m} | E = e, M = m, X = x ] - \mathbb{E}[ Y_{1,m} | E = e, M \ne m, X = x ].$$ When $t \equiv 0$, ignorability holds. For fixed $\lambda$, plug-in sensitivity estimators are constructed, e.g.: $$\widehat{\theta}_0^{\mathrm{doubly}}(\lambda^*) = \mathbb{P}_n\Bigg( \frac{I\{E=1\} \widehat{f}_{M|E,X}(M|E=0,X)} {\widehat{f}_{E|X}(1|X) \widehat{f}_{M|E,X}(M|E=1,X)} [Y - \widehat{\mathbb{E}}(Y|E=1,M,X)] + \widetilde{\eta}^{par}(1,0, X; \lambda^*)\Bigg).$$ Evaluating the mediation effect across a plausible set $\Lambda$ of $\lambda$ values reveals how inferences depend on possible levels of unmeasured mediator-outcome confounding.

5. Empirical Applications and Simulation Studies

Simulations show that all estimators are efficient under full model correctness, but only multiply robust estimators retain consistency under partial misspecification—other approaches (regression-only, weighting-only) can show substantial bias if nuisance models are misspecified. In small samples, multiply robust strategies substantially improve inferential stability.

A key application is to the JOBS II dataset: the framework replicated existing findings with high agreement to prior work (e.g., the natural direct effect from intervention to depression is statistically significant, while the indirect effect via self-efficacy is minor). The method's increased robustness and efficiency over traditional approaches (structural equation modeling or earlier double robust methods) are empirically substantiated.

6. Implications and Methodological Contributions

The main advances established are:

• General semiparametric representation for mediation effects using mixed-model integration (outcome regression under exposure, mediator distribution under control).
• Derivation of the efficient influence function for marginal mediation functionals (benchmarking achievable inference efficiency).
• Multiply robust (especially triply robust) locally efficient estimators that remain consistent under broad model misspecification scenarios and achieve local efficiency under correct specification.
• A sensitivity analysis protocol quantifying the impact of violations of mediator ignorability, with explicit bias function modeling and robust implementation.
• Demonstrated protection against positivity violations and model misspecification in practical scenarios.

This framework systematically strengthens causal mediation analysis in observational studies, giving practitioners a principled, robust, and efficient set of tools for mediational inference under ignorable and non-ignorable mediator-outcome confounding (Tchetgen et al., 2012).