Marginal Mediator Effects in Causal Analysis

• The paper presents a comprehensive semiparametric framework that decomposes the total effect into direct and indirect pathways using an efficient influence function.
• It introduces multiply robust estimators that ensure consistent estimation even when some pairs of nuisance models are misspecified.
• The methodology incorporates double robust sensitivity analysis to quantify mediator ignorability, enhancing causal inferences in high-dimensional settings.

The marginal effect of the mediator refers to the average (population-level) contribution of a mediator variable to the total causal effect of an exposure on an outcome. In causal mediation analysis, estimating this effect requires a statistical framework capable of decomposing the total effect into direct and indirect pathways, while efficiently handling high-dimensional covariates and addressing model misspecification. A central aim is to quantify how much of the exposure’s impact on the outcome is conveyed exclusively via changes in the mediator, integrating over the mediator’s distribution in the relevant counterfactual regime.

1. Semiparametric Identification of Marginal Mediator Effects

The foundational semiparametric framework is built on the “mediation functional” $\theta_0$,

$$\theta_0 = \iint_{\mathcal{S} \times \mathcal{X}} E(Y | E = 1, M = m, X = x) f_{M|E,X}(m | E = 0, X = x) f_X(x) d\mu(m, x)$$

which represents the counterfactual mean outcome had exposure $E$ been set to 1 but the mediator $M$ had been forced to the distribution observed under $E = 0$ (all conditional on pre-exposure confounders $X$).

The marginal natural direct effect (NDE) and natural indirect effect (NIE) are then

$$\mathrm{NDE} = \theta_0 - \delta_0, \qquad \mathrm{NIE} = \delta_1 - \theta_0$$

where $\delta_e = \int_x E(Y | E = e, X = x) f_X(x) dx$. Identification of these functionals hinges on consistency, positivity, and sequential ignorability. The NIE quantifies the marginal effect of the mediator, as it captures the expected difference in the outcome when the mediator is shifted from the distribution under $E=0$ to that under $E=1$ while holding $E$ fixed.

Within this framework, the efficient influence function for $\theta_0$ in the fully nonparametric model is explicitly derived and forms the analytic basis for estimator development: $$\begin{aligned} S^{\mathrm{eff,nonpar}}_{\theta_0}(O; \theta_0) = &\,\frac{I\{E = 1\} f_{M|E,X}(M|E = 0, X)}{f_{E|X}(1|X) f_{M|E,X}(M|E = 1, X)} \left\{ Y - E(Y | E = 1, M, X) \right\} \ & + \frac{I\{E = 0\}}{f_{E|X}(0|X)} \left\{ E(Y | E = 1, M, X) - \eta(1, 0, X) \right\} \ & + \eta(1, 0, X) - \theta_0 \end{aligned}$$ with $$\eta(1, 0, X) = \int E(Y | E = 1, M = m, X) f_{M|E,X}(m|E = 0, X) d\mu(m)$$.

2. Efficiency Bounds and Multiply Robust Consistency

The derivation of the semiparametric efficiency bound establishes that every regular asymptotically linear estimator for the mediation functional attains the variance of the efficient influence function in the nonparametric model. In finite samples and high-dimensional settings, three nuisance quantities must be estimated:

• $\mathbb{E}(Y | E, M, X)$ (outcome regression),
• $f_{M|E,X}(m | E, X)$ (mediator density),
• $f_{E|X}(e|X)$ (exposure propensity).

The paper’s estimators are “multiply robust” (triply robust), guaranteeing consistency if at least one of the following model pairs is correctly specified: (a) outcome and mediator; (b) outcome and exposure; (c) mediator and exposure. Local efficiency (attainment of the efficiency bound under full specification) is also established. These properties are especially relevant in practical situations with limited sample size and model uncertainty.

3. Nonparametric and Multiply Robust Estimators

Estimators for the marginal effect of the mediator fall into three nonparametric classes (“ym”, “ye”, “em”), each relying on two of the three nuisance models. However, only a multiply robust estimator, constructed from the efficient influence function, achieves consistency when any one pair is correct—a critical advantage in real-world data settings.

The triply robust estimator has plug-in form: $$\hat \theta_0^{\mathrm{triply}} = \mathbb{P}_n \left[ \frac{I\{E = 1\} \hat f_{M|E,X}(M|E=0,X)}{\hat f_{E|X}(1|X) \hat f_{M|E,X}(M|E=1,X)} \left\{ Y - \hat E(Y|E=1, M, X) \right\} + \frac{I\{E=0\}}{\hat f_{E|X}(0|X)} \left\{ \hat E(Y|E=1, M, X) - \hat \eta(1, 0, X) \right\} + \hat \eta(1, 0, X) \right]$$ with

$$\hat \eta(1,0,X) = \int \hat E(Y|E=1, M = m, X) \hat f_{M|E,X}(m|E=0, X) d\mu(m)$$

and analogous estimators for the direct and indirect effects via substitution into the appropriate functional differences.

This estimator is regular and asymptotically linear over the union model and achieves the semiparametric efficiency bound when all nuisance models are correct.

4. Double Robust Sensitivity Analysis for Ignorability of the Mediator

To address potential violations of the sequential ignorability assumption for the mediator, the paper proposes a double robust sensitivity analysis using a parametric selection bias function $$t(e, m, x) = E[Y_{1,m} | E=e, M=m, X=x] - E[Y_{1,m} | E = e, M \neq m, X=x]$$. Under this framework, the counterfactual expectation is re-expressed: $$\begin{aligned} E[Y_{1,m} | M_0 = m, X = x] = &\,E(Y | E = 1, M = m, X = x) \ & - t(1, m, x)[1 - f_{M|E,X}(m|E=1, X=x)] \ & + t(0, m, x)[1 - f_{M|E,X}(m|E=0, X=x)] \end{aligned}$$ This leads to an identification formula for $\theta_0$ that incorporates the bias function, which can be indexed by a sensitivity parameter ($\lambda$) through a family $t_\lambda$. The resulting estimator $\hat \theta_0^{\mathrm{doubly}}(\lambda)$ is double robust: if either the outcome regression or the exposure/mediator density is consistently estimated, valid inference about the marginal effect of the mediator is preserved.

Varying the sensitivity parameter enables transparent reporting of how sensitive mediation effect estimates are to departures from the ignorability assumption.

5. Applications and Implications for Practice

These methodologies are directly relevant in fields where mediation analysis is critical for disentangling mechanisms, such as epidemiology, psychology, and health interventions. For example, in the JOBS II randomized trial, the framework allows decomposition of the effect of job training on mental health into components mediated by job search self-efficacy and those direct pathways not involving the mediator.

Key implications include:

• Multiply robust estimation protects against model misspecification, crucial with many pre-exposure covariates.
• Local efficiency ensures statistical optimality when all models are correct.
• Double robust sensitivity analyses enable quantification of inferential robustness to assumptions about mediator ignorability.
• The framework can be extended to handle moderated mediation, survival outcomes, and potentially measurement error in mediators.

These advances guide researchers to more resilient, interpretable, and transparent mediation analyses when assessing the marginal effect of the mediator under realistic, complex confounding structures.

6. Theoretical and Methodological Contributions

The semiparametric theory presented provides:

• A unified identification formula applicable across linear and nonlinear settings.
• Explicit efficient influence function derivations, facilitating construction of optimal estimators.
• Multiply robust (“triply robust”) estimators that combine flexible model specification with strong finite-sample and large-sample properties.
• The first double robust sensitivity analysis for mediator ignorability that operates at the level of the mediation functional, not requiring explicit unmeasured confounder modeling.
• Proofs of regularity and asymptotic linearity in union models spanning multiple combinations of correctly specified nuisance models.

This framework represents a significant advance in mediation analysis, ensuring that the marginal effect of the mediator can be estimated robustly and efficiently in the presence of high-dimensional covariates and possible violations of standard identification assumptions. Researchers are thus equipped to conduct mediation analyses that are both practically feasible and theoretically grounded, critical for causal mechanism investigations in complex data environments.