Sensitivity Analysis for Mediator–Outcome Confounding

Why Assess Mediator–Outcome Confounding?

In genetic mediation, an estimated pathway from an exposure through a mediator to an outcome depends on assumptions about the mediator–outcome relationship. A genetic instrument may help address confounding of the exposure, but it does not by itself ensure that the mediator and outcome are unconfounded. For example, an unmeasured biological process could affect both gene expression and disease. Adjusting for measured covariates cannot remove confounding by causes that were not measured.

Sensitivity analysis makes this limitation quantitative. Rather than treating adjustment as proof that the causal effects are identified, we specify how strongly unmeasured mediator–outcome confounding would have to act to change the estimates, and examine whether conclusions remain stable over a defensible range of values.

These are counterfactual, population-average effects. Their identification generally requires assumptions about exposure confounding, mediator–outcome confounding, consistency, positivity and the counterfactual relationships involved. Here the focus is the mediator–outcome part: we will allow residual mediator and outcome errors to be associated, and ask how that association changes estimates in a linear model.

A Linear Sensitivity Model

For clarity, consider a one-unit exposure contrast, adjusting for measured covariates \(C\). Assume the following additive models, with covariates suppressed after adjustment:

The model assumes the conditional mean relationships are linear, there is no exposure–mediator interaction, and the coefficients do not vary across individuals. Assume also that the exposure is unconfounded for the mediator and outcome given \(C\), and that the errors have mean zero conditional on \(A,C\). The sensitivity analysis targets a violation of mediator–outcome exchangeability: allow \(\varepsilon_M\) and \(\varepsilon_Y\) to be correlated after conditioning on \(A,C\). Write their conditional residual variance and covariance as:

In this model, \(\alpha_A\) is the exposure effect on the mediator, \(\theta\) is the causal outcome effect per mediator unit under the model, and \(\beta_A\) is the direct outcome effect per exposure unit. If \(\tau=0\), the mediator and outcome residuals are uncorrelated conditional on exposure and measured covariates. When \(\tau\ne0\), an ordinary regression of \(Y\) on \(A,M,C\) does not generally estimate \(\theta\) without bias.

This parameter is not an estimated correction that the observed data can determine on their own. Its value must be posited, bounded, or calibrated using scientific knowledge or other information. The sensitivity analysis reports what the effect estimates would be under each chosen value.

Deriving the Bias Adjustment

Let \(b_M\) and \(b_A\) be the coefficients from the usual adjusted regression of \(Y\) on \(A,M,C\), and let \(\widehat{\alpha}_A\) be the exposure coefficient in the mediator regression. The algebra below explains how residual covariance shifts the outcome regression coefficients. For the derivation, treat the covariates as adjusted out and use the population coefficients; estimated coefficients replace them in an applied analysis.

In detail, the population regression coefficient for \(M\), conditional on \(A,C\), is:

The total exposure effect under these additive models is \(\beta_A+\alpha_A\theta\). The fitted regression's total coefficient is \(b_A+\alpha_A b_M\), since the mediator changes by \(\alpha_A\) units per exposure unit. Equating these expressions and substituting \(b_M=\theta+\delta\) gives:

Thus, for a specified \(\delta\), the sensitivity-adjusted estimates are:

For a one-unit exposure contrast, the indirect effect estimate is \(\widehat{\alpha}_A\widehat{\theta}(\delta)\), and the direct effect estimate is \(\widehat{\beta}_A(\delta)\). Their sum remains the fitted total effect \(b_A+\widehat{\alpha}_A b_M\). This decomposition is a feature of the additive, no-interaction model; it should not be carried over automatically to models with interactions or nonlinear outcome scales.

Worked Sensitivity Analyses

Using the Analysis Responsibly

A practical analysis should present a range of sensitivity values, not just one selected value. For example, researchers might report adjusted direct and indirect estimates across negative, zero and positive values of \(\delta\), and identify where the sign or substantive interpretation changes. When a residual-correlation scale is more interpretable, the conversion \(\delta=\rho\sigma_Y/\sigma_M\) can be used with clearly stated residual standard deviations and assumptions.

Uncertainty also matters. The equations above adjust point estimates for a fixed sensitivity value. Confidence intervals should account for sampling uncertainty in the estimated mediation coefficients and, where relevant, uncertainty in \(\delta\) or its calibration. A point estimate that stays positive over a grid does not by itself establish that the effect differs from zero.

This particular sensitivity model addresses one source of uncertainty: residual mediator–outcome confounding in an additive linear model. It does not repair exposure confounding, horizontal pleiotropy, selection bias, measurement error, an incorrect functional form, or an exposure-induced mediator–outcome confounder. It also does not establish the cross-world assumptions required for natural effects. Those concerns require separate design arguments or sensitivity analyses.

Common Mistake

Wrong reasoning: “We adjusted for measured covariates in the outcome regression, so the mediator coefficient is causal and the natural indirect effect is identified.”

Correction: Covariate adjustment only addresses confounding by variables measured and appropriately modeled. An unmeasured common cause can leave \(\tau\ne0\), shifting the mediator coefficient and, in this model, the direct coefficient as well. Report which sensitivity values were considered and how conclusions change; do not interpret adjustment as evidence that unmeasured confounding is absent.