Sensitivity to Unmeasured Mediator-Outcome Confounding

Why Examine Unmeasured Mediator–Outcome Confounding?

Interventional direct and indirect effects compare outcome means under specified exposure levels and mediator distributions. Their usual g-formula identifies those means by relating observed outcome means at each mediator value to outcomes under an intervention on the mediator. That link can fail when a factor not included in the measured covariates affects both the mediator and the outcome. Adjusting for measured baseline covariates does not guarantee that this problem has disappeared.

A sensitivity analysis makes the unverified link explicit. Rather than claiming that measured covariates suffice for mediator–outcome exchangeability, it specifies how far an observed conditional outcome mean might be from the corresponding causal mean under intervention. The data do not generally determine that discrepancy. Analysts vary it over scientifically plausible values and see how the interventional effects change.

Definition: Let $A$ be a binary exposure, $M$ a mediator, $Y$ an outcome, and $C$ measured baseline covariates. Let $Y(a,m)$ denote the outcome under an intervention setting $A=a$ and $M=m$. Define the causal mean $\mu(a,m,c)=E[Y(a,m)\mid C=c]$, and the observed conditional mean $g(a,m,c)=E[Y\mid A=a,M=m,C=c]$. The additive sensitivity function is $$ \delta(a,m,c)=E[Y(a,m)\mid A=a,M=m,C=c]-E[Y(a,m)\mid A=a,C=c]. $$ Under consistency and exposure exchangeability given $C$, $\delta(a,m,c)=g(a,m,c)-\mu(a,m,c)$. A positive value means the observed group with $A=a,M=m,C=c$ has a higher mean of $Y(a,m)$ than the corresponding group with $A=a,C=c$, beyond the difference explained by measured baseline covariates.

This definition makes the target of sensitivity analysis precise: $\delta$ describes residual selection into mediator values in the outcome that would occur under the specified exposure and mediator interventions. It is measured in the units of $Y$. It is not, by itself, a regression coefficient for an unmeasured confounder.

Identification Conditions and the Adjusted Formula

Write $\psi(a,a^*)$ for the mean outcome under an intervention setting the exposure to $a$, then drawing the mediator from its distribution under exposure $a^*$, conditional on each person’s baseline covariates. The stochastic mediator draw is independent of that person’s potential outcomes conditional on $C$. For binary exposure, common contrasts are the interventional direct effect $\psi(1,0)-\psi(0,0)$, the interventional indirect effect $\psi(1,1)-\psi(1,0)$, and the total effect $\psi(1,1)-\psi(0,0)$.

The sensitivity-adjusted formula uses these assumptions: consistency for exposure, mediator, and outcome; no interference between units; exposure exchangeability given $C$, including $M(a)$ and $Y(a,m)$ independent of $A$ conditional on $C$; exposure positivity; and mediator positivity for the mediator values used by the reference distribution. It also requires well-defined interventions and a specified sensitivity function. Unlike the usual identifying formula, it does not assume mediator–outcome exchangeability given $A,C$. Instead, the analyst supplies or varies $\delta$. If exposure exchangeability is not plausible, this adjustment alone does not address that separate source of confounding.

For clarity, the key identity follows from consistency and the definition of $\delta$. Within the observed stratum $A=a,M=m,C=c$, consistency gives $E[Y\mid A=a,M=m,C=c]=E[Y(a,m)\mid A=a,M=m,C=c]$. Subtracting $\delta(a,m,c)$ leaves $E[Y(a,m)\mid A=a,C=c]$. Exposure exchangeability then makes this last quantity equal to $E[Y(a,m)\mid C=c]=\mu(a,m,c)$. Thus:

\mu(a,m,c)=g(a,m,c)-\delta(a,m,c).

Exposure exchangeability also identifies the mediator distribution under exposure $a^*$ with its observed distribution among units having that exposure and covariates. Averaging the corrected causal mean over that distribution and then over the target population’s covariates yields:

\psi_{\delta}(a,a^*)= \int\int \left\{g(a,m,c)-\delta(a,m,c)\right\} \,dF(m\mid A=a^*,C=c)\,dF(c).

When $\delta(a,m,c)=0$ everywhere, this reduces to the usual mediational g-formula. That special case is justified by mediator–outcome exchangeability given $A,C$, alongside the other identification conditions. A nonzero sensitivity function replaces that assumption with a stated model for the departure.

Worked Examples

Worked Example: Adjusting Effects With a Binary Mediator

Suppose a hypothetical study has binary $A$ and $M$, and $Y$ is a symptom score in points. Assume $C$ is empty for this illustration, and suppose exposure exchangeability, positivity, consistency, and no interference are reasonable. The observed mediator probabilities are $P(M=1\mid A=0)=0.30$ and $P(M=1\mid A=1)=0.60$. The observed conditional outcome means are $g(0,0)=10$, $g(0,1)=14$, $g(1,0)=12$, and $g(1,1)=20$ points.

Use the sensitivity model $\delta(a,m)=\gamma am$, where $\gamma$ is measured in symptom-score points. It asserts that the observed mean for $A=1,M=1$ exceeds the corresponding causal mean by $\gamma$ points, while the other three cells have zero discrepancy. This is an illustrative model, not a conclusion supplied by the data. The adjusted means are $\mu(0,0)=10$, $\mu(0,1)=14$, $\mu(1,0)=12$, and $\mu(1,1)=20-\gamma$.

The reference distribution for $\psi(0,0)$ has $P(M=1\mid A=0)=0.30$, so $ \psi_{\delta}(0,0)=0.70(10)+0.30(14)=11.2 $ points. For $\psi(1,0)$, keep the outcome exposure at 1 but draw $M$ from its distribution under exposure 0: $ \psi_{\delta}(1,0)=0.70(12)+0.30(20-\gamma)=14.4-0.30\gamma $ points.

For $\psi(1,1)$, use the mediator distribution under exposure 1: $ \psi_{\delta}(1,1)=0.40(12)+0.60(20-\gamma)=16.8-0.60\gamma $ points. Therefore, $ \text{IDE}_{\delta}=3.2-0.30\gamma $, $ \text{IIE}_{\delta}=2.4-0.30\gamma $, and $ \text{TE}_{\delta}=5.6-0.60\gamma $, all in symptom-score points. At $\gamma=4$, the estimates are respectively 2.0, 1.2, and 3.2 points. Their additive decomposition holds: $2.0+1.2=3.2$.

The parameter value $\gamma=4$ is an assumption for the sensitivity calculation, not something estimated from these observed means. The calculation shows what the effects would be under that specified departure from mediator–outcome exchangeability.

Allowing Covariates to Vary

With measured baseline covariates, the formula is evaluated within covariate strata before averaging over the target distribution of $C$. This matters because both the mediator distribution and the outcome means can vary across strata. A sensitivity function may also vary across strata; any such pattern is part of the analyst’s model.

Worked Example: Standardizing Across Two Covariate Strata

Consider another hypothetical setting with equally common strata $C=L$ and $C=H$. The observed mediator probabilities under $A=0$ are 0.20 and 0.40 in the two strata; under $A=1$, they are 0.50 and 0.80. The observed conditional outcome means, in points, are:

Stratum	$g(0,0,c)$	$g(0,1,c)$	$g(1,0,c)$	$g(1,1,c)$
$L$	5	9	6	12
$H$	8	12	10	16

Use $\delta(a,m,c)=\gamma am$ and standardize equally over the two strata. For $\psi_{\delta}(0,0)$, the stratum-specific means are $0.80(5)+0.20(9)=5.8$ and $0.60(8)+0.40(12)=9.6$. Their average is $7.7$ points.

For $\psi_{\delta}(1,0)$, the mediator distribution is the $A=0$ distribution. In stratum $L$, the mean is $0.80(6)+0.20(12-\gamma)=7.2-0.20\gamma$. In $H$, it is $0.60(10)+0.40(16-\gamma)=12.4-0.40\gamma$. Averaging gives $9.8-0.30\gamma$ points.

For $\psi_{\delta}(1,1)$, use the $A=1$ mediator distribution: the stratum means are $0.50(6)+0.50(12-\gamma)=9.0-0.50\gamma$ and $0.20(10)+0.80(16-\gamma)=14.8-0.80\gamma$. Their average is $11.9-0.65\gamma$ points. The resulting direct, indirect, and total effects are $2.1-0.30\gamma$, $2.1-0.35\gamma$, and $4.2-0.65\gamma$ points. The total equals the direct plus indirect contrast for every $\gamma$.

Choosing and Interpreting Sensitivity Values

The sensitivity parameter is not identified from the observed data without additional assumptions or information. Its plausible values must therefore be justified using subject-matter knowledge, external evidence, or transparent scenarios. The scale is important: with a continuous outcome, $\gamma$ in an additive model is expressed in outcome units. With a binary outcome, an additive difference in risks is also possible, but the model must respect the possible range of risks.

A single constant sensitivity value may be convenient, but it is not automatically plausible. Residual confounding could differ by exposure, mediator value, or covariate stratum. A more general model is $\delta(a,m,c)=\gamma h(a,m,c)$, where the analyst specifies the pattern $h$ and varies its magnitude $\gamma$. That pattern should be stated clearly; the same numerical $\gamma$ can imply different effect changes under different choices of $h$.

Worked Example: Finding a Tipping Value

Return to the binary-mediator example, where $\text{IIE}_{\delta}=2.4-0.30\gamma$ symptom-score points under the specified model. The value at which the estimated indirect effect reaches zero solves $2.4-0.30\gamma=0$, giving $\gamma=2.4/0.30=8.0$ points.

At $\gamma=6$, the indirect effect is $2.4-0.30(6)=0.60$ points. At $\gamma=10$, it is $2.4-0.30(10)=-0.60$ points. Thus, under this particular sensitivity pattern, a discrepancy of 8 points in the $A=1,M=1$ cell would bring the indirect contrast to zero; larger values would make it negative. This is a tipping-point calculation, not evidence that the true discrepancy is 8 points. Whether values near that threshold are credible requires substantive judgment.

Common Mistakes and Practical Interpretation

Equating adjustment for measured covariates with no residual confounding. The ordinary g-formula needs mediator–outcome exchangeability given $A,C$. A sensitivity analysis explicitly relaxes that condition rather than assuming it follows from measuring $C$.
Using exposure exchangeability as if it also removed mediator confounding. Exposure exchangeability identifies the relevant exposure-specific causal mean before conditioning on mediator selection. It does not imply $\delta=0$.
Defining the causal mean conditional on $C$, then using the wrong comparison. In this tutorial, $\mu(a,m,c)=E[Y(a,m)\mid C=c]$. To relate it to the observed mean, the definition of $\delta$ compares the selected group with $A=a,M=m,C=c$ to the group with $A=a,C=c$. Exposure exchangeability is what allows the latter mean to equal the causal mean conditional on $C=c$. Without that assumption, the stated identity need not hold.
Treating $\gamma$ as an estimated parameter. Observed data alone generally do not identify the residual discrepancy. Report the chosen sensitivity values and show how results vary across them.
Reading an adjusted contrast as proof of a mechanism. Sensitivity analysis measures robustness to the specified confounding model. It does not verify that model or establish a biological mechanism.

A useful report gives the ordinary estimate, defines the sensitivity function and its units, explains the rationale for the values examined, and displays the adjusted contrasts across those values. If uncertainty intervals are calculated, uncertainty in the estimated observed-data quantities should be propagated for each fixed sensitivity value; such intervals do not account automatically for uncertainty about the sensitivity model itself.

Key takeaway: Interventional effects can be recalculated under explicit departures from mediator–outcome exchangeability by replacing the observed conditional outcome mean $g$ with $g-\delta$. The result is informative only when the sensitivity function, exposure-exchangeability conditions, and range of plausible values are made clear.

Check Your Understanding

Answer each question using the definitions and assumptions in this tutorial.

In the additive sensitivity function, what does a positive $\delta(a,m,c)$ mean, and what are its units?
Why does the formula for $\mu(a,m,c)$ require exposure exchangeability in addition to consistency?
For $\delta(a,m,c)=\gamma am$, which outcome cell is shifted, and in what direction when $\gamma>0$?
Why is a sensitivity parameter generally varied over plausible values rather than estimated from the observed conditional means alone?
In the two-stratum example, explain why the mediator probabilities and outcome means are calculated within strata before averaging.

Stratum	\(g(0,0,c)\)	\(g(0,1,c)\)	\(g(1,0,c)\)	\(g(1,1,c)\)
\(L\)	5	9	6	12
\(H\)	8	12	10	16