Sensitivity to Mediator Reference Distributions

Why the Mediator Reference Distribution Matters

An interventional mediation analysis evaluates outcomes under a hypothetical intervention that assigns the mediator according to a specified distribution. The distribution is part of the estimand, not merely a computational choice. For example, drawing mediator values from the untreated group’s distribution asks a different question from drawing them from the treated group’s distribution. A policy-defined distribution can ask a different question again.

Consequently, a reported interventional effect should make clear which mediator distribution was used. If the scientific conclusion changes substantially when a defensible alternative distribution is used, that dependence is important to report. This is sensitivity to the mediator reference distribution; it is distinct from sensitivity to unmeasured mediator-outcome confounding.

Definition: Let $A$ be a binary treatment, $M$ a mediator, $Y$ an outcome, and $C$ measured baseline covariates. Let $g(m\mid c)$ be a specified distribution for the mediator given $C=c$. Define $Q(a,m,c)=E[Y\mid A=a,M=m,C=c]$. The interventional mean under treatment level $a$ and mediator reference distribution $g$ is $$ \mu(a;g)=E_C\left[\int Q(a,m,C)\,d g(m\mid C)\right]. $$ For a discrete mediator, the integral is a sum. Contrasting $\mu(1;g)$ and $\mu(0;g)$ compares treatment levels while using the same mediator reference distribution.

The expectation over $C$ uses the covariate distribution of the target population. Thus, when comparing reference distributions, hold the outcome model and target distribution of baseline covariates fixed. Otherwise, a change in the estimate could reflect a change in the target population as well as a change in the mediator reference.

Common references include $F_0(m\mid c)$, the mediator distribution under $A=0$, and $F_1(m\mid c)$, the distribution under $A=1$. A pooled reference can mix these distributions. A policy-defined reference, $G^*(m\mid c)$, is specified by a substantive intervention or target, such as a program designed to increase the probability of a beneficial mediator value.

Effects Under Different References

For a fixed reference $g$, the contrast $\mu(1;g)-\mu(0;g)$ is an interventional treatment contrast with the mediator distribution held common. To examine a mediator-distribution shift at a fixed treatment level, compare $\mu(a;g_1)$ with $\mu(a;g_0)$. These contrasts answer different questions, and each must name its reference distributions.

For example, using $F_0$ and $F_1$, one possible decomposition of the contrast from $\mu(0;F_0)$ to $\mu(1;F_1)$ is

\mu(1;F_1)-\mu(0;F_0) = \{\mu(1;F_0)-\mu(0;F_0)\} + \{\mu(1;F_1)-\mu(1;F_0)\}.

The first term uses the untreated mediator distribution for both treatment levels; the second changes the mediator reference while holding treatment at 1. An alternative decomposition can change the reference first and then treatment. These are bookkeeping identities, not claims that either component is uniquely privileged. The purpose of sensitivity analysis is to show how the component estimates and conclusions vary across reasonable reference choices.

Worked Example: Comparing Treated, Untreated, and Pooled References

Consider hypothetical data with no baseline covariates and a binary mediator. Suppose the conditional outcome means, in outcome units, are as follows. Assume treatment is randomized, all treatment-mediator combinations have positive probability, and there is no unmeasured mediator-outcome confounding. These assumptions allow the conditional means and mediator distributions to identify the interventional means in this illustration.

Treatment $a$	$Q(a,0)$	$Q(a,1)$
0	10	14
1	13	20

Suppose $P(M=1\mid A=0)=0.20$ and $P(M=1\mid A=1)=0.70$. Both mediator values occur under both treatment levels, so the required support is present in this example. With a binary mediator, if the reference probability of $M=1$ is $p$, then

\mu(a;g)=(1-p)Q(a,0)+pQ(a,1).

Under the untreated reference, $p=0.20$:

\mu(0;F_0)=0.80(10)+0.20(14)=8.00+2.80=10.80,

\mu(1;F_0)=0.80(13)+0.20(20)=10.40+4.00=14.40.

Thus, $\mu(1;F_0)-\mu(0;F_0)=14.40-10.80=3.60$ outcome units. Under the treated reference, $p=0.70$:

\mu(0;F_1)=0.30(10)+0.70(14)=3.00+9.80=12.80,

\mu(1;F_1)=0.30(13)+0.70(20)=3.90+14.00=17.90.

The treatment contrast under $F_1$ is $17.90-12.80=5.10$ outcome units, 1.50 units larger than the contrast under $F_0$. This difference reflects that the treatment contrast in the outcome means is larger when $M=1$: it is $20-14=6$ at $M=1$, versus $13-10=3$ at $M=0$.

For an equal mixture of the two mediator distributions, the reference probability is $p=(0.20+0.70)/2=0.45$. Then $\mu(0;g)=0.55(10)+0.45(14)=11.80$, and $\mu(1;g)=0.55(13)+0.45(20)=16.15$. The common-reference treatment contrast is $16.15-11.80=4.35$ outcome units. Each result is internally consistent, but each answers a question tied to a different mediator reference.

Mixtures and a Sensitivity Curve

Rather than compare only two endpoints, define a family of references that moves gradually from the untreated to the treated mediator distribution. For $0\leq\lambda\leq1$, let

g_\lambda(m\mid c)=(1-\lambda)F_0(m\mid c)+\lambda F_1(m\mid c).

Here $\lambda=0$ gives the untreated reference, $\lambda=1$ gives the treated reference, and intermediate values give mixtures. This is a conditional mixture: at each $c$, the same mixture weight is applied to the two conditional mediator distributions. A pooled reference should be defined explicitly; mixing conditional distributions this way need not be identical to pooling all observed mediator values without conditioning on $C$.

For a fixed treatment level $a$, integration is linear in the mediator distribution. Therefore,

\begin{aligned} \mu(a;g_\lambda) &=E_C\left[\int Q(a,m,C)\,d\{(1-\lambda)F_0+\lambda F_1\}(m\mid C)\right]\\ &=(1-\lambda)\mu(a;F_0)+\lambda\mu(a;F_1). \end{aligned}

Subtracting the corresponding means at $a=0$ gives the sensitivity curve for the common-reference treatment contrast:

\mu(1;g_\lambda)-\mu(0;g_\lambda) =(1-\lambda)\{\mu(1;F_0)-\mu(0;F_0)\} +\lambda\{\mu(1;F_1)-\mu(0;F_1)\}.

This derivation shows why the curve is linear in $\lambda$ for this particular mixture definition when the outcome regression and target covariate distribution are held fixed. It does not imply that every policy-defined reference lies on this curve, or that a linear pattern is expected when other parts of the analysis also change.

Worked Example: Quantifying the Mixture Sensitivity

Use the hypothetical outcome means and mediator probabilities in the preceding example. The reference probability of $M=1$ under $g_\lambda$ is

p_\lambda=(1-\lambda)(0.20)+\lambda(0.70)=0.20+0.50\lambda.

At $M=0$, the treatment contrast is $13-10=3$ outcome units; at $M=1$, it is $20-14=6$ units. Averaging these contrasts over $g_\lambda$, the common-reference treatment contrast is

(1-p_\lambda)(3)+p_\lambda(6)=3+3p_\lambda=3.60+1.50\lambda.

At $\lambda=0.25$, the reference probability is $0.20+0.50(0.25)=0.325$, and the contrast is $3.60+1.50(0.25)=3.975$ outcome units. At $\lambda=0.75$, the probability is $0.575$, and the contrast is $3.60+1.50(0.75)=4.725$ outcome units. Across these choices, the estimate changes by $4.725-3.975=0.750$ units. Reporting the values across $\lambda$ makes the reference dependence visible instead of hiding it behind one selected mixture.

Policy-Defined References

A policy-defined reference need not be a mixture of the observed treatment-arm distributions. For example, a prevention program might target a mediator prevalence between the two observed prevalences, or a clinical rule might assign mediator values according to patient characteristics. Define $G^*(m\mid c)$ to represent that target, then evaluate $\mu(a;G^*)$ using the same outcome regression and target distribution of $C$.

Worked Example: Evaluating a Policy Target

Using the same hypothetical outcome means, suppose a feasible policy targets $P(M=1)=0.40$ for everyone in this covariate-free illustration. This is a policy-defined reference, not the equal mixture in the previous example: the equal mixture had prevalence $0.45$. Since both mediator values are supported in both treatment groups, the stated target does not require extrapolation beyond the observed mediator values.

Under the policy reference:

\mu(0;G^*)=0.60(10)+0.40(14)=6.00+5.60=11.60,

\mu(1;G^*)=0.60(13)+0.40(20)=7.80+8.00=15.80.

The treatment contrast under this common policy distribution is $15.80-11.60=4.20$ outcome units. It lies between the untreated-reference contrast of 3.60 and the treated-reference contrast of 5.10 in this example. That ordering follows from the specified outcome means and mediator distributions; it is not guaranteed in general.

If the policy instead specified a distribution conditional on $C$, the calculation would average the conditional outcome means over that policy distribution at each $C$, then standardize over the same target population of $C$. A policy reference that assigns values outside the mediator support available at a given treatment and covariate level may not be identified from the observed data without additional assumptions.

Assumptions and a Practical Comparison

Changing the reference distribution does not remove the identification requirements for the interventional mean. In particular, consistency and a well-defined treatment and mediator intervention are required; treatment assignment must be exchangeable given measured baseline covariates for the target contrast; and mediator-outcome exchangeability must hold conditional on treatment and measured covariates. The mediational g-formula also requires positivity: the relevant treatment and mediator values must be represented in the data at covariate values that occur in the target population. These conditions are not verified merely by calculating several estimates.

For an alternative $g$, check whether its support is compatible with the observed mediator support at relevant values of $C$. A reference that places substantial probability on poorly supported values can make estimates depend heavily on outcome-model extrapolation. If a policy intentionally targets such values, the resulting analysis needs an explicit account of the additional assumptions or data required.

In estimation by g-computation, fit the outcome regression, predict $Q(a,m,c)$ for mediator values drawn from or averaged over each specified reference, and average over the target covariate distribution. Keep those parts fixed across reference choices if the goal is to isolate reference-distribution sensitivity. Recompute uncertainty for each estimand; intervals from different references are generally dependent because they use the same data and fitted models.

Specify the target and references.
Name the target covariate population and define each mediator distribution, including any mixture weight or policy rule.

Check identification and support.
Assess the assumptions for the outcome and mediator models, and whether each reference assigns probability to supported mediator values.

Re-estimate consistently.
Use the same outcome model and covariate standardization across references, changing only the mediator distribution when isolating its influence.

Report the comparison.
Present estimates with units and uncertainty, identify each reference in words, and describe how conclusions change across defensible choices.

Common Mistakes

Treating the reference as a harmless technical detail. The correction is to name it alongside the effect estimate: different distributions define different intervention questions.
Calling a pooled reference uniquely neutral. Pooling requires a stated rule, such as an equal mixture or a mixture weighted by a target population. Different rules can produce different mediator distributions.
Changing the covariate target at the same time. That confounds sensitivity to the mediator reference with a change in population. Hold the target distribution of $C$ fixed when isolating reference sensitivity.
Interpreting a change across references as evidence of unmeasured confounding. Reference sensitivity concerns which mediator intervention is being evaluated. Unmeasured confounding is a separate identification problem.
Using a policy target beyond observed support without comment. Such estimates may rely on extrapolation. State the support limitation and the assumptions needed to extend beyond the data.

Key takeaway: An interventional effect is indexed by its mediator reference distribution. Comparing treated, untreated, pooled, and policy-defined references reveals how much the answer depends on that choice, provided the target population and other modeling choices are held fixed and identification assumptions remain credible.

Check Your Understanding

Answer each question by identifying the relevant reference distribution or calculation.

In $\mu(1;g)-\mu(0;g)$, what is held common across the two treatment levels?
If $g_\lambda=(1-\lambda)F_0+\lambda F_1$, what references correspond to $\lambda=0$ and $\lambda=1$?
Why should the target distribution of baseline covariates be kept fixed when isolating sensitivity to the mediator reference?
What support concern arises if a policy-defined distribution assigns probability to mediator values not observed at relevant covariate levels?
Why is a change in estimates across mediator references not, by itself, evidence of unmeasured mediator-outcome confounding?

Treatment \(a\)	\(Q(a,0)\)	\(Q(a,1)\)
0	10	14
1	13	20