Mediator Positivity and Support

Why Mediator Support Matters

An interventional mediation effect compares expected outcomes under specified treatment levels while drawing the mediator from a chosen distribution. The draw may come from a different treatment group than the one used to define the outcome response. This creates a basic requirement: the data must contain information about the outcome response at the mediator values that the intervention can draw.

For example, suppose the outcome response is to be evaluated under treatment $A=0$, but the mediator is drawn from its distribution under $A=1$. If some mediator values occur under $A=1$ but never occur under $A=0$, the observed data do not reveal the outcome response under $A=0$ at those values. A fitted model might still produce predictions there, but those predictions rely on extrapolation rather than direct support in the data.

Definition: Let $A$ denote treatment, $M$ the mediator, $Y$ the outcome, and $C$ measured baseline covariates. For treatment levels $a$ and $a^*$, define the interventional mean $$ \psi(a,a^*) = E_C\left[ \int E(Y\mid A=a,M=m,C)\,dF_{M\mid A=a^*,C}(m) \right]. $$ The outcome response is evaluated at $A=a$, while the mediator is drawn from its conditional distribution under $A=a^*$. Mediator positivity for this mean requires that, for almost every covariate value $c$ in the target population, the mediator distribution under $A=a^*$ places no probability on values unsupported by the mediator distribution under $A=a$. Formally, $F_{M\mid A=a^*,C=c}$ must be absolutely continuous with respect to $F_{M\mid A=a,C=c}$.

In a discrete mediator setting, this condition says that whenever $P(M=m\mid A=a^*,C=c)>0$, it must also be true that $P(M=m\mid A=a,C=c)>0$. For a continuous mediator, it concerns the supports of the conditional distributions: target values must lie within the support available under $A=a$, apart from sets having zero probability under the target draw. It does not require the two mediator distributions to be identical.

The Positivity Conditions Behind the Formula

Mediator support is one part of the conditions needed to interpret the displayed mean as identified from observed data. The treatment levels being compared must also occur with positive probability within the relevant covariate strata. In addition, identification requires consistency and suitable conditional exchangeability: measured baseline covariates must suffice to address treatment-outcome and treatment-mediator confounding, and there must be no unmeasured mediator-outcome confounding after conditioning on treatment and $C$. This simple formula also presumes that any relevant mediator-outcome confounders are not affected by treatment, or that the identification approach properly accounts for them.

For $\psi(a,a^*)$, treatment positivity requires both $P(A=a\mid C=c)>0$ and $P(A=a^*\mid C=c)>0$ for the covariate values being averaged over. The mediator support condition is directional: it compares the mediator distribution used for the draw, $F_{M\mid A=a^*,C}$, with the mediator distribution in the group supplying the outcome response, $F_{M\mid A=a,C}$. Reversing $a$ and $a^*$ can therefore change whether positivity holds.

Worked Example: A Binary Mediator With an Unsupported Target Value

Consider a hypothetical study with binary treatment $A$, binary mediator $M$, and a baseline covariate $C$ taking values $L$ and $H$. Suppose each covariate stratum makes up half of the target population, both treatment levels occur in both strata, and the conditional mediator probabilities are:

Covariate stratum	$P(M=1\mid A=0,C)$	$P(M=1\mid A=1,C)$
$L$	0.20	0.80
$H$	0	0.60

Assess mediator positivity for $\psi(0,1)$. This mean evaluates the outcome response under $A=0$ while drawing $M$ from its distribution under $A=1$, separately within each covariate stratum.

In stratum $L$, the target draw assigns probability 0.80 to $M=1$, and $M=1$ also occurs under $A=0$ with probability 0.20. Thus both mediator values used by the target draw have support in the $A=0$ group.

In stratum $H$, the target draw assigns probability 0.60 to $M=1$, but $P(M=1\mid A=0,C=H)=0$. The outcome response under $A=0$ at $M=1,C=H$ is unsupported. Since stratum $H$ has population weight 0.50, the overall target draw places probability $0.50(0.60)=0.30$ on this unsupported covariate-mediator combination. Therefore, $\psi(0,1)$ is not identified by the observed conditional distributions without additional assumptions.

By contrast, in stratum $H$, the target draw for $\psi(1,0)$ comes from $A=0$ and only assigns $M=0$, which also occurs under $A=1$. The support requirement for that stratum and direction is satisfied. This contrast in directions illustrates why support must be checked for the particular interventional mean, not merely declared once for the study.

Why the Support Condition Is Necessary

Write $Q(a,m,c)=E(Y\mid A=a,M=m,C=c)$. Under the exchangeability and consistency conditions stated above, the observed data identify this outcome response only where the corresponding treatment-mediator-covariate combinations have support. When the mediator distribution used for the draw is supported by the mediator distribution under $A=a$, the target integral can be expressed using the latter distribution:

\int Q(a,m,c)\,dF_{M\mid A=a^*,C=c}(m) = \int Q(a,m,c) \frac{dF_{M\mid A=a^*,C=c}}{dF_{M\mid A=a,C=c}}(m) \,dF_{M\mid A=a,C=c}(m).

The derivative in this expression is a density ratio (or, in a discrete setting, a ratio of probabilities). Its existence requires the target mediator distribution to be absolutely continuous with respect to the distribution under $A=a$. In the discrete case, a target-positive mediator value with zero probability under $A=a$ would require division by zero.

The same point can be seen without the density-ratio expression. Suppose a set of mediator values has positive probability under the target draw but zero probability under $A=a$, conditional on some covariate stratum. The observed data then contain no outcomes for $A=a$ in that set. Two possible outcome-response functions could agree everywhere the data provide support and disagree on that unsupported set. They would imply the same observed data distribution but different values of $\psi(a,a^*)$. Thus the mean cannot be recovered from the observed data alone.

This argument establishes the direction of the condition: the mediator distribution used for the interventional draw must be supported by the mediator distribution in the treatment group whose outcome response is being averaged. The condition is not that every treatment group have the same mediator distribution, nor that the distributions have identical shapes.

Conditional Support Cannot Be Replaced by Marginal Overlap

Support must be examined conditional on $C$, because the interventional draw is conditional on baseline covariates before the results are averaged over their target distribution. Marginal mediator distributions can conceal a mismatch: a mediator value may be observed in both treatment groups overall, yet occur in different covariate strata in each group.

Worked Example: Marginal Overlap Hides a Conditional Failure

Suppose a hypothetical target population has $P(C=L)=P(C=H)=0.50$, and both treatment levels occur in both strata. The mediator is binary, with these deterministic conditional distributions:

Covariate stratum	Mediator under $A=0$	Mediator under $A=1$
$L$	$M=1$ with probability 1	$M=0$ with probability 1
$H$	$M=0$ with probability 1	$M=1$ with probability 1

Marginally, $M=0$ and $M=1$ each occur with probability 0.50 under both treatment levels. A check that compared only the two marginal mediator distributions might therefore suggest complete overlap.

For $\psi(0,1)$, however, consider stratum $L$. The target draw under $A=1$ assigns probability 1 to $M=0$, while $M=0$ has probability 0 under $A=0,C=L$. In stratum $H$, the target draw assigns probability 1 to $M=1$, while $M=1$ has probability 0 under $A=0,C=H$. Conditional support fails in both strata, even though the marginal distributions are identical. The relevant outcome responses under $A=0$ are not supported at the mediator values the target draw requires.

The conclusion is not that marginal summaries are useless; they can provide an initial description. They cannot establish the conditional positivity needed for the interventional mean. Support checks must retain the covariates used in the identification formula.

Continuous Mediators and Weak Overlap

For a continuous mediator, individual values generally have probability zero, so positivity is not assessed by asking whether the exact same value appears repeatedly in each treatment group. Instead, compare the conditional distributions or their densities and supports. For example, if $M\mid A=0,C=c$ has support $[0,1]$ while $M\mid A=1,C=c$ has support $[0,2]$, then the target draw for $\psi(0,1)$ includes values between 1 and 2 that are unsupported under $A=0$.

Worked Example: Directional Support for a Continuous Mediator

Suppose that in each relevant covariate stratum, a hypothetical continuous mediator is uniformly distributed on $[0,1]$ under $A=0$ and uniformly distributed on $[0,2]$ under $A=1$. For $\psi(0,1)$, the target draw has density $1/2$ over $[0,2]$. The outcome response under $A=0$ is supported only over $[0,1]$. The target draw assigns probability

P(1<M\leq 2\mid A=1,C=c)=\frac{2-1}{2}=0.50

to values outside the $A=0$ support. Consequently, half of the mediator draws in this stratum require an unsupported outcome response. A model may return predictions for those values, but the observations do not identify those predictions without additional modeling assumptions.

For the reverse direction, $\psi(1,0)$, the target distribution is supported on $[0,1]$, which lies within the $A=1$ support $[0,2]$. The support condition is satisfied in that direction. This does not guarantee that the effect is estimated precisely; it says only that the strict support failure described above is absent.

Strict positivity and practical overlap are related but distinct. A strict violation occurs when a target-positive region has zero probability under the outcome-treatment distribution, preventing nonparametric identification of the relevant response there. Weak overlap occurs when the required region technically has positive support but is sparsely represented. In the density-ratio expression, weak overlap can produce very large ratios, so a small number of observations may carry substantial influence. Estimates can then be unstable and sensitive to modeling choices, even when the formal positivity condition holds.

Diagnosing Support and Avoiding Extrapolation

A useful support assessment follows the structure of the estimand: identify the outcome treatment level, identify the treatment level defining the mediator draw, and compare the corresponding conditional mediator distributions within the target covariate strata. The diagnostic should not stop at an overall comparison.

Specify the direction.
For $\psi(a,a^*)$, the mediator is drawn from $A=a^*$, while the outcome response comes from $A=a$. Check that particular pairing.

Condition on baseline covariates.
Compare mediator distributions within relevant values or regions of $C$, not only after pooling across strata.

Locate unsupported or sparse regions.
For discrete mediators, inspect conditional probabilities. For continuous mediators, examine conditional ranges, densities, and regions with little data.

State what the diagnosis implies.
Separate a structural zero, which violates support, from weak overlap, which can make estimates unstable and model-dependent.

A common mistake is to say, “Both treatment groups contain low and high mediator values overall, so the positivity assumption is satisfied.” This reasoning ignores the covariates that define the conditional draw. The correction is to compare the mediator distribution in the target draw group with the outcome-treatment group within each relevant covariate stratum. Another mistake is to treat a model prediction outside observed support as new empirical information. A prediction can be computed there, but its validity comes from the model’s extrapolation assumptions, not from observed outcomes at those mediator values.

Key takeaway: For an interventional mediator draw, conditional support is directional and covariate-specific. The draw distribution must not assign positive probability to mediator values where the outcome-treatment group provides no support. Weak overlap may not destroy identification, but it can make estimates fragile and heavily dependent on extrapolation.

Check Your Understanding

Use the direction of the mediator draw and the conditioning on baseline covariates in each answer.

For $\psi(1,0)$, which treatment group supplies the outcome response, and which supplies the mediator draw?
In a discrete mediator setting, state the conditional probability condition required for mediator positivity.
Why can identical marginal mediator distributions under $A=0$ and $A=1$ coexist with a conditional support violation?
If a target mediator distribution has support $[0,2]$ and the outcome-treatment group has support $[0,1]$, which direction of the interventional mean violates support?
How does weak overlap differ from a structural positivity violation, and why can either matter for an estimate?

Covariate stratum	Mediator under \(A=0\)	Mediator under \(A=1\)
\(L\)	\(M=1\) with probability 1	\(M=0\) with probability 1
\(H\)	\(M=0\) with probability 1	\(M=1\) with probability 1