Conditional Versus Marginal Mediator Distributions

Two Ways to Draw a Reference Mediator

A stochastic mediator intervention replaces an individual's mediator value with a draw from a specified distribution. The reference distribution matters: should the draw reflect the mediator distribution among people with the individual's baseline covariates, or the distribution across a target population as a whole? These choices can assign different mediator values to the same person and can therefore produce different mean outcomes.

Let \(A\) denote treatment, \(L\) baseline covariates measured before treatment, \(M\) a mediator, and \(Y\) the outcome. Fix a target population with covariate distribution \(F_L\). For a reference treatment level \(a^*\), write \(f(m\mid A=a^*,L=l)\) for the mediator distribution among people with covariates \(l\) at that treatment level. The target population must be specified because it determines how covariate-specific mediator distributions are combined.

The conditional intervention retains a link between a recipient's baseline covariates and the mediator draw: people with different \(L\) can receive draws from different distributions. The marginal intervention instead uses the same distribution for every recipient. In particular, the draw is not tailored to the recipient's \(L\), even though \(L\) can still predict the outcome.

The marginal reference distribution here is standardized to \(F_L\). It is not necessarily the crude distribution of \(M\) among observed people with \(A=a^*\). If treatment groups have different covariate distributions, that crude distribution mixes the conditional mediator distributions using the treatment group's covariate mix, not necessarily the target population's mix.

Mean-Outcome Estimands

Before interpreting either quantity causally, assume that \(L\) is pre-treatment, treatment and mediator interventions are well defined, and consistency holds. For identification from observational data, assume adequate control of treatment-outcome and treatment-mediator confounding by \(L\), and adequate control of mediator-outcome confounding by \(A,L\). In particular, there must be no unmeasured common causes of \(M\) and \(Y\) after conditioning on \(A,L\), including no treatment-induced mediator-outcome confounder left uncontrolled. The relevant treatment and mediator probabilities or densities must also be positive in the covariate strata used. These are identification requirements, not consequences of choosing a conditional or marginal distribution.

Let \(\mu(a,m,l)=E[Y\mid A=a,M=m,L=l]\). Under those assumptions, the mean outcome under treatment level \(a\) and the conditional reference intervention is

For the marginal reference intervention, the corresponding mean is

Both estimands average outcomes over the target population's \(L\) distribution and use the outcome response under treatment \(a\). Their distinction is the mediator draw: the first uses the recipient's \(L\), whereas the second uses the target-standardized marginal distribution. Neither formula pairs a person's outcome under one treatment with that same person's mediator value under a different treatment. Thus, these stochastic-intervention estimands do not require the cross-world independence assumption used for natural effects.

For either choice of reference distribution, a contrast such as \(\psi(a,a_1)-\psi(a,a_0)\) compares two mediator interventions while holding the outcome treatment at \(a\). A contrast such as \(\psi(a_1,a^*)-\psi(a_0,a^*)\) compares outcome treatment levels while holding the mediator intervention fixed. The choice between \(\psi_C\) and \(\psi_M\) is part of the estimand: it is not just a choice of how to calculate the same quantity.

Why the Difference Arises

A short derivation makes the source of the difference precise. For a binary mediator, define \(q(l)=P(M=1\mid A=a^*,L=l)\), and suppose the outcome mean under treatment \(a\) is linear in \(M\) within each stratum: \(\mu(a,m,l)=\alpha(l)+\beta(l)m\). Let \(\bar q=\int q(l)\,dF_L(l)\). Under the conditional intervention, the mean is the target average of \(\alpha(l)+\beta(l)q(l)\). Under the marginal intervention, \(M\) has probability \(\bar q\) in every stratum, so the mean is the target average of \(\alpha(l)+\beta(l)\bar q\). Subtracting yields

Here the covariance is over the target distribution of \(L\). Thus, under this outcome model, the two means agree if the mediator effect \(\beta(L)\) is constant across strata or if the reference probability \(q(L)\) is constant. More generally, they may also agree through cancellation. In other settings, equality depends on the full outcome response and mediator distributions; it should not be presumed just because both interventions use the same reference treatment.

Choosing and Reporting the Intervention

Neither choice is universally preferable. A conditional reference distribution asks what mean outcome would result if each person received a mediator draw like those of reference-treatment individuals with the same baseline covariates. A marginal reference distribution asks what would result if everyone received a draw from the target population's overall reference mediator distribution, regardless of their own covariates. The substantive question determines which intervention is relevant.

State the target population, reference treatment, and whether the mediator draw is conditional on recipient covariates. Report the resulting contrast in outcome units and avoid describing one intervention as a mere adjustment to the other. In particular, a marginal draw can break the association between \(L\) and \(M\) that the conditional intervention preserves. That change can matter when covariates modify the effect of the mediator on the outcome.

• Pitfall: “Both interventions use the mediator distribution under \(a^*\), so they estimate the same effect.” This overlooks whether the draw is tailored to each recipient's \(L\).
• Correction: Write down the distribution used to draw \(M\), including how it is averaged over covariates. Then evaluate the corresponding mean-outcome functional. Equality requires conditions on the distributions and outcome response, not just a shared reference treatment.
• Pitfall: Calling the crude mediator distribution in the observed \(A=a^*\) group the target marginal distribution without checking its covariate mix.
• Correction: Standardize the conditional mediator distributions over the intended target \(F_L\), unless the observed reference group itself is explicitly the target population.