Deriving the Mediational G-Formula

From a Stochastic Mediator Intervention to an Observed-Data Formula

Interventional mediation effects compare outcomes under specified treatment levels while assigning the mediator a distribution associated with a chosen treatment level. To calculate such an effect from observed data, two ingredients must be combined: the expected outcome at a given treatment, mediator, and covariate value, and the distribution from which the mediator is drawn. The resulting average is the mediational g-formula.

Let \(A\) denote treatment, \(M\) a mediator measured after treatment, \(Y\) the outcome, and \(C\) a set of baseline covariates. For simplicity, take \(A\in\{0,1\}\). Write \(Y(a,m)\) for the outcome that would occur if treatment were set to \(a\) and the mediator to \(m\), and \(M(a)\) for the mediator value that would occur under treatment \(a\).

We use the following contrasts: the interventional direct effect at the reference mediator distribution is \(\psi(1,0)-\psi(0,0)\); the interventional indirect effect under treatment level 1 is \(\psi(1,1)-\psi(1,0)\). Their sum is the total contrast \(\psi(1,1)-\psi(0,0)\). These are additive contrasts on the mean outcome scale.

Identification Conditions and the Observed-Data Ingredients

The formula below is an identification result: it expresses a causal estimand in terms of the observed-data distribution under stated assumptions. It is not, by itself, a prescription for fitting models or calculating uncertainty. Assume that \(C\) is measured before treatment and is sufficient to control the relevant confounding of treatment with the mediator and outcome, and of the mediator with the outcome.

The exchangeability conditions rule out unmeasured confounding for the relations needed in this calculation. In particular, the stated setup does not accommodate an unmeasured mediator-outcome confounder or an exposure-induced mediator-outcome confounder left outside the covariate history. Positivity is needed because the formula combines an outcome regression at \(A=a\) with mediator values drawn from the distribution under \(A=a^*\). That combination must be supported by observed data.

Define the observed conditional outcome mean as \(\mu(a,m,c)=E(Y\mid A=a,M=m,C=c)\). Define \(F(m\mid a^*,c)\) as the conditional distribution of \(M\) given \(A=a^*,C=c\), and \(F_C\) as the target population distribution of \(C\). In the discrete case, write \(p(m\mid a^*,c)\) for the corresponding conditional probability.

Deriving the Mediational G-Formula

The derivation first conditions on \(C\), then averages over the mediator draw. By the definition of the stochastic intervention, conditional on \(C=c\), the draw has distribution \(F_{M(a^*)\mid C=c}\). Treatment exchangeability and consistency identify that distribution with the observed mediator distribution \(F(m\mid a^*,c)\). For the outcome, treatment exchangeability, mediator exchangeability, and consistency identify the mean under fixed \(a,m,c\) with \(\mu(a,m,c)\). Thus the conditional intervention mean is the outcome regression averaged over the chosen mediator distribution:

Finally, average this conditional mean over the covariate distribution of the target population. This gives the observed-data g-formula:

For discrete \(C\) and \(M\), the same expression is a pair of sums:

The inner average uses the outcome regression from treatment arm \(a\) and the mediator distribution from arm \(a^*\), both conditional on the same covariate value. The outer average uses the covariate distribution in the target population. The formula therefore does not generally reduce to evaluating the outcome regression at an average mediator value. Nor can the conditional mediator distribution generally be replaced by a marginal distribution that ignores \(C\).

Why the Order of Averaging Matters

The g-formula makes the target population and the mediator intervention explicit. First, within each covariate stratum, it averages the outcome regression over the selected mediator distribution; then it averages those results over \(F_C\). This order respects the fact that both the mediator distribution and the conditional outcome mean can vary with baseline covariates.

A common mistake is to replace \(P(M=m\mid A=a^*,C=c)\) with \(P(M=m\mid A=a^*)\), then combine that marginal mediator distribution with outcome regressions conditional on \(C\). This generally changes the intervention: it no longer assigns mediator values according to the chosen treatment-specific distribution within covariate strata. Another mistake is to substitute \(E(M\mid A=a^*,C=c)\) into \(\mu(a,m,c)\). That is valid only in special cases, such as a regression linear in \(m\) with no features that make other aspects of the mediator distribution relevant. The formula requires integration over the distribution, not a plug-in at its mean.

The formula also clarifies the roles of its components. The outcome regression describes how the mean outcome varies with treatment, mediator, and covariates. The mediator distribution specifies which mediator values are assigned under the stochastic intervention. The covariate distribution determines how stratum-specific means are combined for the target population. Identification depends on the assumptions and support conditions, not merely on having written down these three ingredients.