Why the Choice of Mediator Intervention Matters
A natural indirect effect asks what would happen to an individual’s outcome under one treatment level if that same individual’s mediator took the value it would have taken under another treatment level. This is a scientifically meaningful comparison, but it links counterfactuals from different treatment conditions for the same person. Data do not directly reveal that cross-condition link.
An interventional effect uses a different comparison. Instead of assigning each person their own counterfactual mediator value, it draws a mediator value from a specified distribution—for example, the distribution that would be observed under the control treatment. The draw is made independently of the person’s outcome counterfactual, conditional on baseline covariates. This change in the estimand avoids requiring a particular association between a person’s natural mediator and their outcome under another treatment.
That does not make interventional effects assumption-free. Identifying them from observed data still requires suitable treatment and mediator exchangeability conditions, along with consistency and positivity. The distinction is about which counterfactual relationship must be assumed, not whether causal assumptions are needed at all.
Natural Effects and the Cross-World Link
For binary treatment, one natural direct effect on the mean-difference scale is \(E[Y(1,M(0))-Y(0,M(0))]\). One natural indirect effect is \(E[Y(1,M(1))-Y(1,M(0))]\). Both involve nested counterfactuals: the outcome under one intervention is evaluated using a mediator counterfactual from another intervention.
A standard identifying condition for natural effects is the cross-world independence assumption: conditional on baseline covariates, the potential outcome under a fixed treatment and mediator value is independent of the mediator that would occur under another treatment. In notation, for relevant \(a,m,a'\),
This is not merely ordinary treatment randomization or a claim that the mediator is unrelated to the observed outcome. It concerns the joint distribution of counterfactuals that cannot be observed together for the same person. The assumption supplies a cross-world link needed to express a natural nested mean using separate distributions of outcomes and mediators.
Worked Example: Same Observed Distributions, Different Natural Effects
Consider a hypothetical population with no baseline covariates and binary \(A\), \(M\), and \(Y\). In each of two possible counterfactual worlds, half the people have \(Z=Y(1,0)=1\), and half have \(Z=0\). Set \(Y(1,1)=0\) for everyone and \(Y(0,m)=0\) for both mediator values. Let \(M(0)\) be equally likely to be 0 or 1, and let \(M(1)\) be a fair binary draw independent of \(Z\) and \(M(0)\).
The worlds differ only in how \(M(0)\) pairs with \(Z\). In World 1, \(M(0)=1-Z\); in World 2, \(M(0)=Z\). In both worlds, the marginal probability that \(M(0)=1\) is \(0.5\), and the marginal probability that \(Z=1\) is \(0.5\).
These worlds produce the same observed distributions under either treatment. Under \(A=0\), the outcome is always 0 and the mediator is equally likely to be 0 or 1. Under \(A=1\), the mediator is an independent fair draw: when \(M=0\), the outcome has mean \(0.5\); when \(M=1\), the outcome is 0.
Now calculate the natural nested mean \(E[Y(1,M(0))]\). In World 1, \(M(0)=0\) exactly when \(Z=1\), so half the population has \(Y(1,M(0))=Y(1,0)=1\), and the rest has outcome 0. The mean is \(0.5\). In World 2, whenever \(M(0)=0\), \(Z=0\), so \(Y(1,0)=0\); when \(M(0)=1\), \(Y(1,1)=0\). The mean is therefore 0.
The observed distributions do not determine which cross-world pairing is correct. In contrast, the interventional mean using the control mediator distribution is fixed by the marginals:
The stochastic draw combines the mediator distribution under control with the outcome response under treatment without asserting how each person’s \(M(0)\) pairs with \(Y(1,m)\). This is the specific cross-world independence requirement the interventional estimand avoids.
Interventional Effects and Their Identification
For binary treatment, an interventional direct effect holding the mediator distribution at level \(a'\) is \(\psi(1,a')-\psi(0,a')\). An interventional indirect effect evaluated at outcome-treatment level \(a\) is \(\psi(a,1)-\psi(a,0)\). These contrasts compare means under specified interventions; they are not generally the same quantities as natural direct or indirect effects.
A standard set of conditions for identifying these interventional means from observed data is as follows. Consistency means that observed outcomes and mediators agree with their counterfactual values under the interventions actually received. Treatment exchangeability requires no unmeasured confounding of treatment and the relevant potential mediators and outcomes, conditional on \(C\). Mediator-outcome exchangeability requires no unmeasured confounding of the mediator and potential outcomes, conditional on treatment and \(C\). Positivity requires that the treatment and mediator values used in the calculation have support at the relevant covariate values. The covariates \(C\) must be pre-treatment for this standard formula.
The first condition is treatment exchangeability; it can follow from randomization of \(A\), possibly conditional on \(C\). The second condition is mediator-outcome exchangeability. It does not follow just because treatment was randomized: a randomized treatment does not necessarily randomize the mediator. If an unmeasured common cause of \(M\) and \(Y\) remains after conditioning on \(A,C\), this condition fails. The simple formula also does not cover mediator-outcome confounders caused by treatment; the stated conditions and formula presume no such post-treatment confounder is needed.
Under these conditions, the mediational g-formula identifies the interventional mean:
For discrete variables, the integrals are replaced by sums. The outcome regression is evaluated at treatment \(a\); the mediator distribution is taken at treatment \(a'\); and the result is averaged over the target population’s baseline covariate distribution.
Why the G-Formula Does Not Need Cross-World Independence
The identification argument uses two separate observed-data links. Treatment exchangeability and consistency identify the conditional outcome mean \(E[Y(a,m)\mid C=c]\) from \(E[Y\mid A=a,M=m,C=c]\), given mediator exchangeability as well. Treatment exchangeability and consistency identify the conditional mediator distribution of \(M(a')\) from \(f(m\mid A=a',C=c)\). These are within-regime identifications: each links a counterfactual to observations under the same intervention values.
The remaining step is part of the interventional definition. Conditional on \(C=c\), the auxiliary draw \(G_{a'}\) is independent of the outcome counterfactual. Therefore, averaging the response at each \(m\) over the draw’s distribution gives:
Averaging both sides over \(f(c)\) yields the displayed g-formula. The factorization uses the independence built into the auxiliary stochastic intervention; it does not assert that the naturally occurring \(M(a')\) is independent of \(Y(a,m)\) for each person. This is the key distinction: the interventional target defines a new independent draw, whereas the natural target uses the person’s own counterfactual mediator.
The exchangeability assumptions remain essential to the identification steps. If mediator-outcome exchangeability fails, the observed conditional outcome mean may not equal the mean response under an intervention setting \(M=m\). Calling the mediator draw “random” does not repair confounding in the data used to estimate that response.
Worked Example: Interventional Effects Without Covariates
Suppose a hypothetical randomized study has a binary mediator and a continuous outcome measured in symptom points. Assume consistency, treatment randomization, mediator-outcome exchangeability conditional on \(A\), and positivity for both mediator values. The observed mediator probabilities are \(P(M=1\mid A=0)=0.3\) and \(P(M=1\mid A=1)=0.6\). The outcome means are 3 points for \((A=0,M=0)\), 5 for \((A=0,M=1)\), 4 for \((A=1,M=0)\), and 8 for \((A=1,M=1)\).
The interventional mean \(\psi(1,0)\) applies the treatment-1 outcome response to the control mediator distribution. Since \(P(M=0\mid A=0)=0.7\),
Similarly, \(P(M=0\mid A=1)=0.4\), so
The interventional indirect effect at \(a=1\) is \(6.4-5.2=1.2\) symptom points. For comparison, the control-outcome mean under the control mediator distribution is \( \psi(0,0)=0.7(3)+0.3(5)=2.1+1.5=3.6 \) points. The direct effect holding the mediator distribution at control is \(5.2-3.6=1.6\) points. Their sum, \(1.2+1.6=2.8\) points, equals the total interventional contrast \(6.4-3.6=2.8\) points. This calculation identifies the specified stochastic contrasts under the stated assumptions; it does not establish the corresponding natural effects without an additional cross-world condition.
Baseline Covariates and Standardization
When baseline covariates influence treatment, the mediator, or the outcome, the interventional formula averages within covariate strata before averaging over the target population. This matters because the mediator distribution and the outcome response can both differ across those strata. The example below uses the same \(C\) distribution for each interventional mean and explicitly checks the standardization.
Worked Example: Standardizing Across Two Baseline Strata
In a hypothetical population, half the individuals have \(C=L\) and half have \(C=H\). Assume consistency, treatment exchangeability conditional on \(C\), mediator-outcome exchangeability conditional on \(A,C\), and positivity for the treatment and mediator values shown. The probabilities of \(M=1\) under \(A=0\) are 0.2 for \(C=L\) and 0.8 for \(C=H\); under \(A=1\), they are 0.6 and 0.9, respectively. The outcome means, in outcome units, are:
| Stratum | \(E[Y\mid A=0,M=0,C]\) | \(E[Y\mid A=0,M=1,C]\) | \(E[Y\mid A=1,M=0,C]\) | \(E[Y\mid A=1,M=1,C]\) |
|---|---|---|---|---|
| \(C=L\) | 1 | 4 | 2 | 6 |
| \(C=H\) | 4 | 8 | 5 | 9 |
First calculate \(\psi(1,0)\). In stratum \(L\), the control mediator distribution gives \(0.8(2)+0.2(6)=1.6+1.2=2.8\). In stratum \(H\), it gives \(0.2(5)+0.8(9)=1.0+7.2=8.2\). Standardizing over the equally weighted strata,
For \(\psi(1,1)\), use the treatment-1 mediator probabilities. The stratum means are \(0.4(2)+0.6(6)=4.4\) and \(0.1(5)+0.9(9)=8.6\). Thus \( \psi(1,1)=0.5(4.4)+0.5(8.6)=6.5 \), and the interventional indirect effect at \(a=1\) is \(6.5-5.5=1.0\) outcome unit.
For a comparison direct effect, calculate \(\psi(0,0)\). Its stratum means are \(0.8(1)+0.2(4)=1.6\) and \(0.2(4)+0.8(8)=7.2\). Therefore, \( \psi(0,0)=0.5(1.6)+0.5(7.2)=4.4 \). The direct effect holding the mediator distribution at control is \(5.5-4.4=1.1\) outcome units. The total contrast is \(6.5-4.4=2.1\), equal to \(1.0+1.1\). This arithmetic depends on the stated exchangeability and positivity conditions; stratifying by \(C\) does not by itself guarantee that they hold.
Common Pitfalls and Why the Distinction Matters
- Claiming interventional effects need no assumptions. They avoid the natural cross-world independence condition, but still require valid treatment and mediator exchangeability, consistency, and positivity for the g-formula identification described here.
- Calling the stochastic draw the person’s natural mediator. \(G_{a'}\) is an auxiliary draw from a specified distribution. It is not \(M(a')\) for that same individual.
- Assuming treatment randomization controls mediator confounding. Randomizing \(A\) addresses treatment exchangeability, but a common cause of \(M\) and \(Y\) can still violate mediator-outcome exchangeability.
- Treating natural and interventional effects as interchangeable. They are different estimands. Their values may coincide under additional conditions, but that is not guaranteed by their definitions.
- Ignoring covariate strata. If \(C\) affects the mediator or outcome, plugging in an overall mediator distribution and outcome mean can fail to implement the stated conditional stochastic intervention. The formula standardizes within \(C\) and then averages over its target distribution.
Interventional effects are useful when a mediator distribution is a meaningful target for intervention and the natural cross-world pairing is not defensible or not desired. Their advantage is a different, often more policy-oriented estimand—not a shortcut around confounding control. Any causal interpretation remains conditional on the identification assumptions being credible in the application.
Check Your Understanding
Answer each question by distinguishing the estimand from the assumptions used to identify it.
- In \(Y(1,M(0))\), which counterfactual is evaluated under treatment 1, and what makes the expression cross-world?
- What does the auxiliary draw \(G_{0}\) represent in \(\psi(1,0)\), and how does it differ from an individual’s \(M(0)\)?
- State the cross-world independence condition commonly used to identify natural effects, and explain why the interventional g-formula does not require it.
- If treatment is randomized but an unmeasured variable confounds the mediator-outcome relation, which exchangeability condition can fail?
- In the two-stratum worked example, why are the stratum-specific mediator probabilities and outcome means combined before averaging over \(C\)?