Exposure-Induced Mediator-Outcome Confounding

How Treatment Can Create Confounding

A mediator–outcome confounder is a common cause of the mediator and outcome. When that common cause is measured before treatment, it is a baseline variable. A different problem arises if treatment itself changes the common cause: the variable is then both downstream of treatment and a cause of the mediator and outcome. This is called exposure-induced mediator–outcome confounding.

The timing matters. Treatment may alter a patient’s early clinical state, which can affect both whether the patient adheres to a later treatment component (the mediator) and the final outcome. The early state is not simply a baseline characteristic to condition on: its value may differ depending on treatment. Natural effects combine counterfactuals from different treatment conditions, so the relevant mediator and outcome counterfactuals can involve different counterfactual values of that state.

Definition: An exposure-induced mediator–outcome confounder is a post-treatment variable that is causally affected by the exposure and is also a cause of both the mediator and the outcome. In the graph below, $L$ has this role: $$ A \to L,\qquad L \to M,\qquad L \to Y. $$ The graph may also contain $A \to M$, $A \to Y$, and $M \to Y$. These arrows represent possible causal effects, not necessarily effects that are present in every application.

Here $A$ denotes exposure or treatment, $L$ the post-treatment variable, $M$ the mediator, and $Y$ the outcome. We omit baseline covariates to keep the notation focused; such covariates can be included in a fuller analysis. The key feature is not merely that $L$ predicts $M$ and $Y$. It is that $A$ can change $L$, and $L$ can then affect both $M$ and $Y$.

Counterfactuals With a Post-Treatment Confounder

Write $L(a)$ for the value of $L$ under treatment level $a$. Write $M(a,l)$ for the mediator value if treatment were set to $a$ and $L$ to $l$, and $Y(a,l,m)$ for the outcome under those same settings and mediator value $m$. These are potential outcomes: each describes a counterfactual intervention, whether or not that intervention is observed for a particular person.

Under the natural course of treatment $a$, the mediator is $M(a,L(a))$. The outcome with treatment set to $a$ and the mediator set to $m$, while $L$ follows its natural value under $a$, is $Y(a,L(a),m)$. Thus a natural indirect effect at treatment level 1 compares the outcome with the mediator under treatment 1 against the outcome with the mediator under treatment 0, while retaining treatment 1 for the outcome and its post-treatment variable.

Definition: In this notation, the natural indirect effect at treatment level 1 is $$ \operatorname{NIE}(1) = E\!\left[ Y\{1,L(1),M(1,L(1))\} - Y\{1,L(1),M(0,L(0))\} \right]. $$ The second term combines the outcome-world value $L(1)$ with the mediator-world value $M(0,L(0))$. This cross-world combination is central to the identification difficulty.

The corresponding natural direct effect at the reference mediator value is $E[Y\{1,L(1),M(0,L(0))\}-Y\{0,L(0),M(0,L(0))\}]$. The precise decomposition depends on the chosen natural-effect definitions and scale. For the present issue, the important point is that the mediator counterfactual under treatment 0 depends on $L(0)$, while the outcome under treatment 1 naturally involves $L(1)$.

Worked Example: Tracing One Person’s Counterfactuals

Consider a hypothetical patient with $L(0)=0$ and $L(1)=1$. Suppose the mediator responds to $L$ according to $M(a,l)=l$, and the outcome is given by $Y(a,l,m)=10a+4l+6m$, measured in hypothetical outcome points. These simple structural rules are stipulated for illustration; they are not estimates from a real study.

Under treatment 1, the patient has $L(1)=1$, so $M(1,L(1))=M(1,1)=1$. The outcome with the treatment-1 mediator is $Y(1,1,1)=10+4+6=20$ points.

Under treatment 0, the mediator is $M(0,L(0))=M(0,0)=0$. To form the second term of the natural indirect effect at treatment 1, keep the outcome-world treatment and $L$ at their treatment-1 values, but use this treatment-0 mediator: $Y(1,1,0)=10+4+0=14$ points.

For this patient, the natural indirect contrast is $20-14=6$ points. Notice the combination in the second outcome: it uses $L(1)=1$ alongside a mediator value generated using $L(0)=0$. This is a well-defined nested counterfactual in the stipulated model, but it is not the patient’s observed configuration under either treatment condition.

Why the Natural-Effect Target Is Difficult to Identify

A natural effect is a population average of counterfactual outcomes. Randomization, if present, can support comparisons between treatment groups, but it does not reveal both $L(0)$ and $L(1)$ for the same person. Nor does it directly reveal the outcome under treatment 1 with the mediator value that would have arisen under treatment 0. The target therefore involves cross-world information: counterfactual quantities indexed by different treatment assignments for the same individual.

The usual mediation formula for natural effects relies on conditions that rule out mediator–outcome confounding after accounting for baseline covariates, together with the relevant treatment and positivity conditions. An exposure-induced confounder violates the intended structure of that formula. In particular, the treatment-0 mediator distribution reflects $L(0)$, but outcomes under treatment 1 reflect $L(1)$. The data do not generally provide the joint counterfactual information needed to connect those two values within individuals.

The issue can be seen by expanding the indirect-effect contrast:

\begin{aligned} \operatorname{NIE}(1) &= E\!\left[ Y\{1,L(1),M(1,L(1))\} \right]\\ &\quad - E\!\left[ Y\{1,L(1),M(0,L(0))\} \right]. \end{aligned}

The first expectation concerns treatment-1 counterfactuals. The second requires the mediator under treatment 0, which depends on $L(0)$, inside an outcome that uses $L(1)$. Observed data can inform distributions under treatment 0 and distributions under treatment 1, subject to the design and assumptions. Those separate distributions do not, by themselves, determine how $L(0)$ and $L(1)$ are paired across individuals or the corresponding cross-world outcome.

Worked Example: Same Observed Data, Different Natural Indirect Effects

Consider a hypothetical population with two equally common types. Treatment is randomized. For simplicity, define $M(a,l)=l$ and $Y(a,l,m)=lm$, with binary $L$, $M$, and $Y$. Compare two possible counterfactual populations:

Model	Type 1: $(L(0),L(1))$	Type 2: $(L(0),L(1))$
Model I	(0, 0)	(1, 1)
Model II	(0, 1)	(1, 0)

In both models, half the population has $L=0$ and half has $L=1$ under either treatment. Under the observed treatment $a$, consistency gives $L=L(a)$, $M=L(a)$, and $Y=L(a)^2=L(a)$. Thus, in each treatment group, both models produce the same observed distribution: half have $(L,M,Y)=(0,0,0)$ and half have $(1,1,1)$.

For the natural indirect effect at treatment 1, the first outcome term is $Y\{1,L(1),M(1,L(1))\}=L(1)$, whose mean is $0.5$ in both models. The second term is $Y\{1,L(1),M(0,L(0))\}=L(1)L(0)$. In Model I, its mean is $(0+1)/2=0.5$, so the indirect effect is $0.5-0.5=0$. In Model II, its mean is $(0+0)/2=0$, so the indirect effect is $0.5-0=0.5$.

The observed treatment-specific distributions are identical, yet the natural indirect effects differ. This deliberately simple construction makes the missing cross-world pairing visible; because $M=L$, it also has a positivity limitation. It illustrates why the observed distributions alone do not generally identify the natural effect in this structure, rather than serving as a general identification theorem.

What This Does—and Does Not—Imply

An exposure-induced mediator–outcome confounder threatens identification of natural effects; it does not automatically mean that every causal quantity in the study is unidentified. For example, under suitable treatment-exchangeability and positivity conditions, the total effect comparing $E[Y(1)]$ with $E[Y(0)]$ may still be identified. That total-effect question does not require splitting the effect using a mediator value from one treatment world inside an outcome from another.

Nor does the graph alone tell us the size or direction of any effect. It describes causal structure, not numerical results. Identification requires a target quantity, a causal model, and assumptions that connect its counterfactual definition to observed data. A natural-effect estimate obtained by applying a standard mediation formula without addressing the post-treatment common cause may therefore have no justified causal interpretation, even if the model fits the observed data well.

Common Mistake

Wrong reasoning: “$L$ is measured, so I can treat it like an ordinary baseline confounder and use the usual natural-effect formula.”
Correction: $L$ is caused by treatment, so its value under treatment 0 need not equal its value under treatment 1. The natural indirect effect combines $M(0,L(0))$ with an outcome involving $L(1)$. Measurement alone does not supply that cross-world information or make the baseline-confounder formula valid.
Another overstatement: “Because this graph has an exposure-induced confounder, no causal effect can be identified.” The justified conclusion is narrower: standard identification of natural effects is generally threatened. Other targets may remain identifiable under their own assumptions.

Key takeaway: When treatment causes a common cause of the mediator and outcome, the mediator under one treatment can depend on a different counterfactual value of that common cause than the value governing the outcome. This cross-world structure is why natural-effect identification cannot generally proceed as if mediator–outcome confounding were purely baseline.

Check Your Understanding

Use the counterfactual definitions and graph in this tutorial to answer the following.

What arrows in a causal graph identify $L$ as an exposure-induced mediator–outcome confounder?
In $M(0,L(0))$, what does each argument represent?
Why can the second outcome term in $\operatorname{NIE}(1)$ involve both $L(1)$ and $L(0)$?
In the two-model example, calculate the mean of $L(1)L(0)$ in each model and explain how this changes the indirect effect.
Does the presence of an exposure-induced mediator–outcome confounder alone establish that the total treatment effect is unidentified? Explain.

Exposure-Induced Mediator-Outcome Confounding

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