Mediation Identification Case Workshop

From a Clinical Question to an Identification Decision

Suppose a hypothetical trial evaluates a treatment intended to reduce a clinical outcome partly by changing a biological marker. A common question is how much of the treatment effect operates through that marker. Answering it requires more than fitting a model for the outcome: natural direct and indirect effects refer to outcomes under combinations of treatment and mediator values that may not occur together for the same person. We must define the target first, then assess whether the study assumptions connect it to the observed data.

Let $A\in\{0,1\}$ denote treatment, $M$ a mediator measured after treatment, $Y$ an outcome, and $C$ measured baseline covariates. Write $M(a)$ for the mediator value that would be observed under treatment $a$, and $Y(a,m)$ for the outcome under treatment $a$ if the mediator were set to $m$. For a binary outcome, effects below are expressed as differences in risks.

Definition: The natural direct effect with the mediator set to its value under control is $E[Y(1,M(0))-Y(0,M(0))]$. The natural indirect effect with treatment fixed at 1 is $E[Y(1,M(1))-Y(1,M(0))]$. Their sum is the total effect $E[Y(1,M(1))-Y(0,M(0))]$. These are population-average contrasts; the natural indirect effect compares outcomes under two treatment-specific natural mediator values.

The direct and indirect effects depend on the chosen reference treatment. The definitions above use control as the mediator reference for the direct effect and treatment 1 as the outcome-treatment level for the indirect effect. Their additive decomposition follows by adding and subtracting $E[Y(1,M(0))]$. It does not imply that either component is identified from observed data.

What Must Hold for Point Identification?

Point identification means that the target estimand has a unique value determined by the observed-data distribution together with stated assumptions. For the usual covariate-adjusted mediation functional, the covariates $C$ must be measured before treatment and suffice to control the relevant confounding. The following assumptions are a standard sufficient set for this setting.

Identification assumptions:

Consistency and no interference: a person's observed mediator and outcome equal the corresponding potential values under that person's received treatment and mediator; one person's treatment does not affect another person's outcome.
Treatment exchangeability given $C$: conditional on $C$, treatment is independent of the potential mediator and outcome values. Random assignment can support this condition, subject to adherence, missingness, and the treatment definition.
Mediator-outcome exchangeability: within treatment and $C$ strata, there is no unmeasured common cause of mediator and outcome. In potential-outcome terms, the mediator assignment is independent of $Y(a,m)$ among individuals with $A=a,C$, for the relevant $a,m$.
Cross-world independence: conditional on $C$, $Y(a,m)$ is independent of $M(a')$ for the treatment levels used in the estimand. This links an outcome under one treatment–mediator setting to a mediator value under another treatment setting.
Positivity: treatment levels have positive probability within covariate strata in the target population, and the mediator values needed to evaluate the outcome regressions have support within the corresponding treatment and covariate strata.

The cross-world condition is not a comparison of two observed groups: no individual reveals both $Y(a,m)$ and $M(a')$ when those settings differ. It is therefore a substantive assumption, not something established merely by randomizing treatment or by measuring many covariates. Likewise, recording a potential confounder does not control it if it was omitted, measured poorly, or arose after treatment in a way that makes the ordinary natural-effect formula inappropriate.

Deriving the Mediation Functional

Define $q(a,m,c)=E[Y\mid A=a,M=m,C=c]$ and $g(a\mid c)=P(M\mid A=a,C=c)$, where $g$ denotes the conditional mediator distribution. Under the assumptions above, the mean outcome under treatment $a$ and the natural mediator value under treatment $a'$ is identified by

\mu(a,a')=E[Y(a,M(a'))] =\sum_c\sum_m q(a,m,c)\,P(M=m\mid A=a',C=c)\,P(C=c).

For continuous covariates or a continuous mediator, the corresponding sums are integrals. The formula averages the outcome regression under treatment $a$ over the mediator distribution under treatment $a'$, then averages over the target population's covariate distribution. It does not simply condition on the mediator and compare treatment groups.

The key identification step can be seen by conditioning on $C$ and $M(a')$. Cross-world independence allows the conditional mean of $Y(a,m)$ to be combined with the distribution of $M(a')$ given $C$. Treatment exchangeability and consistency identify those pieces from observed conditional distributions: $E[Y(a,m)\mid C=c]$ is represented by $E[Y\mid A=a,M=m,C=c]$, and $P(M(a')=m\mid C=c)$ by $P(M=m\mid A=a',C=c)$. Multiplying these conditional pieces and averaging over $C$ yields the displayed functional. Mediator-outcome exchangeability is needed for the outcome regression to represent the relevant potential outcome mean rather than a confounded association.

Result: With $\mu(a,a')=E[Y(a,M(a'))]$, the natural direct effect is $\mu(1,0)-\mu(0,0)$, the natural indirect effect is $\mu(1,1)-\mu(1,0)$, and the total effect is $\mu(1,1)-\mu(0,0)$. The decomposition is exact on the additive scale: $$ \{\mu(1,0)-\mu(0,0)\}+\{\mu(1,1)-\mu(1,0)\} =\mu(1,1)-\mu(0,0). $$

Worked Study and Estimand

Worked Example: Defining the Natural Effects in a Hypothetical Trial

A hypothetical randomized study evaluates a treatment for a chronic condition. The mediator is a binary biomarker measured after treatment, and the outcome is a binary indicator of a clinical event by one year. Baseline disease severity and age group are recorded as $C$. The investigators want to distinguish the treatment effect through the biomarker from the remaining effect.

Let $A=1$ represent the active treatment and $A=0$ usual care. $M(a)$ is the biomarker status that would occur under treatment $a$, and $Y(a,m)$ is the one-year event outcome under treatment $a$ and biomarker status $m$. The natural direct effect is $E[Y(1,M(0))-Y(0,M(0))]$: the risk difference if treatment changes while each person's biomarker is set to the value that person would have had under usual care.

The natural indirect effect is $E[Y(1,M(1))-Y(1,M(0))]$: the risk difference associated with changing the biomarker from its usual-care value to its active-treatment value while holding treatment at 1. These interpretations concern counterfactual contrasts, not groups that can be directly observed. Randomization supports treatment exchangeability, but it does not by itself establish mediator-outcome exchangeability or cross-world independence. The study can use the functional only if those additional assumptions and positivity are defensible.

Worked Calculation Under Stated Assumptions

Worked Example: Computing Natural Effects From Hypothetical Risks

Consider a separate hypothetical study with binary $C$, $A$, $M$, and $Y$. Suppose $P(C=0)=P(C=1)=0.5$. Assume the following conditional mediator probabilities and outcome risks are the population quantities, and suppose consistency, no interference, the exchangeability conditions, cross-world independence, and positivity all hold.

Covariate stratum	$P(M=1\mid A=0,C)$	$P(M=1\mid A=1,C)$	$q(0,0,C)$	$q(0,1,C)$	$q(1,0,C)$	$q(1,1,C)$
$C=0$	0.20	0.60	0.10	0.30	0.20	0.50
$C=1$	0.40	0.80	0.20	0.40	0.30	0.60

First calculate $\mu(0,0)$, using the mediator distribution under control and the outcome regression under control. In stratum $C=0$, the risk is $0.80(0.10)+0.20(0.30)=0.14$. In stratum $C=1$, it is $0.60(0.20)+0.40(0.40)=0.28$. Thus

\mu(0,0)=0.50(0.14)+0.50(0.28)=0.210.

For $\mu(1,0)$, retain the control mediator distribution but use the treated outcome regression. The two stratum-specific risks are $0.80(0.20)+0.20(0.50)=0.26$ and $0.60(0.30)+0.40(0.60)=0.42$. Therefore

\mu(1,0)=0.50(0.26)+0.50(0.42)=0.340.

For $\mu(1,1)$, use both the treated mediator distribution and treated outcome regression. The stratum-specific risks are $0.40(0.20)+0.60(0.50)=0.38$ and $0.20(0.30)+0.80(0.60)=0.54$. Hence

\mu(1,1)=0.50(0.38)+0.50(0.54)=0.460.

The natural direct effect is $0.340-0.210=0.130$, or 13 percentage points in one-year event risk. The natural indirect effect is $0.460-0.340=0.120$, or 12 percentage points. Their sum, $0.130+0.120=0.250$, equals the total effect $0.460-0.210=0.250$, or 25 percentage points. These are causal interpretations only under the assumptions stated for this example; arithmetic alone cannot verify those assumptions.

When a Formula Does Not Justify Identification

A formula may be algebraically available yet unsupported by the study. In particular, an outcome model cannot recover a treatment–mediator combination absent from the data without extrapolation. The following hypothetical example shows how this problem can make natural effects differ across causal models that agree on every observed quantity.

Worked Example: Same Observed Data, Different Natural Effects

Let treatment $A$ be randomized, let an unmeasured binary factor $U$ have probability $0.5$ of being 1, and suppose the mediator follows $M(a)=\max(a,U)$. Thus under control $M(0)=U$, while under treatment $M(1)=1$. In the observed data, $Y=M$. Consequently, for $A=0$, half the participants have $M=Y=0$ and half have $M=Y=1$; for $A=1$, everyone has $M=Y=1$.

Two causal outcome models can generate these same observed data. In Model A, set $Y(a,m)=m$. In Model B, set $Y(a,m)=1$ if $U=1$ or if $a=1,m=0$, and otherwise set $Y(a,m)=m$. Under observed treatment and mediator values, both models give $Y=M$: for $A=0$, $M=U$; for $A=1$, $M=1$. Thus the observed distributions of $A,M,Y$ are identical.

In Model A, $E[Y(1,M(1))]=1$ and $E[Y(1,M(0))]=E[U]=0.5$, while $E[Y(0,M(0))]=0.5$. The natural indirect effect is $1-0.5=0.5$, and the natural direct effect is $0.5-0.5=0$.

In Model B, $Y(1,M(1))=1$ for everyone, and $Y(1,M(0))=1$ for everyone: when $U=0$, $a=1,m=0$ triggers the value 1, and when $U=1$, the first condition does. Also $Y(0,M(0))=U$, with mean 0.5. Its natural indirect effect is $1-1=0$, and its natural direct effect is $1-0.5=0.5$. Both models have the same total effect, 0.5, but different natural-effect components.

The data have no treated participants with $M=0$, so positivity fails for the treated outcome regression at that mediator value. In addition, $U$ is a common cause of $M$ and $Y$ in Model B and is unmeasured. The observed data therefore do not choose between the two natural-effect decompositions. A numerical estimate from an outcome model would require unsupported extrapolation; point identification is not justified.

Identification Is a Causal Claim, Not a Modeling Output

For an applied study, a defensible conclusion separates three questions: what natural-effect contrast is scientifically intended, whether the assumed causal structure supports the identification conditions, and whether the observed data have the needed support. Measured baseline covariates can help control confounding, but their inclusion in a regression does not establish that all relevant confounding has been measured. Nor can a good model fit establish a cross-world assumption.

Define the target.
Specify the treatment contrast, mediator reference, outcome scale, and target population before choosing a model.

Assess the causal assumptions.
Check whether treatment and mediator-outcome confounding are addressed, whether the cross-world condition is being assumed, and whether variables used for adjustment are pre-treatment.

Check support.
Verify that the mediator values needed for each outcome regression occur within the relevant treatment and covariate strata in the target population.

State the conclusion at the right strength.
If the assumptions and support are defensible, the functional identifies the estimand. If not, report which condition fails rather than presenting the functional as an identified causal effect.

A common mistake is to argue, “Treatment was randomized and the model adjusts for all measured covariates, so the natural indirect effect is identified.” Randomization addresses treatment assignment, not necessarily mediator-outcome confounding or cross-world independence; measured covariates do not guarantee that unmeasured causes are absent, and adjustment cannot repair a positivity failure. The correction is to state each identifying assumption explicitly and decide whether the study design and substantive knowledge support it. When they do not, the natural-effect calculation may still be a model-based quantity, but it is not thereby point-identified as a causal effect.

Check Your Understanding

Use the definitions and assumptions in the study context when answering.

Write the natural direct and indirect effects using $\mu(a,a')$, and state why their sum equals the total effect.
In the mediation functional, which treatment level determines the mediator distribution in $\mu(1,0)$, and which determines the outcome regression?
Why does random treatment assignment alone not establish identification of a natural indirect effect?
What does mediator positivity require for the calculation of $\mu(1,0)$?
If two causal models agree on the observed-data distribution but give different natural direct effects, what does that imply about point identification from that distribution under the assumptions considered?

Covariate stratum	\(P(M=1\mid A=0,C)\)	\(P(M=1\mid A=1,C)\)	\(q(0,0,C)\)	\(q(0,1,C)\)	\(q(1,0,C)\)	\(q(1,1,C)\)
\(C=0\)	0.20	0.60	0.10	0.30	0.20	0.50
\(C=1\)	0.40	0.80	0.20	0.40	0.30	0.60