Identification Conditions for Interventional Mediation

Why Identification Conditions Matter

An interventional mediation effect compares outcomes under specified treatment levels while assigning the mediator according to a chosen distribution. Its interpretation is causal, but the data record only the treatment, mediator, outcome, and measured covariates for each person. Identification conditions explain when those observed distributions contain enough information to determine the causal contrast.

The conditions do different jobs. Consistency links observed values to the outcomes defined by the interventions. Treatment exchangeability supports learning mediator distributions and outcome responses without treatment confounding. Mediator-outcome exchangeability supports learning outcome responses under mediator assignments. Positivity ensures that the observed data include the treatment and mediator values the target intervention requires. These are substantive assumptions, not conclusions established simply by fitting a model.

Interventional Means and the Target Distributions

Let \(A\) denote treatment, \(M\) a mediator, \(Y\) the outcome, and \(C\) measured baseline covariates. Consider two treatment levels, denoted 0 and 1. Write \(Y(a,m)\) for the potential outcome if treatment were set to \(a\) and the mediator to \(m\). Let \(G_{a^*}\) denote a stochastic mediator intervention that, conditional on \(C\), draws a mediator value from the distribution of \(M\) under treatment level \(a^*\). The draw is randomized according to that distribution, rather than being the particular person's natural mediator value.

Definition: The interventional mean \(\psi(a,a^*)\) is the mean outcome if treatment were set to \(a\) and the mediator were drawn from \(G_{a^*}\), the mediator distribution under treatment level \(a^*\) conditional on baseline covariates. The first argument specifies the outcome treatment level; the second specifies the mediator reference distribution.

For binary treatment, contrasts such as \(\psi(1,0)-\psi(0,0)\) compare treatment levels while holding the mediator reference distribution fixed, and \(\psi(1,1)-\psi(1,0)\) compare mediator reference distributions while holding treatment fixed. This tutorial concerns the assumptions that identify such means and contrasts. In particular, identification does not follow from the fact that a contrast has an interventional interpretation; it also requires a defensible connection between the intervention and the observed data.

The conditions below use \(C\) for covariates measured before treatment. A variable caused by treatment is not a baseline covariate. Adjusting for an exposure-induced mediator-outcome confounder as though it were baseline can change the target or introduce bias; the causal role and timing of every adjustment variable must therefore be clear.

Four Conditions and What Each Supports

Consistency: If a person receives treatment \(A=a\), then the observed mediator equals \(M(a)\). If that person also has mediator value \(M=m\), the observed outcome equals \(Y(a,m)\). The intervention must be sufficiently well defined that the observed treatment and mediator values correspond to the versions specified by these potential outcomes.

Consistency is the link from observed records to potential outcomes. It also encompasses the absence of interference for the usual individual-level formulation: one person's outcome is not changed by other people's treatment or mediator assignments. If treatment versions differ in ways relevant to the outcome, merely recording the same treatment label does not guarantee consistency.

Treatment exchangeability: Conditional on baseline covariates \(C\), treatment assignment is independent of the potential mediators and outcomes relevant to the target. In compact form, for the treatment levels under consideration, \(A \perp \{M(a),Y(a,m):a,m\}\mid C\).

This condition says that, after accounting for \(C\), treatment groups are comparable with respect to the counterfactual quantities needed here. It supports using the observed mediator distribution among people with \(A=a^*\) to represent the mediator distribution that would arise under treatment \(a^*\), conditional on \(C\). It also supports interpreting observed outcome responses within treatment levels as corresponding to the relevant treatment-specific potential outcomes, subject to the mediator condition below. Random assignment can justify treatment exchangeability by design; in an observational study, it requires that the measured baseline covariates control all relevant treatment-outcome and treatment-mediator confounding.

Mediator-outcome exchangeability: Within each treatment level \(a\), conditional on baseline covariates \(C\), the observed mediator is independent of the potential outcomes under that treatment: \(Y(a,m)\perp M\mid A=a,C\), for the mediator values \(m\) relevant to the intervention.

This condition rules out unmeasured common causes of the mediator and outcome after conditioning on treatment and \(C\). It supports using the observed outcome response among people with \(A=a\), \(M=m\), and \(C=c\) to represent the mean outcome that would result from setting the mediator to \(m\) for people in that stratum. Treatment exchangeability alone does not provide this link: treatment may be unconfounded while mediator assignment remains confounded.

Positivity: Every treatment and mediator value required by the target intervention must have support in the corresponding observed-data strata. Treatment positivity requires \(P(A=a\mid C=c)>0\) for treatment levels used in the contrast and covariate strata in the target population. Mediator positivity requires that values drawn from the reference distribution under \(a^*\), at a given \(c\), also have support among people with treatment \(a\) at that \(c\).

For a discrete mediator, the latter requirement means that if \(P(M=m\mid A=a^*,C=c)>0\), then \(P(M=m\mid A=a,C=c)>0\) whenever the outcome response under \(A=a\) is needed. For a continuous mediator, the corresponding requirement is support overlap: the reference distribution must not place mass in regions where the outcome treatment group provides no mediator data. Treatment positivity is needed to learn each treatment-specific distribution; mediator positivity is needed to evaluate outcome responses at mediator values that the reference distribution can generate.

How the Conditions Link to Observed Data

Under these conditions, the interventional mean is identified by combining two observed-data distributions: the outcome mean conditional on treatment, mediator, and baseline covariates; and the mediator distribution conditional on treatment and baseline covariates. In the expression below, the integral is replaced by a sum for a discrete mediator.

\psi(a,a^*) = \int \int E(Y\mid A=a,M=m,C=c)\, dF(m\mid A=a^*,C=c)\, dF(c)

Here \(F(c)\) is the baseline covariate distribution in the target population. The outcome regression \(E(Y\mid A=a,M=m,C=c)\) is supported by consistency and mediator-outcome exchangeability, with treatment exchangeability connecting the observed treatment group to its counterfactual counterpart. The mediator distribution \(F(m\mid A=a^*,C=c)\) is supported by consistency and treatment exchangeability. Positivity ensures both pieces are available where the expression calls for them.

A short justification makes the roles clearer. First average the potential outcome under treatment \(a\) over the mediator draw and the target covariate distribution. Given \(C=c\), the draw has distribution \(F(m\mid A=a^*,C=c)\) once treatment exchangeability and consistency identify the mediator distribution under \(a^*\). For each drawn \(m\), mediator-outcome exchangeability and consistency identify the mean potential outcome under \(a,m\) with \(E(Y\mid A=a,M=m,C=c)\). Averaging these conditional means over the specified mediator and covariate distributions gives the displayed observed-data expression. Positivity is what permits those conditional means and distributions to be used at all required values.

Worked Examples: Applying the Conditions

Worked Example: Computing a Mean When Support and Exchangeability Are Assumed

Suppose a hypothetical study has a binary baseline covariate \(C\), with \(P(C=0)=P(C=1)=0.5\). Treatment levels 0 and 1 both occur in each stratum. Assume consistency, treatment exchangeability, mediator-outcome exchangeability, and the required positivity conditions are substantively plausible. The observed outcome means under \(A=1\) are 10 and 14 for \(M=0\) and \(M=1\) when \(C=0\), and 20 and 26 when \(C=1\). Under \(A=0\), the probability of \(M=1\) is 0.25 for \(C=0\) and 0.50 for \(C=1\). Find \(\psi(1,0)\).

The mediator reference distribution is the one under \(A=0\), while the outcome response is evaluated under \(A=1\). For \(C=0\), the mean response averaged over that reference distribution is \(0.75(10)+0.25(14)=7.50+3.50=11.00\). For \(C=1\), it is \(0.50(20)+0.50(26)=10.00+13.00=23.00\).

Since each covariate stratum has probability 0.5, the target mean is \(0.5(11.00)+0.5(23.00)=5.50+11.50=17.00\) outcome units. The calculation uses outcome means in the \(A=1\) group and mediator probabilities in the \(A=0\) group. Its causal interpretation depends on the stated assumptions; the arithmetic itself cannot establish those assumptions.

Worked Example: Diagnosing Mediator-Outcome Confounding

In a hypothetical setting with treatment held fixed, let an unmeasured binary variable \(U\) have probability 0.5 of being 1. Suppose \(Y=U\), so changing the mediator does not change the outcome, but the mediator is associated with \(U\): \(P(M=1\mid U=1)=0.8\) and \(P(M=1\mid U=0)=0.2\). Treatment is randomized, but \(U\) is not measured or included in \(C\).

Among those with \(M=1\), the probability of \(U=1\) is \(0.8(0.5)/[0.8(0.5)+0.2(0.5)]=0.4/0.5=0.8\). Thus the observed outcome mean is \(E(Y\mid M=1)=0.8\). Similarly, \(P(U=1\mid M=0)=0.2(0.5)/[0.2(0.5)+0.8(0.5)]=0.1/0.5=0.2\), so \(E(Y\mid M=0)=0.2\).

Those observed means differ by 0.6, even though setting \(M\) to either value leaves \(Y=U\), whose mean is 0.5 under either intervention. The observed mediator groups differ in \(U\), a common cause of \(M\) and \(Y\). Therefore mediator-outcome exchangeability fails, and the observed conditional outcome means do not identify the causal response to mediator assignment. Randomizing treatment does not repair this failure.

Worked Example: A Mediator Positivity Failure

Consider a hypothetical binary mediator. At a fixed covariate value, the reference group \(A=0\) has \(P(M=1\mid A=0)=0.5\). In the outcome treatment group \(A=1\), however, everyone has \(M=0\). Suppose \(Y\) is binary and the observed mean under \(A=1,M=0\) is 0.2. The target \(\psi(1,0)\) requires averaging the \(A=1\) outcome response over the mediator distribution under \(A=0\).

That average would be \(0.5(0.2)+0.5E(Y\mid A=1,M=1)\), but the second conditional mean is not observed because no one in the \(A=1\) group has \(M=1\). Since \(Y\) is binary, the unobserved mean could be anywhere from 0 to 1. The target mean is therefore only bounded by \(0.5(0.2)+0.5(0)=0.10\) and \(0.5(0.2)+0.5(1)=0.60\).

Even if the other assumptions were plausible, the data would not identify a unique value of \(\psi(1,0)\). The failure is specifically mediator positivity: the reference distribution calls for a mediator value with no outcome-group support. A fitted model might extrapolate a value, but extrapolation does not turn the missing support into observed information.

Practical Interpretation and Common Pitfalls

The assumptions should be assessed in relation to a clearly specified target population, treatment contrast, mediator intervention, and covariate set. Exchangeability is not directly verified by a nonsignificant test or by a good-fitting outcome model. It is a claim about the absence of relevant unmeasured confounding after conditioning on \(C\), supported by study design and subject-matter knowledge. Positivity can be investigated by examining treatment frequencies and mediator overlap across covariate strata, although sparse data can make practical support weaker than a formal nonzero probability suggests.

A common mistake is to treat “adjust for baseline covariates” as a guarantee of identification. The correction is to ask what each condition contributes. Treatment exchangeability addresses confounding of treatment; mediator-outcome exchangeability addresses confounding of the mediator-outcome relation; positivity addresses whether the required groups are represented; consistency addresses whether the observed variables correspond to the interventions being defined. A failure in one role is not repaired merely by satisfying another.

Key takeaway: Interventional mediation effects are identified only when the observed data can represent both the mediator reference distributions and the outcome responses that the intervention combines. Consistency, treatment exchangeability, mediator-outcome exchangeability, and treatment and mediator positivity each support a distinct part of that connection.

Check Your Understanding

For each question, distinguish the causal assumption from what can be learned directly from observed data.

Which observed-data distribution is used to represent the mediator reference distribution under \(a^*\), and which assumptions support that representation?
Why does random assignment of treatment not by itself guarantee mediator-outcome exchangeability?
At a covariate value \(c\), the reference distribution assigns positive probability to \(M=m\), but no individuals with \(A=a\) have that mediator value. Which condition fails, and which part of the identification expression is unsupported?
How can treatment versions that share one recorded label create a consistency problem?
Why is mediator-outcome exchangeability a substantive assumption rather than something established by observing that \(E(Y\mid A,M,C)\) can be estimated?