Identification Assumptions for Genetic Mediation

Why Mediation Requires More Than an Exposure Effect

A genetic instrument can sometimes help estimate the effect of an exposure \(X\) on an outcome \(Y\). Mediation asks a more detailed question: how much of that effect operates through a mediator \(M\)? Answering it requires information about how exposure changes the mediator and how the mediator relates causally to the outcome. An instrument that addresses confounding of \(X\) and \(Y\) does not automatically resolve confounding of \(M\) and \(Y\).

Let \(M(x)\) be the mediator value that would occur if exposure were set to \(x\), and let \(Y(x,m)\) be the outcome that would occur if exposure were set to \(x\) and the mediator to \(m\). The natural outcome \(Y(x,M(x'))\) combines the outcome under exposure \(x\) with the mediator value that would naturally arise under exposure \(x'\). This cross-condition quantity is central to natural direct and indirect effects.

Definition: For two exposure levels \(x_0\) and \(x_1\), define \(\psi(x,x')=E[Y(x,M(x'))]\). The natural direct effect at mediator reference \(x_0\) is \(\psi(x_1,x_0)-\psi(x_0,x_0)\). The natural indirect effect at exposure reference \(x_1\) is \(\psi(x_1,x_1)-\psi(x_1,x_0)\). Their sum is the total effect \(E[Y(x_1,M(x_1))-Y(x_0,M(x_0))]\). These are population-average contrasts; their units are the units of \(Y\).

These definitions specify causal contrasts, not an estimation procedure. In particular, observing that \(M\) predicts \(Y\), or that a genetic variant predicts both, does not establish a natural indirect effect. Identification asks whether the counterfactual quantities are determined by the distribution of observed data under stated assumptions.

Assumptions for Identification

Let \(C\) denote measured pre-exposure covariates that may jointly affect exposure, mediator, or outcome. The following assumptions are stated for the exposure levels and mediator values relevant to the contrasts. They are substantive conditions, not facts established merely by fitting a model.

Consistency and no interference: If a person receives exposure \(x\) and mediator value \(m\), the observed outcome equals \(Y(x,m)\); if exposure is \(x\), the observed mediator equals \(M(x)\). One person's exposure or mediator does not alter another person's outcome, and the interventions represented by \(x\) and \(m\) are sufficiently well-defined for these counterfactuals to be meaningful.

Exposure exchangeability: Conditional on \(C\), exposure is independent of the relevant potential mediators and outcomes: \(\{M(x),Y(x',m)\}\perp X\mid C\) for the exposure and mediator values under consideration. This rules out unmeasured common causes of exposure and mediator, and of exposure and outcome, after conditioning on \(C\).

Mediator–outcome exchangeability: Conditional on exposure and \(C\), the observed mediator is independent of the relevant potential outcomes: \(Y(x,m)\perp M\mid X=x,C\). This rules out unmeasured common causes of mediator and outcome within exposure and covariate strata. It must hold for the outcome response being used, including when the outcome exposure level differs from the level that generated the mediator distribution.

Positivity: Each exposure level being compared must have positive probability at covariate values in the target population. Also, mediator values that occur under the reference exposure must be supported among people at the outcome exposure level, within the relevant covariate strata. Without this mediator support, the needed outcome response cannot be learned from observed data without extrapolation.

Natural effects additionally involve counterfactuals from different exposure conditions in the same expression. One sufficient condition used for the mediation formula is a cross-world independence condition: \(Y(x_1,m)\perp M(x_0)\mid C\). It says that, conditional on \(C\), the outcome that would occur under \(x_1,m\) is independent of the mediator value that would arise under \(x_0\). This is not directly testable from observed data. In standard mediation arguments it is supported by a suitable sequential exchangeability structure and the absence of an exposure-induced mediator–outcome confounder; it should not be treated as a consequence of instrument relevance.

A particularly important complication is a variable \(L\) caused by exposure that then affects both mediator and outcome. Such an \(L\) is an exposure-induced mediator–outcome confounder. Simply adjusting for it does not generally identify the natural effects defined above: adjustment can change the counterfactual comparison, and the ordinary mediation formula need not apply. The causal structure and target effect must be reconsidered rather than assuming that adding \(L\) to a regression solves the problem.

How the Mediation Formula Follows

For a discrete mediator, define \(\mu(x,m,c)=E[Y\mid X=x,M=m,C=c]\) and \(g(m\mid x,c)=P(M=m\mid X=x,C=c)\). Under consistency, exchangeability, positivity, and the cross-world condition described above, the nested mean can be written as:

\psi(x_1,x_0) = \sum_c \sum_m \mu(x_1,m,c)\, g(m\mid x_0,c)\, P(C=c).

For continuous \(C\) or \(M\), the corresponding sums are replaced by integrals. The formula combines an outcome response under \(x_1\) with a mediator distribution under \(x_0\), then averages over the covariate distribution of the target population.

The justification makes clear why several assumptions are needed. First, condition on \(C\) and on the mediator value that would arise under \(x_0\). Cross-world independence lets the conditional mean of \(Y(x_1,m)\) be separated from the distribution of \(M(x_0)\). Exposure exchangeability and consistency identify the latter distribution as \(g(m\mid x_0,c)\). For the outcome component, consistency alone is not enough: consistency, exposure exchangeability, and mediator–outcome exchangeability together identify \(E[Y(x_1,m)\mid C=c]\) by \(\mu(x_1,m,c)\). Positivity ensures both observed-data quantities are available on the required support. Averaging their product over \(m\) and \(c\) gives the displayed expression.

Represent the target counterfactual.
For each \(C=c\), average the outcome response \(Y(x_1,m)\) over the mediator values that would occur under \(x_0\).

Replace counterfactual quantities with observed-data quantities.
Use consistency and exposure exchangeability for the mediator distribution; use consistency, exposure exchangeability, and mediator–outcome exchangeability for the outcome mean.

Average over the target covariate distribution.
Weight each conditional nested mean by \(P(C=c)\), provided exposure and mediator positivity hold.

Worked Examples

Worked Example: Locating the Required Exchangeability Assumptions

Suppose a hypothetical study examines genetically predicted lipid exposure \(X\), a blood inflammatory marker \(M\), and a cardiovascular outcome \(Y\). Age, sex, and ancestry are measured in \(C\). The aim is to estimate natural direct and indirect effects of changing \(X\) from \(x_0\) to \(x_1\).

Exposure–mediator and exposure–outcome confounding. If a pre-exposure factor such as a health behavior affects lipid levels and inflammation or cardiovascular risk, it can confound the exposure relations. The analysis must measure and appropriately control relevant common causes in \(C\), or justify exposure exchangeability by another design. Merely observing that a variant predicts \(X\) does not show that all exposure confounding is absent.

Mediator–outcome confounding. Suppose an unmeasured inflammatory illness affects both the marker \(M\) and cardiovascular risk \(Y\). Then, even within levels of \(X\) and \(C\), people with different observed \(M\) may differ in their potential outcomes. Mediator–outcome exchangeability fails, so \(E[Y(x_1,m)\mid C=c]\) cannot be replaced by the observed conditional mean \(\mu(x_1,m,c)\) on that assumption set.

Exposure-induced confounding. If \(X\) changes a treatment \(L\), and \(L\) subsequently affects both \(M\) and \(Y\), \(L\) is an exposure-induced mediator–outcome confounder. The ordinary natural-effect formula is not justified by simply adding \(L\) to \(C\). Thus, the required assumptions concern distinct causal links; evidence for one does not verify the others.

Worked Example: Standardizing a Natural-Effect Mean

Consider hypothetical data with binary \(X\) and \(M\), no covariates, and a continuous outcome measured in arbitrary outcome units. Suppose the outcome means are \(\mu(1,0)=6\) and \(\mu(1,1)=10\), while the mediator probabilities are \(P(M=1\mid X=0)=0.20\) and \(P(M=0\mid X=0)=0.80\). Assume consistency, exchangeability, positivity, the cross-world condition, and no exposure-induced mediator–outcome confounder.

The nested mean \(\psi(1,0)\) uses the outcome response under \(X=1\), weighted by the mediator distribution under \(X=0\):

\psi(1,0) = \mu(1,1)P(M=1\mid X=0) + \mu(1,0)P(M=0\mid X=0)

Substituting the hypothetical values gives \(10(0.20)+6(0.80)=2.00+4.80=6.80\) outcome units. This calculation is an identification illustration, not proof that the assumptions hold. In particular, the outcome means are usable for the counterfactual response only under the stated mediator–outcome exchangeability as well as consistency and exposure exchangeability. If an unmeasured cause of \(M\) and \(Y\) remains, the arithmetic does not repair the identification failure.

Worked Example: What an Exposure Instrument Does Not Identify

Suppose, in a hypothetical genetic-instrument analysis, one allele is associated with a \(0.20\)-standard-deviation higher exposure and a \(0.10\)-standard-deviation higher outcome. Under the instrumental-variable assumptions of relevance, independence from exposure–outcome confounders, and exclusion of pathways to the outcome other than through \(X\), the Wald ratio is:

\frac{\beta_{GY}}{\beta_{GX}} = \frac{0.10\ \text{SD of }Y}{0.20\ \text{SD of }X} = 0.50\ \text{SD of }Y\text{ per SD of }X.

This ratio is an exposure-effect illustration under those assumptions; it is not a natural indirect effect. Even if the instrument is valid for the total effect of \(X\), an unmeasured factor \(U\) may still cause both mediator \(M\) and outcome \(Y\). The instrument's association with \(X\) does not randomize \(M\) within exposure strata and does not by itself establish mediator–outcome exchangeability. A mediation claim therefore needs additional identification assumptions and an analysis that targets the mediated contrast.

What Genetic Instruments Can and Cannot Contribute

A genetic instrument can address exposure confounding only under its own assumptions: it must predict \(X\) (relevance), be independent of causes of \(X\) and \(Y\) after appropriate control (independence), and affect \(Y\) only through the exposure pathway represented by the analysis (exclusion). Population structure, family-level effects, or genetic pathways to the outcome outside the specified exposure can threaten those assumptions. Passing instrument diagnostics does not establish the mediation assumptions.

For mediation, there are at least two separate causal questions: how changing \(X\) changes \(M\), and how changing \(M\) changes \(Y\) at a specified exposure level. Instruments for \(X\) may help with the first exposure-related question, but they do not automatically remove confounding between \(M\) and \(Y\). A variant used as an instrument for \(M\) would need its own relevance, independence, and exclusion justification; an association with the mediator alone is insufficient. Neither instrument strategy, by itself, supplies the cross-world assumptions required for natural effects.

The practical implication is that genetic evidence can strengthen some parts of a mediation argument while leaving other parts dependent on measured covariates, design, or substantive assumptions. A defensible analysis states which assumptions support each link and identifies where those assumptions are least credible.

Common Mistake

Wrong reasoning: “The genetic instrument removes confounding, so the indirect effect is identified.”
Correction: An instrument may identify an exposure effect under its IV assumptions, but it does not generally remove mediator–outcome confounding or establish cross-world independence. Natural direct and indirect effects require the relevant exchangeability, positivity, consistency, and cross-world conditions in addition to any instrumental-variable assumptions.
Wrong reasoning: “Consistency and exposure exchangeability identify the outcome mean at a fixed mediator value.”
Correction: Identifying \(E[Y(x,m)\mid C=c]\) by \(E[Y\mid X=x,M=m,C=c]\) also requires mediator–outcome exchangeability. Selection into observed mediator values can otherwise make the two means differ.

Key takeaway: Direct and indirect effects are identified only when the counterfactual contrasts can be connected to observed data under explicit assumptions. Exposure exchangeability, mediator–outcome exchangeability, positivity, consistency, and suitable cross-world conditions address different parts of that connection; a genetic instrument for exposure does not replace them all.

Check Your Understanding

Use the assumptions and distinctions in this tutorial to answer the following questions.

What causal relation is addressed by exposure exchangeability, and how does it differ from mediator–outcome exchangeability?
Why is consistency alone insufficient to identify \(E[Y(x,m)\mid C=c]\) from an observed conditional outcome mean?
What role does the cross-world condition play in identifying a natural effect?
Describe a situation in which an exposure-induced variable confounds the mediator–outcome relation, and explain why ordinary covariate adjustment may not solve it.
Under what assumptions might a genetic instrument estimate an exposure effect, and which key mediation assumption does that fact not establish?