Assumptions for identifying natural effects

Fix a pair of treatment levels $a,a^{\mathrm{\prime }}$ in the simple Markovian mediation model of A simple Markovian mediation model. The assumptions used for the natural-effect formula are:

1. Consistency and treatment versions. The treatment versions and joint treatment/mediator interventions have the meanings fixed by Controlled and natural mediation effects. The observed responses satisfy $M=M(A)$ and $Y=Y(A,M(A))$ by the structural recursion.

2. No unmeasured treatment confounding. Conditional on baseline $L$, the assignment error $U_A$ is independent of the mediator and outcome response errors. Equivalently in this model, for every supplied $b,c,m$, $A\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp M(b)\mid \sigma (L)$ and $A\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp Y(c,m)\mid \sigma (L)$.

3. No unmeasured mediator–outcome confounding. Conditional on $L$, the mediator and outcome response errors are independent. This excludes an unmeasured common cause of $M$ and $Y$ in the stated graph.

4. No treatment-induced mediator–outcome confounder. There is no post-treatment endogenous variable other than $M$ that is a common cause of $M$ and $Y$. If such a variable is present, the four-equation model above does not apply.

5. Cross-world restriction. For all supplied $b,c,m$, $M(b)\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp Y(c,m)\mid \sigma (L)$. In particular this includes $M(a^{\mathrm{\prime }})\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp Y(a,m)\mid \sigma (L)$ for the nested response in the natural direct effect. The lemma Independent errors imply mediation cross-world independence proves this restriction for the simple NPSEM-IE model. The same-world instances $b=c$ express mediator–outcome exchangeability; the instances $b\mathrm{\ne }c$ are additional cross-world assumptions.

6. Treatment positivity. $P(A=b)>0$ and $P(A=b\mid \sigma (L))>0$ almost surely for each treatment level $b$ used in the contrast, in the target-support convention of Positivity for discrete treatment.

7. Mediator positivity. Let $Q_b(l,\cdot )$ be a conditional law of $M$ given $(A,L)=(b,l)$, in the sense of Conditional law given a random element. For the cross-treatment response $Y(a,M(a^{\mathrm{\prime }}))$, require $Q_{a^{\mathrm{\prime }}}(l,\cdot )\ll Q_a(l,\cdot )$ for almost every $l$ under the law $P_L$ of $L$ from Law or distribution of a random element. This support-domination condition ensures that the conditional outcome law observed at treatment $a$ is defined at mediator values drawn under treatment $a^{\mathrm{\prime }}$. It is a measure-level condition; for a countable mediator it says every $m$ with $Q_{a^{\mathrm{\prime }}}(l,\{m\})>0$ also has $Q_a(l,\{m\})>0$.

8. Integrability. Every nested outcome $Y(b,M(c))$ used in the displayed effects is integrable Integrable real and complex functions, and their integrals under the target population law.

These are sufficient full-data assumptions for the theorem on this page, not empirical facts certified by a regression fit. The NPSEM-IE independent-error restriction supplies the cross-world condition. FFRCISTG means “Finest Fully Randomized Causally Interpretable Structured Tree Graph”; in the single-world convention described by Hernán–Robins, Chapter 23, its regime-specific laws do not impose a joint independence between counterfactuals from incompatible regimes. Thus FFRCISTG alone does not supply the cross-world condition above. This item uses that source distinction only to name the missing assumption; the explicit two-law counterexample is a separate B-page result. Standardizing or fitting a regression does not by itself establish the assumptions above.