Mediation and cross-world assumptions

Fix a target population, two treatment versions $a,a^{\mathrm{\prime }}$, a mediator random element $M$ in a standard-Borel space Standard Borel spaces, and a real random variable $Y$ Random elements and real random variables. Let $M(a)$ be the mediator response under $do(A=a)$ and let $Y(a,m)$ be the outcome response under the joint intervention $do(A=a,M=m)$, with the intervention semantics of Recursive structural causal models on finite DAGs. These are full-data random variables; consistency alone does not identify their joint law from observed data.

When the displayed differences are integrable Integrable real and complex functions, and their integrals, the controlled direct effect at mediator setting $m$ is ${\mathrm{CDE}}_m(a,a^{\mathrm{\prime }})=E[Y(a,m)-Y(a^{\mathrm{\prime }},m)]\mathrm{.}$ It requires that setting the mediator to $m$ be a meaningful intervention. The natural direct effect and natural indirect effect, with $a^{\mathrm{\prime }}$ the reference treatment level, are $\mathrm{NDE}(a,a^{\mathrm{\prime }})=E[Y(a,M(a^{\mathrm{\prime }}))-Y(a^{\mathrm{\prime }},M(a^{\mathrm{\prime }}))],$ $\mathrm{NIE}(a,a^{\mathrm{\prime }})=E[Y(a,M(a))-Y(a,M(a^{\mathrm{\prime }}))]\mathrm{.}$ The first argument is the active outcome treatment level and the second is the reference level that supplies the mediator response. When $a\mathrm{\ne }{a}^{\mathrm{\prime }}$, these targets involve nested counterfactuals from different intervention regimes. They are defined on a full-data law whether or not they are identified by the observed law. The total mean contrast is $\tau (a,a^{\mathrm{\prime }})=E[Y^a-Y^{a^{\mathrm{\prime }}}],$ using the treatment-only potential outcomes of Potential outcomes and consistency. For $a=1,a^{\mathrm{\prime }}=0$, this agrees with the ATE convention in Average and conditional average treatment effects.

Let $L$ be an observed baseline random element in a standard-Borel space Standard Borel spaces, $A$ a treatment random element Random elements and real random variables taking values in a nonempty countable set Finite, countably infinite, countable, uncountable equipped with the discrete sigma-algebra (the full power set), following the treatment-version convention of Potential outcomes and consistency. Let $M$ be an observed mediator random element in a standard-Borel space Standard Borel spaces, and let $Y$ be an observed real random variable Random elements and real random variables. A simple Markovian mediation SCM is a recursive structural causal model in the sense of Recursive structural causal models on finite DAGs whose endogenous variables are exactly $(L,A,M,Y)$ and whose equations have the form $L=f_L(U_L),A=f_A(L,U_A),M=f_M(A,L,U_M),Y=f_Y(A,M,L,U_Y)\mathrm{.}$ The functions are jointly measurable in the sense of A measurable function between measurable spaces, and the exogenous random elements $U_L,U_A,U_M,U_Y$ are mutually independent Independent random elements, as required by the Markovian condition in that SCM definition. The graph therefore has no latent common cause outside $L$ and no additional post-treatment variable that causes both $M$ and $Y$. This model is a special case of an NPSEM-IE (a nonparametric structural equation model with independent errors); it is a substantive model restriction, not a consequence of the observed distribution. It is a single-unit model; if units interfere, the treatment version must include the relevant joint assignment or exposure, as required by Potential outcomes and consistency.

The mediator and outcome responses are the recursive responses under the interventions from Controlled and natural mediation effects: $M(a)=f_M(a,L,U_M),Y(a,m)=f_Y(a,m,L,U_Y)\mathrm{.}$ In particular, recursion gives $Y^a=Y(a,M(a))$. The model's absence of an additional post-treatment common cause is part of the hypothesis; if such a variable is present, this definition does not cover that causal system.

Suppose an expanded structural causal model decomposes a treatment into two substantively meaningful components $N$ and $O$ that can in principle be set separately. Write $M(n,o)$ and $Y(n,o)$ for the mediator and outcome under $do(N=n,O=o)$, and $Y(n,o,m)$ for the outcome under the additional intervention $do(M=m)$, using the intervention semantics of Recursive structural causal models on finite DAGs. Here $Y(n,o)$ leaves the mediator equation active. Mixed component values must correspond to meaningful intervention protocols; an arbitrary mathematical relabeling of $A$ is not a component decomposition. For reference component levels $n,n^{\mathrm{\prime }},o,o^{\mathrm{\prime }}$, the separable component contrasts are ${\mathrm{SE}}_N(n,n^{\mathrm{\prime }};o)=E[Y(n,o)-Y(n^{\mathrm{\prime }},o)],$ ${\mathrm{SE}}_O(o,o^{\mathrm{\prime }};n)=E[Y(n,o)-Y(n,o^{\mathrm{\prime }})],$ whenever the contrasts are integrable Integrable real and complex functions, and their integrals. They refer to feasible component interventions, not to the nested individual mediator value $M(a^{\mathrm{\prime }})$ in a natural effect.

For the simple single-mediator structural system, the relevant dismissible-component restrictions are the structural exclusions $M(n,o)=M(n),Y(n,o,m)=Y(n^{\mathrm{\prime }},o,m)\text{for all }n,n^{\mathrm{\prime }},o,m\mathrm{.}$ The first says that the outcome-only component $O$ has no effect on the mediator. The second says that the mediator-only component $N$ has no direct effect on the outcome when the mediator is fixed. In other causal systems the formal dismissible-component conditions are model-specific. Meaningful component interventions, consistency, positivity, and appropriate exchangeability conditions are required before these contrasts can be identified. No identification theorem for separable effects is asserted here; they define a different target from the natural direct and indirect effects in Controlled and natural mediation effects.

Assume the Axiom of Choice for the conditional-expectation conventions. In a simple Markovian mediation SCM, for every supplied treatment levels $a,a^{\mathrm{\prime }}$ and mediator value $m$, $M(a^{\mathrm{\prime }})\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp Y(a,m)\mid \sigma (L),$ $A\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp M(a^{\mathrm{\prime }})\mid \sigma (L),A\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp Y(a,m)\mid \sigma (L)\mathrm{.}$ Thus independent structural errors supply the cross-world restriction needed by the natural-effect formula, in addition to the single-world treatment-outcome, treatment-mediator, and mediator-outcome independences.

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.

Assume AC. Let $A$ be the countable-discrete treatment random element in A simple Markovian mediation model, let $L$ and $M$ be standard-Borel random elements Standard Borel spaces Random elements and real random variables, and let $Y$ be real-valued and integrable Integrable real and complex functions, and their integrals. Suppose the full-data law is a simple Markovian mediation SCM from A simple Markovian mediation model, with consistency, treatment positivity, mediator positivity, and the integrability conditions of Assumptions for identifying natural effects. Fix treatment levels $a,a^{\mathrm{\prime }}$. Let $Q_b(l,dm)$ be a conditional law Conditional law given a random element of $M$ given $(A,L)=(b,l)$, and let $R((b,l,m),dy)$ be a conditional law Conditional law given a random element of $Y$ given $(A,L,M)=(b,l,m)$. Define the zero-filled conditional mean ${\mu }_b(l,m)=\{\begin{array}{ll}\int yR((b,l,m),dy),& \int \mathrm{\mid }y\mathrm{\mid }R((b,l,m),dy)<\mathrm{\infty },\\ 0,& \text{otherwise.}\end{array}$ Then the nested mean $\theta (a,a^{\mathrm{\prime }})=E[Y(a,M(a^{\mathrm{\prime }}))]$ is the following functional of the observed-data law: $\theta (a,a^{\mathrm{\prime }})={\int }_{\mathcal{L}}{\int }_{\mathcal{M}}{\mu }_a(l,m)Q_{a^{\mathrm{\prime }}}(l,dm)P_L(dl),$ where $P_L$ is the target-population law Law or distribution of a random element of $L$. The support-domination hypothesis $Q_{a^{\mathrm{\prime }}}(l,\cdot )\ll Q_a(l,\cdot )$ makes the integral independent of the chosen conditional-mean version. Consequently, $\mathrm{NDE}(a,a^{\mathrm{\prime }})=\theta (a,a^{\mathrm{\prime }})-\theta (a^{\mathrm{\prime }},a^{\mathrm{\prime }}),\mathrm{NIE}(a,a^{\mathrm{\prime }})=\theta (a,a)-\theta (a,a^{\mathrm{\prime }})$ are identified Identifiability of an estimand in this model. The total mean contrast satisfies $\tau (a,a^{\mathrm{\prime }})=\mathrm{NDE}(a,a^{\mathrm{\prime }})+\mathrm{NIE}(a,a^{\mathrm{\prime }})\mathrm{.}$ The formula is a population identification result; a regression coefficient alone does not identify these effects.

If $M$ is countable discrete, the conditional-law formula reduces to ${\sum }_m\int {\mu }_a(l,m)Q_{a^{\mathrm{\prime }}}(l,\{m\})P_L(dl)$. In either formulation the support condition is essential for the observed conditional mean to be defined where it is integrated. The independent-error and no-post-treatment- confounder restrictions are substantive; the observed-data law does not certify them.