Assumptions for Identifying Natural Effects

Why Natural Effects Need Identification Assumptions

Natural direct and indirect effects compare outcomes under combinations of treatment and mediator values that may never occur together for the same person. For example, a natural effect can involve the outcome under treatment \(a\) with the mediator value that person would have had under treatment \(a'\). Observed data record one treatment, one mediator, and one outcome per person; they do not reveal all those counterfactuals. Identification therefore requires assumptions that connect the observed distribution to the counterfactual quantities.

Let \(A\) denote treatment, \(M\) the mediator, \(Y\) the outcome, and \(C\) a set of measured baseline covariates. Write \(M(a)\) for the mediator value under treatment \(a\), and \(Y(a,m)\) for the outcome under treatment \(a\) with the mediator set to \(m\). A natural outcome is \(Y(a,M(a'))\): the outcome under treatment \(a\) paired with the mediator value the same person would have under treatment \(a'\). For example, a natural direct effect on the mean-difference scale compares \(E[Y(a,M(a'))]\) with \(E[Y(a',M(a'))]\). The assumptions below concern whether such counterfactual quantities can be related to observed data.

Definition: An identification assumption is a condition under which a target counterfactual quantity is uniquely determined by the distribution of the observed variables. Identification is not the same as estimation: it specifies what the data distribution would determine if it were known exactly, before considering sampling uncertainty or a particular estimator.

Consistency and Well-Defined Interventions

Consistency connects observed values to counterfactual values. If a person receives treatment \(A=a\), then their observed mediator equals \(M(a)\). If, in addition, their observed mediator is \(m\), their observed outcome equals \(Y(a,m)\). In compact form, the observed values satisfy \(M=M(A)\) and \(Y=Y(A,M)\). These equalities presume that the interventions represented by \(a\) and \(m\) are sufficiently well-defined: materially different versions of “treatment \(a\)” or “setting the mediator to \(m\)” should not be silently treated as the same intervention.

Consistency also relies on a suitable no-interference condition: one person’s observed outcome under their treatment and mediator is not changed by other people’s treatment or mediator values. If interference is important, the counterfactuals must instead specify the relevant treatment or mediator assignments among other people. Consistency is not a claim that treatment causes the observed outcome; it is the link between an observed assignment and the corresponding counterfactual.

Worked Example: Applying Consistency to a Hypothetical Study

Suppose a hypothetical study defines \(A=1\) as receiving a specified counseling program, \(M\) as the number of follow-up visits attended, and \(Y\) as a symptom score at three months. One participant receives the program, attends four visits, and has a symptom score of 12.

For this participant, consistency gives \(M=M(1)=4\) and \(Y=Y(1,4)=12\). It does not tell us the participant’s \(M(0)\), \(Y(0,4)\), or \(Y(1,M(0))\). Those are counterfactual values under assignments or combinations not observed for this participant.

The interpretation depends on how the interventions are defined. If “the program” includes versions with substantially different counseling content, the label \(A=1\) may not specify a single treatment well enough for \(Y(1,m)\) to have a clear meaning. Likewise, “setting visits to four” needs a meaningful intervention interpretation. The recorded values alone do not resolve those design questions.

Positivity: The Required Treatment and Mediator Support

Even with exchangeability assumptions, comparisons cannot be recovered from strata in which the needed treatment or mediator values never occur. Treatment positivity requires every treatment level being compared to have positive probability at each relevant baseline covariate value. For natural effects, mediator positivity additionally requires support for the mediator values drawn from one treatment condition when evaluating outcomes under the other condition.

Definition: For discrete \(A\) and \(C\), treatment positivity requires \(P(A=a\mid C=c)>0\) for every treatment level \(a\) of interest and every \(c\) with \(P(C=c)>0\). A corresponding mediator support condition requires that mediator values possible under \(A=a'\) at \(C=c\) also have positive probability under \(A=a\) at that \(C=c\), for each treatment pair needed. For continuous variables, the condition is stated in terms of support or densities rather than positive probability at an exact value.

For a binary mediator, requiring both mediator values to occur within each relevant treatment and covariate stratum is a simple sufficient check. With a continuous mediator, the relevant requirement is support overlap: the mediator values whose counterfactual outcomes are needed must not lie outside the values represented under the treatment condition used to learn those outcomes. Positivity concerns the population and the intervention being considered, not just whether a finite sample happens to contain a few observations in every cell.

Worked Example: Checking Positivity in Hypothetical Strata

Consider 400 hypothetical participants in two baseline age strata. There are 240 younger participants and 160 older participants. Among the younger participants, 120 receive \(A=1\) and 120 receive \(A=0\); among the older participants, 80 receive each treatment. The treatment probabilities in the two strata are therefore:

P(A=1\mid C=\text{younger})=\frac{120}{240}=0.500, \qquad P(A=1\mid C=\text{older})=\frac{80}{160}=0.500.

Both treatment levels occur in both age strata, so the observed counts satisfy the basic treatment-positivity check for these strata. Now suppose the hypothetical mediator is binary, and the mediator counts are as follows:

Age stratum	Treatment	M=0	M=1	Total
Younger	A=0	72	48	120
Younger	A=1	60	60	120
Older	A=0	56	24	80
Older	A=1	40	40	80

Within the younger \(A=0\) group, the mediator proportions are \(72/120=0.600\) for \(M=0\) and \(48/120=0.400\) for \(M=1\); within the younger \(A=1\) group they are \(60/120=0.500\) for each value. Both mediator values occur under both treatments. In the older stratum, the corresponding proportions are \(56/80=0.700\) and \(24/80=0.300\) under \(A=0\), and \(40/80=0.500\) for each value under \(A=1\). Again, both mediator values occur under both treatments.

Thus the displayed sample counts show treatment and binary-mediator overlap in both age strata. They do not prove population positivity: a zero-probability value could be missed in a finite sample, and sparse cells can still make estimation unstable.

Exchangeability and the Cross-World Restriction

Exchangeability expresses the absence of relevant unmeasured differences between groups being compared, conditional on \(C\). Treatment exchangeability requires treatment assignment to be independent of the potential mediators and outcomes, given baseline covariates. In this setting, it is a statement about treatment assignment, not a consequence of having measured a large number of covariates.

Definition: Treatment exchangeability requires \(\{M(a),Y(a,m)\}\perp A\mid C\) for the treatment and mediator values of interest. In words, among people with the same baseline covariates, treatment assignment carries no further information about their potential mediator or outcome values.

Random assignment can justify treatment exchangeability by design, possibly conditional on the randomization strata. In an observational study, it is a substantive claim that the measured baseline covariates suffice to control common causes of treatment and the relevant potential outcomes. It cannot be established merely by observing that measured covariates look balanced.

Mediator-outcome exchangeability concerns a different comparison. Given treatment and baseline covariates, the observed mediator must not carry information about the potential outcome under a specified treatment and mediator intervention. One standard form is \(Y(a,m)\perp M\mid A=a,C\). This condition is threatened by unmeasured common causes of the mediator and outcome. Randomizing \(A\) alone does not randomize \(M\), and therefore does not automatically guarantee this condition.

Definition: Mediator-outcome exchangeability requires \(Y(a,m)\perp M\mid A=a,C\) for the relevant \(a\) and \(m\). A further cross-world restriction used for general natural-effect identification is \(Y(a,m)\perp M(a')\mid C\) for the relevant treatment pairs and mediator values. The latter relates potential outcomes under one mediator intervention to potential mediators under another treatment assignment.

The cross-world restriction is not the same as mediator-outcome exchangeability. The latter relates a potential outcome to the observed mediator within a treatment group; the former relates two counterfactual quantities that may refer to different treatment assignments for the same person. It is especially consequential because a natural outcome \(Y(a,M(a'))\) combines those different worlds. A structural model with independent sources of variation for the mediator and outcome can imply the restriction, but the restriction is not generally testable from the observed distribution.

Why the Cross-World Restriction Enters

A short factorization shows what the cross-world condition contributes. For clarity, take a discrete mediator and condition on \(C=c\). Partition the expectation of the natural outcome by the value of the counterfactual mediator:

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

Without an assumption connecting \(Y(a,m)\) and \(M(a')\), the conditional mean inside the sum is not determined by the separate marginal distributions of those two counterfactuals. Under the cross-world restriction \(Y(a,m)\perp M(a')\mid C\), it simplifies to \(E[Y(a,m)\mid C=c]\). This is a justification for why the restriction matters, not a derivation of an observed-data mediation formula: further assumptions and steps are needed to connect the counterfactual means to observed quantities.

In particular, observing \(Y\), \(M\), and \(A\) does not reveal whether a person who would have had \(M(a')=m\) would also have had a high or low \(Y(a,m)\). The cross-world restriction supplies a particular relationship between these unobserved quantities. It should therefore be defended using substantive knowledge or a stated causal model, rather than described as something that the data have confirmed.

Worked Example: A Hypothetical Model That Satisfies the Cross-World Restriction

Consider a hypothetical binary treatment, binary mediator, and binary outcome. Let \(U_M\) and \(U_Y\) be independent uniform random variables on \([0,1]\), and let treatment \(A\) be randomized independently of both. Define the counterfactual mediator by \(M(a)=1\{U_M\leq p_a\}\), with \(p_0=0.300\) and \(p_1=0.600\). Define \(Y(a,m)=1\{U_Y\leq q_{am}\}\), where \(q_{10}=0.200\).

For the particular cross-world pair \(Y(1,0)\) and \(M(0)\), independence of \(U_M\) and \(U_Y\) gives:

P\{Y(1,0)=1,M(0)=1\} =q_{10}p_0 =(0.200)(0.300) =0.0600.

The marginal probabilities are \(P\{Y(1,0)=1\}=0.200\) and \(P\{M(0)=1\}=0.300\), whose product is also \(0.0600\). In this specified model the pair is independent. The same argument applies to any \(a,m,a'\): the first counterfactual is a function of \(U_Y\), the second is a function of \(U_M\), and those variables are independent.

This example explains one way a cross-world restriction can arise from a model. It does not show that real data establish the restriction. The observed distribution alone does not reveal whether the underlying mediator and outcome variation have the independent structure stipulated here.

Putting the Assumptions Together

For general natural effects, identification requires a coherent set of conditions, not just one successful diagnostic. The assumptions concern different links in the argument:

Define the interventions and counterfactuals.
Specify treatment, mediator, outcome, and baseline covariates. Require consistency and a suitable no-interference interpretation.

Check support.
Require treatment positivity across relevant baseline strata and mediator support for the values needed across treatment conditions.

Address treatment confounding.
Justify treatment exchangeability given \(C\), through randomization or a credible argument that measured baseline covariates suffice.

Address mediator-outcome confounding.
Justify mediator-outcome exchangeability given treatment and \(C\); randomization of treatment alone does not provide it.

Defend the cross-world link.
State why the potential outcome and counterfactual mediator are assumed independent conditional on \(C\), or otherwise justify the required cross-world restriction.

These assumptions are generally not all empirically testable. Observed data can reveal a lack of overlap or a contradiction with consistency as operationally defined, but cannot prove that there is no unmeasured confounding or establish the joint distribution of counterfactuals from different worlds. A careful analysis states which conditions are supported by design, which rely on subject-matter arguments, and which may be examined only through sensitivity analysis.

Common Mistake: Treating Randomization as Sufficient

A common but incorrect argument is: “Treatment was randomized, so the natural effects are identified.” Randomization can justify treatment exchangeability, but it does not ensure mediator positivity, eliminate unmeasured mediator-outcome confounding, establish consistency, or imply the cross-world restriction. For instance, a baseline characteristic that affects both mediator and outcome can still make mediator-outcome exchangeability implausible even when treatment was assigned at random.

The correction is to assess each assumption separately. Ask whether the treatment and mediator interventions are well-defined; whether every needed treatment and mediator value has support; whether treatment assignment is exchangeable given \(C\); whether mediator-outcome confounding is controlled; and what justifies the cross-world relation. Passing one check does not substitute for the others.

Key takeaway: Natural effects require assumptions that connect observed assignments to counterfactual outcomes and that relate counterfactuals from different treatment worlds. Positivity and exchangeability are distinct requirements, and the cross-world restriction is substantive rather than something verified by the observed data.

Check Your Understanding

For each question, identify the assumption involved and explain what it does or does not establish.

A treatment has probability zero among people in one baseline covariate stratum. Which positivity condition fails, and why can adjustment not create the missing treatment comparison?
Why does randomizing \(A\) not, by itself, guarantee \(Y(a,m)\perp M\mid A=a,C\)?
In your own words, distinguish mediator-outcome exchangeability from \(Y(a,m)\perp M(a')\mid C\).
What consistency statement applies to a person observed with \(A=1\), \(M=2\), and outcome \(Y=8\)? Which natural counterfactual remains unobserved?
Why does the cross-world restriction simplify the expectation of \(Y(a,M(a'))\), and why can it not generally be verified from observed treatment, mediator, and outcome data?