Nonidentification From Unmeasured Mediator-Outcome Confounding

Why Observed Mediation Data May Not Determine Natural Effects

A treatment may change a mediator, and the mediator may in turn affect an outcome. But an observed association between mediator and outcome does not by itself establish how changing the mediator would change the outcome. An unmeasured variable can affect both: it can help determine which mediator value a person receives and also predict that person’s outcome. In that case, the observed mediator–outcome relationship can mix causal effects with confounding.

The central issue here is point identification. A causal quantity is point identified from observed data under specified assumptions if every causal model satisfying those assumptions and producing the observed data gives the same value for that quantity. To show nonidentification, it is enough to construct two compatible models that agree on the observed distribution but disagree on the target effect.

We will use binary treatment \(A\), binary mediator \(M\), binary outcome \(Y\), and an unmeasured binary variable \(U\). Potential outcomes \(M(a)\) and \(Y(a,m)\) denote, respectively, the mediator under treatment \(a\) and the outcome under treatment \(a\) with the mediator set to \(m\). The nested counterfactual \(Y(a,M(a'))\) combines the outcome treatment setting \(a\) with the mediator value that would occur under \(a'\).

For a concrete decomposition, define \(\mu(a,a')=E[Y(a,M(a'))]\). We use the natural direct effect at the mediator value under control, \(\mu(1,0)-\mu(0,0)\), and the natural indirect effect at treatment level 1, \(\mu(1,1)-\mu(1,0)\). Their sum is the total effect \(\mu(1,1)-\mu(0,0)\). The counterexample below shows that these means need not be determined by the observed distribution.

A Shared Treatment and Mediator Model

In both causal models, let \(A\) be randomized and independent of \(U\). Set \(P(A=1)=0.5\) and \(P(U=1)=0.25\). The mediator is determined by

where “xor” is 1 when its two binary inputs differ and 0 when they agree. Thus \(U\) affects the mediator, and treatment also affects the mediator. The two models differ only in the outcome response to treatment, mediator, and \(U\).

Two Outcome Models, Two Sets of Natural Effects

In Model 1, define the outcome under interventions by \(Y(a,m)=U\). The unmeasured variable causes both \(M\) and \(Y\), so it is a mediator–outcome confounder. The outcome does not respond causally to treatment or to an intervention on the mediator in this particular model.

In Model 2, define the outcome by the structural equation

where \(\mathbf{1}\{\cdot\}\) equals 1 when the stated condition holds and 0 otherwise. Here \(U\) affects the outcome as well as the mediator. For example, at \(a=0,m=1\), setting \(u=0\) gives \(Y=0\), whereas setting \(u=1\) gives \(Y=1\). Thus \(U\) is a common cause of mediator and outcome in this model too.

Why the Observed Mediation Formula Does Not Resolve the Difference

Under the usual conditions for mediation identification, the nested mean can be written as an observed-data functional:

This expression combines the outcome mean at treatment level \(a\) and mediator level \(m\) with the mediator distribution under \(a'\). Its use requires conditions that rule out unmeasured mediator–outcome confounding, along with the relevant consistency and positivity conditions. Randomizing treatment alone does not ensure those conditions.

In the example, all treatment–mediator combinations have positive probability: \(P(M=0\mid A=0)=0.75\), \(P(M=1\mid A=0)=0.25\), \(P(M=0\mid A=1)=0.25\), and \(P(M=1\mid A=1)=0.75\). But \(U\) remains unmeasured and confounds the mediator–outcome relationship. Applying the displayed expression mechanically gives values that are not the natural nested means.

The Nonidentification Argument

The construction establishes nonidentification directly. Both models use randomized treatment, the same distribution of \(U\), and the same mediator equation. In both, the observed mediator always equals \(A\mathbin{\mathrm{xor}}U\). Model 1 has \(Y=U\); Model 2 also has \(Y=U\) on every naturally observed treatment–mediator combination. Hence they yield exactly the same probabilities for every observed \((A,M,Y)\) value.

Yet the models disagree about outcomes under mediator interventions that depart from \(M=A\mathbin{\mathrm{xor}}U\). Those unobserved responses enter \(Y(a,M(a'))\) when \(a\ne a'\). Model 1 assigns the natural direct and indirect effects the value 0. Model 2 assigns them \(-0.25\) and \(0.25\). No calculation based only on the common observed distribution can choose between these answers. Therefore the natural direct and indirect effects are not point identified from that distribution without additional assumptions or information.

Common Mistake and Practical Implication

A common mistake is to reason: “Treatment was randomized, so mediator–outcome confounding is controlled.” Randomization makes treatment independent of pre-treatment causes, but it does not randomize the mediator. In the example, \(U\) is independent of \(A\) but affects both \(M\) and \(Y\). Conditioning on the mediator therefore does not remove the confounding by \(U\).

The correction is to distinguish treatment exchangeability from mediator–outcome exchangeability. A randomized treatment can identify a total treatment effect while leaving natural direct and indirect effects unidentified. Measuring plausible common causes of mediator and outcome, defending the required assumptions, or changing the causal estimand may be necessary. The observed-data pattern alone cannot establish that the natural effects are identified.