Potential-Outcome Estimands for Biomarker Mediation

Why Define Mediation Effects With Potential Outcomes?

A treatment can change a post-treatment biomarker, and the biomarker can in turn be related to a clinical outcome. To ask how much of the treatment effect operates through that biomarker, we need to define the comparisons precisely. A regression coefficient for the biomarker, or a correlation between biomarker change and clinical response, does not by itself identify a causal mediated effect.

Potential outcomes make the comparisons explicit. They also expose a central challenge: a natural effect compares outcomes under one treatment condition while assigning the biomarker value that the same individual would have had under another treatment condition. That comparison is not directly observed for any individual, so its causal interpretation depends on assumptions beyond treatment randomization.

For example, if \(Y\) is a symptom score where larger values indicate worse symptoms, a negative treatment effect is beneficial. If \(Y\) is a response score where larger values are better, the same numerical sign has a different clinical meaning. Effect direction must therefore be interpreted using the outcome’s scale and coding.

Natural Direct and Indirect Effects

The total effect compares the outcome under each treatment along with the biomarker value naturally produced by that treatment. It can be decomposed in two standard ways. One holds the biomarker at its control-treatment potential value for the direct-effect comparison; the other holds it at its active-treatment potential value.

These are two exact decompositions of the same total effect, but their components need not be equal. When treatment and biomarker interact in their effects on the outcome, the direct effect can depend on which biomarker value is held fixed, and the indirect effect can depend on which treatment condition is used to evaluate it. Each effect is an average over the target population.

Assumptions for Causal Identification

Potential-outcome definitions specify the estimands; they do not guarantee that observed data identify them. A common identification approach uses baseline covariates \(L\) to standardize outcome and biomarker distributions. Its assumptions should be considered before calculating the mediation formula.

• Consistency and no interference: the observed outcome and biomarker agree with the potential outcomes under the treatment actually received, and one individual’s treatment does not alter another individual’s outcome. Treatment and biomarker values must also be sufficiently well-defined for these potential outcomes to be meaningful.
• Treatment exchangeability: conditional on measured baseline covariates \(L\), treatment assignment is independent of the relevant potential biomarker and outcome values. Randomization generally supports this condition for treatment, subject to adherence, missing data, and the target population considered.
• Mediator–outcome exchangeability: conditional on treatment and baseline covariates, there are no unmeasured common causes of the biomarker and clinical outcome. Randomizing treatment does not randomize the biomarker, so this condition is not supplied by a randomized trial alone.
• No treatment-induced mediator–outcome confounder: there is no post-treatment variable caused by treatment that then confounds the biomarker–outcome relation. Ordinary adjustment for such a variable does not generally identify the natural effects defined above.
• Positivity: within covariate strata represented in the target population, both treatment conditions must be possible, and the biomarker values needed for the comparisons must have support under the relevant treatment conditions. Extrapolating to biomarker values absent from an arm is not justified by the formula.
• Cross-world condition: identifying natural effects requires linking potential outcomes across treatment conditions—for example, a suitable conditional independence between \(Y(a,m)\) and \(M(a')\) given \(L\). This is an untestable assumption about joint potential outcomes, often motivated by a structural causal model with appropriate independent errors.

Under these conditions, define \(\psi(a,a')=E[Y(a,M(a'))]\). For a discrete biomarker and covariates, the mediation formula is

For continuous variables, the sums are replaced by integrals. In this expression, the outcome regression comes from treatment condition \(a\), while the biomarker distribution comes from condition \(a'\). The natural effects follow by taking differences: \(\mathrm{NDE}_0=\psi(1,0)-\psi(0,0)\) and \(\mathrm{NIE}_1=\psi(1,1)-\psi(1,0)\), for example.

Worked Decomposition With a Binary Biomarker

Standardization Across Baseline Covariates

Baseline characteristics can affect both biomarker distributions and outcomes. The following example shows how the mediation formula averages within covariate strata before combining results in the target population.

Interaction Changes the Components

A simple linear outcome model illustrates why the choice of decomposition matters when the treatment effect varies with the biomarker. It is a model-based illustration, not a substitute for the identification assumptions above.

Diagnostics, Alternatives, and Limitations

The mediation formula depends on causal and support assumptions that cannot all be checked from the observed data. Diagnostics can reveal some problems, but passing them does not prove the assumptions. Examine treatment and biomarker overlap across baseline covariate strata; very small probabilities or empty combinations make the standardized estimates unstable or extrapolative. Check the temporal ordering and measurement definitions, inspect outcome-model fit, and consider whether plausible common causes of biomarker and outcome were measured before treatment.

A common mistake is to say, “Treatment was randomized, so the estimated biomarker-mediated effect is causal.” Randomization can support treatment exchangeability, but the biomarker itself is usually not randomized. If an unmeasured factor affects both biomarker and outcome, or if treatment creates a confounder of their relationship, the natural-effect calculation may not have a causal interpretation. The correction is to state the additional mediator–outcome and cross-world assumptions, assess their plausibility, and avoid presenting the decomposition as established solely by the trial design.

Another pitfall is to multiply a treatment-to-biomarker coefficient by a biomarker-to-outcome coefficient and label the product a natural indirect effect without checking the model. Such a product can correspond to a mediated effect in particular linear models, for example with no treatment–biomarker interaction and suitable causal assumptions, but it is not a general definition. Likewise, natural effects are distinct from controlled direct effects, which compare treatment outcomes after setting the biomarker to a specified fixed value. The latter answer a different intervention question.

Natural effects are useful when the scientific question concerns pathways through the biomarker as it would naturally respond to treatment. They are nevertheless assumption-dependent, may be sensitive to unmeasured mediator–outcome confounding, and can be difficult to interpret when biomarker values are not manipulable or when treatment-induced confounders are present. Report the outcome scale, target population, effect decomposition, covariates used for standardization, and key identifying assumptions. The result is a causal estimand under those assumptions, not proof that the biomarker is a validated surrogate endpoint.