Identification of Hypothetical Estimands

What a Hypothetical Estimand Asks

Suppose a trial compares two treatments, but some participants discontinue assigned treatment, take rescue medication, or switch to another therapy. A hypothetical estimand asks about outcomes under a specified scenario in which an intercurrent event is prevented. It is not simply a comparison among participants who happened not to experience the event. The target includes the defined population, with each treatment considered under the same hypothetical event-prevention strategy.

Let \(A\) denote assigned treatment, with values \(0\) and \(1\). Let \(L\) denote measured baseline covariates, \(E\) indicate whether the intercurrent event occurs during the relevant period, and \(Y\) denote the outcome at the target time. Write \(Y^{a,g_0}\) for the outcome that would be observed if treatment were set to \(a\) and the event-prevention intervention \(g_0\) were applied. The intervention must be specified in words: for example, “rescue medication is not initiated through week 12.” The notation does not, by itself, say how such prevention would be achieved.

This definition separates the estimand—the counterfactual quantity of interest—from an estimator calculated using observed data. It also clarifies that “prevent the event” is part of the question, not a statistical trick. If preventing the event would change other care or behavior, the hypothetical intervention needs enough detail to make its consequences interpretable.

Assumptions That Connect the Target to Observed Data

The observed data contain outcomes under the treatment and event histories that actually occurred, not every participant’s \(Y^{a,g_0}\). A standard identification argument uses baseline covariates \(L\) to connect the hypothetical means to observed outcomes among participants with \(E=0\). The assumptions below are substantive; randomization alone does not establish all of them.

• Well-defined interventions and consistency. Treatment levels and \(g_0\) must describe sufficiently clear interventions. For a participant with \(A=a\) and \(E=0\), the observed outcome equals the counterfactual outcome under the corresponding treatment and event-prevention strategy: \(Y=Y^{a,g_0}\). This consistency statement assumes that the observed event-free history represents the relevant intervention history; it is not a claim that every way of preventing the event has identical effects.
• Treatment exchangeability. Conditional on \(L\), treatment assignment carries no additional information about \(Y^{a,g_0}\): \(Y^{a,g_0}\mathrel{\perp\!\!\!\perp} A\mid L\). In a randomized trial this can follow from the randomization scheme, with appropriate attention to the variables used in randomization. In an observational study it is a no-unmeasured-confounding assumption.
• Event exchangeability. Among people with treatment \(a\) and covariates \(L\), event occurrence carries no additional information about the outcome under \(g_0\): \(Y^{a,g_0}\mathrel{\perp\!\!\!\perp} E\mid A=a,L\). This assumption rules out residual differences in the hypothetical outcome between observed event-free and event-experiencing participants after conditioning on \(L\). It is often strong and cannot be verified from the observed data alone.
• Positivity. Each treatment must be possible at covariate values represented in the target population, and the event-free state must have positive probability in each treatment-covariate group used for identification: \(P(A=a\mid L=l)>0\) and \(P(E=0\mid A=a,L=l)>0\) for relevant \(l\).

The event-exchangeability condition is specific to the hypothetical question. If event occurrence is associated with prognosis even after conditioning on \(L\), then observed event-free outcomes may not represent what event-experiencing participants would have experienced had the event been prevented. Randomization of \(A\) does not randomize \(E\).

Identification by Standardization

Under these assumptions, the treatment-specific mean can be identified by averaging the observed conditional mean among event-free participants over the baseline covariate distribution of the target population:

Here \(F_L\) is the distribution of \(L\) in the target population. The formula says to estimate the mean outcome for treatment \(a\) among event-free participants within each covariate pattern, then average those means using the target population’s covariate distribution. It does not average using the covariate distribution among observed event-free participants, which may differ across treatment groups.

A short derivation makes the role of the assumptions explicit. First, average the counterfactual mean over \(L\). Treatment exchangeability lets us replace the treatment counterfactual mean at \(L=l\) with the corresponding mean among those assigned \(a\). Event exchangeability then lets us restrict that conditional mean to \(E=0\); consistency equates that observed event-free outcome with \(Y^{a,g_0}\):

The equalities require the stated exchangeability and consistency conditions, as well as positivity so the observed conditional means exist for the relevant covariate values. In practice, standardization may be implemented by estimating the conditional mean model and predicting under \(A=a,E=0\) for each target-population participant, then averaging those predictions.

Inverse-Probability Representation and Treatment Assignment

The same target mean has a weighted representation. Define the treatment propensity \(P(A=a\mid L)\) and the conditional event-free probability \(P(E=0\mid A=a,L)\). Under the identification assumptions and positivity,

The weights reweight observed participants who received treatment \(a\) and remained event-free to represent the target population under that treatment and event-prevention scenario. The treatment probability must be conditional on \(L\) unless treatment assignment is known to have a constant probability independent of \(L\). Randomization does not generally imply a constant assignment probability: probabilities can vary across randomization strata.

When the Observed Data Do Not Identify the Target

Positivity is not a technical detail that can be repaired by choosing a more elaborate estimator. If no one in a target-population subgroup is observed event-free under treatment \(a\), that subgroup’s mean under the hypothetical intervention is not learned from event-free outcomes. A model can extrapolate into the unsupported subgroup, but the resulting answer depends on untestable modeling assumptions.

Diagnostics, Alternatives, and Limits

Diagnostics cannot prove exchangeability, but they can reveal when an analysis is relying on fragile support or modeling choices. Before interpreting a hypothetical estimate, examine:

• Event-free support: tabulate event-free counts across treatment and important baseline covariates. Empty or very small cells warn of positivity problems.
• Estimated probabilities and weights: inspect treatment and event-free probabilities, the range of combined weights, and whether a few observations dominate. Extreme weights often signal limited overlap or unstable estimates.
• Model dependence: compare reasonable conditional-mean specifications and check whether predictions for sparse covariate groups drive the result. Large changes indicate substantial extrapolation.
• Assumption plausibility: consider whether measured \(L\) captures common causes of event occurrence and the hypothetical outcome. This is a scientific assessment, not a test settled by a good model fit.

The main alternatives answer different questions. A treatment-policy estimand includes outcomes regardless of the intercurrent event and therefore does not ask what would happen if the event were prevented. A comparison restricted to observed event-free participants conditions on a post-treatment event and generally compares selected groups; it is not automatically the hypothetical contrast. Standardization and inverse-probability weighting are alternative identification-based approaches to the same hypothetical target when their assumptions hold. Their finite-sample behavior and sensitivity to model specification can differ, but neither method removes the need for exchangeability and positivity.

The hypothetical estimand also has limits of interpretation. It does not describe the effect among only those people who would remain event-free under either treatment. Its target is the population outcome under the specified prevention intervention. If that intervention is implausible, poorly specified, or changes the clinical context in ways not captured by the question, the resulting contrast may have limited practical relevance. When the event is frequent or near-deterministic in important groups, identification may be weak even when the target is clearly stated.

• Common mistake: “Treatment was randomized, so the event-free participants in each arm identify the hypothetical means.”
• Correction: Randomization supports treatment exchangeability, but event occurrence is generally not randomized. Identification also requires event exchangeability conditional on measured covariates and positive event-free probability in relevant groups. A crude event-free comparison can additionally have different covariate distributions across arms.