Hypothetical Estimands

What Does “If the Event Had Not Occurred” Mean?

A participant may start rescue medication, discontinue assigned treatment, or experience another event that changes how a trial outcome is observed or interpreted. A hypothetical estimand asks a counterfactual clinical question: what would the outcome, and the treatment contrast, have been under a specified scenario in which that event did not occur? The phrase “did not occur” is not yet a complete clinical question. It must be made clear what is imagined to happen instead, and what treatment and outcome are being considered.

For example, asking what symptoms would have been at week 24 if rescue medication had not been started is different from asking what symptoms would have been if the participant had continued assigned treatment without rescue medication. The first leaves treatment after rescue unspecified; the second describes more of the scenario but may or may not be clinically plausible. A useful hypothetical estimand makes the imagined scenario precise enough that clinicians can interpret the result.

Here \(Y^{a,g}\) is a counterfactual outcome: it is the outcome for a participant under treatment \(a\) in the stated scenario, whether or not that participant actually experienced the event. It is not automatically the outcome observed among participants who happened not to experience the event. The scenario could, for example, specify that rescue medication is not initiated while assigned treatment continues according to a defined regimen. It should say what happens to treatment, follow-up, and other relevant care as far as needed to make the outcome interpretable.

Specify the Clinical Scenario Before the Contrast

The five attributes of an estimand provide a useful structure: population, treatment conditions, variable (outcome), handling of intercurrent events, and population-level summary. For a hypothetical estimand, the event-handling attribute must state the scenario that replaces the event. The other attributes remain necessary. A statement such as “the treatment effect if there were no rescue” is incomplete if it does not say which population, which outcome and time point, and which treatment contrast are intended.

A clinically meaningful scenario should answer questions such as these:

• Which event is being prevented or set aside?
• What treatment and care continue instead, and does the imagined scenario apply in both randomized arms?
• What outcome is defined at the target time under that scenario?
• Is the scenario plausible enough to inform the clinical decision at hand?

The answer need not claim that the imagined course is routinely achievable. It does need to be clear about what is being imagined. If avoiding the event would require withholding necessary treatment, for instance, the estimand may be mathematically expressible but clinically difficult to interpret or ethically unhelpful. A hypothetical estimand concerns the stated scenario; it does not establish that preventing the event would be safe, feasible, or a recommended clinical action.

Why the No-Event Subgroup Is Not the Hypothetical Population

It is tempting to compare outcomes only among participants who did not experience the event. That comparison answers a different question. Event occurrence can be related to prognosis, treatment response, or other participant characteristics. In a randomized trial, randomization balances treatment groups at baseline, but it does not guarantee that the subsets who remain event-free are comparable: membership in those subsets is determined after randomization and may differ between arms.

In notation, an observed no-event comparison might compare \(E[Y\mid A=1,E=0]\) with \(E[Y\mid A=0,E=0]\), where \(E\) records whether the event occurred. A hypothetical contrast instead compares \(E[Y^{1,g}]\) with \(E[Y^{0,g}]\). The former conditions on observed event status; the latter refers to outcomes under a defined scenario. These quantities are generally not equal.

From a Counterfactual Question to an Estimable Quantity

The definition of the estimand does not, on its own, tell us how to estimate it from trial data. For participants who experience the event, the outcome under the hypothetical no-event scenario is not directly observed. Estimation therefore requires assumptions and an analysis model or other method that connects observed data to the counterfactual outcomes. This is a point where the clinical scenario and statistical assumptions must be considered together: a model cannot rescue a scenario that has no coherent clinical interpretation.

A basic justification for the mean-difference form is linearity of expectation. For each participant, imagine the two scenario-specific potential outcomes \(Y^{1,g}\) and \(Y^{0,g}\). The population average of their individual differences satisfies

This identity defines a population treatment contrast; it does not say that both potential outcomes are observed for any participant. In a randomized trial, randomization supports comparison of assigned treatment groups for outcomes defined by the actual trial course. It does not by itself reveal the unobserved outcomes under \(g\) for participants whose event occurred. Additional assumptions are needed for that step.

Among the assumptions to consider are:

• Well-defined scenario and consistency: the scenario and treatment conditions are sufficiently specified, and observed outcomes agree with the corresponding potential outcome when a participant’s actual course matches those conditions.
• Randomization and appropriate analysis: treatment assignment follows the trial’s randomization scheme, and the analysis respects the assignment and target population used to define the estimand.
• Support for the scenario: the observed data contain enough relevant information to estimate outcomes under the scenario. If particular kinds of participants almost always experience the event, their hypothetical outcomes may depend heavily on extrapolation.
• Correct assumptions for unobserved outcomes: the model or other estimation method must adequately represent the relationship between observed information and outcomes under \(g\). Randomization alone does not ensure this.

When the scenario involves decisions over time, such as continuing or changing treatment as symptoms evolve, additional assumptions about the time-varying history and treatment decisions may be required. The details depend on the scenario and analysis. The central distinction remains: the estimand states the clinical quantity; the estimation method and its assumptions determine how the available data are used to learn about it.

Checks, Comparisons, and Limitations

Before interpreting an analysis, check that its scenario matches the clinical question and that the reported result is a contrast for that scenario. A useful review should also examine how much of the result relies on estimating unobserved counterfactual outcomes.

• Scenario ambiguity: If reasonable readers can imagine different treatment or care after the event, the scenario needs more detail.
• Weak support in the data: If few participants with relevant histories follow a course resembling \(g\), the analysis may depend substantially on modeling assumptions.
• Changing conclusions: Compare results under defensible alternative assumptions about outcomes after the event. Large changes indicate that the conclusion is sensitive to those assumptions, not that one scenario has automatically been proven correct.
• Mismatch between label and analysis: An analysis restricted to observed event-free participants does not become a hypothetical estimand analysis simply because the report says “as if the event had not occurred.”

A treatment-policy estimand, introduced in the earlier material on Treatment-Policy Estimands, asks about outcomes regardless of whether the event occurs, so the event is retained as part of the treatment experience being summarized. A hypothetical estimand instead targets outcomes under a specified scenario that prevents the event. These are different clinical questions and can yield different answers. Choosing between them is not a technical preference: it depends on which treatment decision the result is intended to inform.

A hypothetical estimand also has limitations. The imagined scenario may be clinically unrealistic, ethically unacceptable to implement, or weakly supported by observed data. Even when it is clearly defined, its estimate can depend on unverifiable assumptions about outcomes that were not observed under that scenario. Reporting should therefore state the scenario, give the treatment contrast and its units, and make important assumptions and sensitivity to them visible. The result should not be interpreted as a direct description of what happened to participants who avoided the event.