Hypothetical Estimands and Their Counterfactual Questions

Why a Hypothetical Scenario Must Be Precise

A clinical trial can ask what treatment effect would have been observed if an intercurrent event had not occurred. For example, after treatment discontinuation, the endpoint may still be collected, but the result reflects a treatment history that includes discontinuation. A hypothetical question instead targets the outcome in a specified counterfactual scenario in which discontinuation does not occur. That is a different clinical question, not merely a different way to analyze the same observed endpoint.

A phrase such as “the effect assuming no discontinuation” is not yet a complete definition. It leaves open whether participants continue their assigned treatment, whether other care is unchanged, and what happens to other events. The scenario must be defined closely enough that the target outcome has a clear clinical interpretation. The estimand’s population, treatment conditions, endpoint, and population-level summary must also be specified, as in the earlier material on clinical trial estimands.

Writing the Counterfactual Question

Let \(A\) denote the assigned treatment, with \(a=1\) for the experimental treatment and \(a=0\) for control. For participant \(i\), write \(Y_i(a;H)\) for the endpoint that would be observed under treatment condition \(a\) in the scenario \(H\). Here \(H\) is a precisely described hypothetical condition, such as “the participant does not discontinue assigned treatment before the endpoint assessment and otherwise receives the specified care.” This notation distinguishes the target outcome from the outcome actually observed under the participant’s realized treatment history.

For a continuous endpoint where larger values are better, one possible target is the population mean difference:

For a binary endpoint, a corresponding risk difference is \(P\{Y_i(1;H)=1\}-P\{Y_i(0;H)=1\}\). These are examples of summaries, not part of the definition of the hypothetical strategy itself. The endpoint scale and direction must be stated; for example, if lower symptom scores are better, a negative mean difference may favor the experimental treatment.

The target population generally remains the population specified by the estimand, not only participants who happened to avoid the event. Restricting analysis to observed event-free participants changes the population and can select people with different prognoses. Nor does “hypothetical” mean that the event is ignored while retaining the post-event outcome: the target is the outcome under the stated counterfactual scenario.

A Short Justification for the Population Contrast

A hypothetical estimand is usually defined for a target population that includes participants who might experience the event. The contrast is therefore formed from each treatment condition’s counterfactual outcome over that population, rather than by dropping participants who experienced the event. To see the distinction, suppose the target population consists of two baseline groups, \(G=1\) and \(G=0\), with proportions \(0.4\) and \(0.6\). If the hypothetical mean outcomes are 8 and 5 in group \(G=1\), and 6 and 4 in group \(G=0\), under experimental treatment and control respectively, then:

This follows by partitioning the target population by \(G\): \(E\{Y(a;H)\}=\sum_g P(G=g)E\{Y(a;H)\mid G=g\}\). The weights are the proportions in the target population, not the proportions observed in an event-free subset. If event occurrence changes who remains observed, averaging only those observed can change the target population and hence the quantity being summarized.

What Must Be Specified, and What Must Be Assumed

Defining an estimand and estimating it from trial data are separate tasks. A well-defined hypothetical estimand states the target, but observed data do not automatically reveal outcomes in a scenario that did not occur. Randomization supports comparisons of assigned treatments in the trial; by itself, it does not identify outcomes under the counterfactual removal of a post-randomization event.

For a hypothetical scenario to be clinically interpretable, specify at least the following:

• The intercurrent event: identify what counts as the event and when it can occur.
• The counterfactual condition: say whether the event is prevented, delayed, or replaced by a particular action.
• Treatment and care after the event time: clarify what participants receive in the scenario. “No discontinuation” may imply continued assigned treatment, but that implication must be explicit.
• Other events and the endpoint: state whether they remain as they would otherwise occur and when and how the outcome is defined.
• The target population and summary: state whose counterfactual outcomes are summarized and on what scale.

Identification from observed data requires additional assumptions that depend on the design and analysis. A common requirement is that, conditional on measured treatment history and relevant prognostic variables, event occurrence is not related to unobserved counterfactual outcomes in a way left unaccounted for by the model. There must also be adequate data support for the histories used to predict outcomes in the no-event scenario; if nearly everyone with a particular history discontinues, the no-discontinuation outcome for that history may be an extrapolation. Correctly representing relevant predictors, treatment history, and outcome relationships is important. These assumptions cannot be verified simply by fitting a model, and their credibility is a scientific question.

Comparison, Diagnostics, and Limitations

A treatment-policy estimand, covered in the earlier material on treatment-policy strategies, targets outcomes under the treatment conditions as assigned while allowing the event and its consequences to occur. A hypothetical estimand instead asks about outcomes under a specified no-event scenario. Neither is automatically more scientifically correct: the choice depends on the clinical question. The same trial can support distinct estimands that answer distinct questions.

Several warning signs indicate that a hypothetical estimand or its estimation may be going wrong:

• Vague counterfactual: “assume no discontinuation” is given without saying what treatment continues. Different plausible versions may imply different outcomes.
• Event-free subgroup substitution: the analysis compares only participants who avoided the event and calls this the hypothetical effect. This generally targets a selected group, not the stated population’s counterfactual outcomes.
• Unsupported extrapolation: the scenario requires predicting outcomes for treatment histories barely or never observed. The numerical result may then depend heavily on model structure and assumptions.
• Conflating the event with missingness: an event can occur even when the endpoint is observed. Conversely, the endpoint can be missing whether or not the event occurred. The estimand defines the target; missing outcome data concern what was actually recorded.
• Unclear scope: other post-randomization events are left unspecified, although they may affect the counterfactual endpoint. The scenario must clarify which features are changed and which are retained.

The main limitation is that counterfactual outcomes under an event-free scenario are not directly observed for participants who experience the event. Their estimation therefore relies on design, data, and assumptions, and uncertainty about those assumptions can be substantial. A Bayesian analysis can represent uncertainty conditional on its model and prior, but a posterior distribution does not make an ill-defined scenario precise or remove nonidentification. The clinical meaning of the counterfactual question must be settled before choosing an analysis.