Worked Estimand Analysis for a Rescue-Therapy Trial

Why Rescue Therapy Changes the Question

In a randomized trial, some participants may need rescue therapy because their symptoms remain uncontrolled or worsen. The outcome measured after rescue is still clinically relevant, but it does not describe what would have happened had rescue been unavailable. A trial analysis must therefore make explicit how rescue use is handled. Different choices define different estimands, and consequently answer different clinical questions.

Consider a hypothetical 12-week trial comparing an investigational treatment with control for a condition measured on a 0-to-10 symptom scale, where lower scores indicate fewer symptoms. Participants may use a specified rescue medication before week 12. The examples below use symptom score at week 12 as the outcome unless a composite responder outcome is explicitly defined. All numerical values are hypothetical.

Let \(A=1\) denote assignment to the investigational treatment and \(A=0\) assignment to control. Let \(R=1\) indicate rescue-therapy use before week 12 and \(R=0\) indicate no use. Write \(Y\) for the week-12 symptom score. A treatment contrast on the mean-difference scale will be written as treatment minus control; a negative value favors the investigational treatment because lower scores are better.

Three Ways to Define the Rescue Strategy

The same trial can support several clinically coherent questions. The key is not to treat these questions as interchangeable. For each estimand, the treatment conditions and outcome scale should stay clear while the handling of rescue changes.

For the mean symptom score, the treatment-policy contrast is \(E(Y\mid A=1)-E(Y\mid A=0)\), for the randomized target population. In a randomized trial, assignment supports comparison of these marginal outcomes, provided outcome data are adequately observed or appropriately handled. Rescue users remain part of their assigned groups. Their measured week-12 outcomes, including any benefit from rescue, contribute to the result.

This estimand asks what the treatment contrast would be if participants did not use rescue therapy. It is not the same as comparing observed participants who happened not to use rescue: that subset may differ systematically between arms and from participants who required rescue. Outcomes \(Y^{a,0}\) are not observed for participants who actually used rescue, so the hypothetical contrast requires assumptions and an appropriate estimation approach.

The composite estimand could be the difference between treatment and control in the probability of composite success. Its meaning is not simply “the effect on symptoms.” It concerns a combined outcome: symptom response without rescue. The threshold and the rule for rescue use must be stated in advance and justified clinically.

Worked Example: Specify the Three Clinical Questions

Treatment-Policy Analysis: Include Rescue Outcomes

For a treatment-policy estimand, rescue use is not a reason to remove a participant or disregard their outcome. The marginal mean in each randomized arm can be expressed as a weighted average of the means among rescue users and nonusers. This identity helps show why an analysis limited to participants who avoided rescue changes the question.

The weighted-mean identity follows by partitioning each arm into the mutually exclusive groups \(R=0\) and \(R=1\). Each group's contribution to the arm mean equals its proportion in that arm multiplied by its group mean. Adding the two contributions recovers the mean for the full arm. Omitting rescue users, or averaging only the \(R=0\) group means, does not recover this treatment-policy mean.

Hypothetical Analysis: Estimate Outcomes Under No Rescue

A hypothetical estimand requires a model or other identification strategy for outcomes that would have been observed if rescue had not occurred. Randomization makes treatment assignment independent of baseline potential outcomes in expectation, but it does not randomize rescue use. Participants who need rescue may have worse symptoms, different prognostic characteristics, or different clinical histories. Therefore, the observed non-rescue group is generally not a valid stand-in for the full population under a no-rescue scenario.

A typical identification argument requires a clearly defined no-rescue intervention, consistency between observed outcomes and the relevant potential outcomes when rescue is not used, adequate probability of remaining rescue-free across relevant histories, and sufficient control of predictors of both rescue and outcome. In longitudinal settings, the analysis must account for measured histories that affect later rescue and outcomes. The needed assumptions are substantive; a fitted model does not make them true.

Composite Analysis: Make Rescue Part of the Outcome

A composite outcome is useful when needing rescue is itself an important unfavorable event and the clinical question concerns achieving benefit without that event. For example, let \(B=1\) indicate at least a 30% reduction in symptom score from baseline, and define \(C=1\) only if \(B=1\) and \(R=0\). Equivalently, \(C=0\) if rescue is used or if the response threshold is not met. The treatment effect can then be summarized as a risk difference, \(P(C=1\mid A=1)-P(C=1\mid A=0)\).

Compare Interpretations and Check the Analysis

Several checks help reveal whether the estimand and its analysis are coherent:

• Confirm the event definition. Specify what counts as rescue, the relevant time window, and how repeated or partial use is handled. Ambiguous event coding can make the estimand ambiguous.
• Check outcome availability. Treatment policy still requires an outcome definition after rescue. If week-12 outcomes are missing, the missing-data approach must target the same estimand rather than silently excluding participants.
• Inspect hypothetical-model support. If some clinically relevant histories almost always lead to rescue, there may be little information about their outcomes without rescue. Predictions in such regions rely heavily on extrapolation.
• Check composite clinical meaning. A threshold and rescue rule should reflect the clinical decision of interest. If rescue is frequent but symptom response remains valuable, treating every rescue user as failure may obscure that distinction.
• Keep the scale visible. A difference of symptom-score points and a difference of success probabilities have different units and interpretations; neither is automatically the preferred measure.

The main alternative is not a statistical trick but a different estimand. A treatment-policy analysis is often directly aligned with a pragmatic question about assignment under available care. A hypothetical analysis targets a no-rescue scenario that may be useful for understanding treatment effects absent rescue, but depends on untestable or partly testable assumptions. A composite analysis answers a combined benefit-and-rescue question and may be more directly relevant when avoiding rescue is part of treatment success. The choice should follow the clinical question, not which result appears most favorable.

Common Pitfall

The distinction matters because rescue therapy can be both a consequence of treatment and a marker of participants' evolving health. Treatment-policy, hypothetical, and composite estimands preserve different parts of that clinical story. Reporting the event strategy alongside the outcome and summary measure makes clear what the treatment contrast can—and cannot—support.