Death in the Estimand Framework

Why Death Changes the Estimand

Suppose a trial measures a symptom score at 90 days, with lower scores indicating fewer symptoms. A participant who dies before day 90 has no day-90 symptom score: this is not simply a score that was recorded but lost. The outcome, as defined for a living participant, does not exist after death. Therefore, an analysis must not silently treat death as an ordinary missing value. It must first specify what clinical question is to be answered when death prevents measurement.

The estimand framework makes that choice explicit. The three strategies considered here answer different questions: a composite strategy incorporates death into the endpoint; a hypothetical strategy asks about an outcome under a specified scenario in which death is prevented; and a principal-stratum strategy targets people who would survive under either treatment. These are not alternative calculations of one universal effect. They define different outcomes or populations and therefore different treatment effects.

Composite Strategy: Death Is Part of the Endpoint

A composite strategy defines an outcome that is meaningful for everyone, including participants who die. One simple example is success at day 90, defined as being alive and having achieved a prespecified clinical improvement. A participant who dies before day 90 is classified as not achieving this composite success. The treatment contrast then concerns the probability of success in the full target population, not symptom change among survivors alone.

This approach requires a clear endpoint definition: the status for death must be specified, along with the clinical success criterion for those alive. A binary endpoint such as “alive and improved” avoids needing to assign a numerical symptom score to someone who died. It does, however, combine survival and clinical improvement. A difference in composite success can reflect a treatment difference in survival, in improvement among survivors, or in both.

The justification for this contrast follows directly from the endpoint definition. If \(C^a\) equals 1 when a person would be alive and improved under assignment \(a\), and 0 otherwise, then \(C^a\) is defined for every person. The population contrast is \(E(C^1)-E(C^0)\), which, for a binary endpoint, equals the difference between the two treatment-specific probabilities of composite success. Random assignment supports a comparison of those assignment groups, subject to the trial’s design and appropriate handling of outcome ascertainment. No symptom value after death is required.

The main clinical cost is that the composite may obscure which component drives the result. Two treatments could have the same composite success rate but different survival and improvement patterns. Component summaries can help describe the result, but the prespecified composite contrast remains the effect for this estimand. The clinical importance of the endpoint depends on whether its definition matches the decision at hand.

Hypothetical Strategy: Specify a World Without Death

A hypothetical strategy asks about the outcome under a clearly specified counterfactual scenario in which death does not occur. It targets the trial’s treatment population but changes the event-handling question: what would the clinical outcome be under each assigned treatment in the specified no-death scenario? This is not obtained merely by removing participants who died. Nor is it enough to say “assume no one dies” without clarifying what scenario that phrase represents.

For example, the question might posit a specified intervention that prevents otherwise fatal events while leaving the rest of the treatment context as defined in the estimand. Let \(Y^{a,g}\) denote the outcome under treatment assignment \(a\) and a specified death-prevention scenario \(g\). A mean contrast is \(E(Y^{1,g})-E(Y^{0,g})\). The scenario \(g\) must be sufficiently clear to give this contrast a clinical interpretation: different ways of preventing death could also affect symptoms, so they need not imply the same hypothetical outcome.

The key assumptions concern the counterfactual scenario and the information used to estimate its outcomes. The scenario must be clinically interpretable, and the model or design must provide a defensible basis for predicting outcomes that were not observed because death occurred. Randomization establishes comparability of assigned groups for outcomes that can be observed under the trial conditions; it does not, by itself, reveal each deceased participant’s outcome under \(g\). The estimate can consequently depend strongly on modeling choices and assumptions about how the hypothetical intervention would affect outcomes.

A diagnostic question is whether the proposed no-death scenario is concrete enough that clinicians could understand what it means. If not, the hypothetical estimand may be too vague to guide a decision. Another warning sign is treating the model’s extrapolation as if it were observed data. The result should be reported as an estimate under stated assumptions, with sensitivity to plausible alternatives considered where appropriate.

Principal-Stratum Strategy: Target People Who Would Survive Either Way

A principal-stratum strategy defines the target population by potential survival under both treatment assignments. The always-survivor stratum consists of people for whom \(S^1=1\) and \(S^0=1\): they would be alive at the outcome time under either assignment. A mean treatment contrast in this stratum is

This asks about the treatment effect on the clinical outcome among people who would survive under either treatment. The definition does not assign a post-death score: both potential outcomes in this stratum are defined because its members would survive under both assignments. But stratum membership is counterfactual. For any one participant, the trial reveals survival under the assigned treatment, not survival under both assignments. The principal stratum therefore cannot be identified just by selecting observed survivors in each treatment group.

This strategy is most relevant when the scientific question is specifically about treatment effects among people who would survive either way. Its target is a subgroup defined by potential outcomes, not the full trial population and not the observed survivors under either arm. Comparing observed survivors directly can answer a different question: treatment may affect who survives, so the two groups of observed survivors need not consist of comparable people.

A practical diagnostic is to ask whether the clinical decision genuinely concerns the always-survivor population and whether the assumptions used to estimate that latent subgroup are defensible. If the target subgroup cannot be characterized or the necessary assumptions are implausible, a precise-looking estimate may still have a fragile interpretation. Detailed identification questions require special care; the defining estimand alone does not solve them.

Compare the Questions Before Choosing a Strategy

The strategies are not interchangeable. The composite asks about an endpoint defined for everyone; the hypothetical strategy changes the scenario; the principal-stratum strategy changes the target population. Their estimates can disagree without contradiction because they address different clinical questions.

Common Mistakes and Limits

• Calling death ordinary missingness. A symptom score after death is undefined under the stated outcome definition. Do not impute it as though it were merely unrecorded.
• Calling observed survivors the always-survivor stratum. Observed survival reveals only the outcome under the assigned treatment. It does not reveal survival under both assignments.
• Describing a hypothetical result without naming its scenario. A no-death contrast requires a specified counterfactual scenario; different scenarios can imply different effects.
• Interpreting a composite contrast as a symptom-only effect. If death is part of the endpoint, the result can reflect both survival and clinical improvement.
• Assuming randomization solves every problem. Randomization does not alone identify outcomes under an unobserved death-prevention scenario or membership in a counterfactual survival stratum.

The selection should follow the clinical decision, not which strategy produces the most favorable result. If the decision concerns being alive and improved, a prespecified composite may be direct. If it concerns an explicitly defined scenario without death, the hypothetical question may be relevant but assumption-dependent. If it concerns effects among those who would survive either way, the principal-stratum target is conceptually precise but difficult to identify. In all cases, state the target population and interpretation alongside the estimate.