Why Death Requires a Separate Decision
In a stroke trial, functional status might be assessed at a fixed follow-up time using a disability scale. Some participants may die before that assessment; others may be alive but have no recorded assessment. These situations are not interchangeable. For a participant who dies before the assessment, functional status at that time is generally undefined, not a value that was measured but lost. For a living participant with no recorded score, the outcome is defined but unobserved.
This distinction matters before any imputation or comparison is attempted. Imputing a functional score after death silently assigns meaning to a quantity that the chosen outcome does not define. Conversely, treating an absent score from a living participant as if the participant had died changes the endpoint. A sound analysis states what outcome is being compared, how death enters that outcome, and which living participants' missing assessments require assumptions.
The coding of death depends on the estimand: the precisely specified quantity the trial is intended to estimate. One option is a composite outcome that gives death an explicit status, such as “favorable function while alive” versus “death or unfavorable function.” Another is to describe functional status among those alive at the assessment time. That latter quantity is conditional on survival in each treatment group. Since treatment can affect survival, the survivors in the two groups need not represent comparable sets of participants. A survivor-only contrast therefore does not, by itself, estimate the effect in a shared population of survivors.
A treatment-policy strategy for handling treatment switching does not resolve this issue: following participants regardless of switching does not make a functional score defined after death. The analysis still needs an outcome definition that addresses death. The decision is scientific, not a missing-value coding detail.
A Hypothetical Trial and an Observed-Data Audit
Consider a hypothetical randomized stroke trial with 100 participants assigned to each group. A favorable functional outcome is defined in advance for participants alive at the scheduled assessment. In the treatment group, 10 participants died before assessment, 9 were alive but had no recorded assessment, and 81 were assessed: 54 had favorable function and 27 had unfavorable function. In the control group, 16 died, 4 were alive without an assessment, and 80 were assessed: 40 had favorable function and 40 had unfavorable function.
Before analyzing scores, verify that the categories are exhaustive and non-overlapping. For treatment, \(10+9+81=100\); for control, \(16+4+80=100\). Deaths are identified separately from living participants with missing assessments. If death status is uncertain, resolving that status is a data-quality task, not an imputation choice.
Worked Example: What the Assessed Participants Show
Among assessed participants, calculate the observed favorable proportions and the treatment-to-control odds ratio. These are summaries of assessed participants only, not yet a complete analysis of all randomized participants.
In the treatment group, the observed favorable proportion is \(54/81=0.6667\), and the observed odds of favorable rather than unfavorable function are \(54/27=2\). In control, the observed favorable proportion is \(40/80=0.5000\), and the odds are \(40/40=1\). Thus the assessed-participant odds ratio is \(2/1=2\).
The treatment-group log odds are calculated from the exact counts:
This calculation is internally consistent because it uses the exact fractions, not rounded decimal proportions. The estimate describes the observed functional outcomes among assessed participants. It does not account for deaths or living participants without assessments, and differential survival or assessment can make this selected group contrast differ from the target trial contrast.
Define Death in the Outcome Before Handling Missing Scores
For an all-randomized comparison, one possible estimand defines success as being alive with favorable function at the assessment time. Death is then explicitly a non-success, whatever the participant's unobserved functional status would otherwise have been. This is a composite definition: it combines survival and function into one outcome. It does not claim that death is a poor score on the functional scale.
Under this definition, deaths are not missing functional scores to be imputed. The 9 treatment-group and 4 control-group participants who were alive but unassessed remain unresolved for the composite outcome: their success status depends on their unobserved functional status. A sensitivity analysis can vary assumptions about those living participants while keeping the death convention fixed.
Worked Example: A Composite Outcome Under a Conservative Assumption
Suppose the analysis defines success as alive with favorable function, and as a simple sensitivity case assumes every living participant without an assessment would have had unfavorable function. The assumption applies to the 9 and 4 living participants with missing assessments. Deaths are non-successes by the composite definition, not by imputation.
The treatment group has 54 successes out of 100, so its success proportion is \(54/100=0.54\), or 54.0%. The remaining 46 are 27 observed unfavorable outcomes, 9 living participants assumed unfavorable for this sensitivity case, and 10 deaths. Control has 40 successes out of 100, giving \(40/100=0.40\), or 40.0%; the other 60 comprise 40 observed unfavorable outcomes, 4 living participants assumed unfavorable, and 16 deaths.
The estimated risk difference, treatment minus control, is
Under this particular missing-outcome assumption and composite definition, the treatment group has an estimated 14.0 percentage-point higher success proportion. This is a descriptive point estimate from hypothetical data; it is not a confidence interval or a guarantee of benefit. The conclusion is conditional on the endpoint and on the stated assumption about the living participants with missing assessments.
Sensitivity Analysis for Living Participants
For living participants with unrecorded assessments, the outcome is defined but unavailable. An analysis may make assumptions about its distribution. A transparent sensitivity analysis varies those assumptions and shows how the target estimate changes. The assumptions should concern the missing outcomes among living participants; they should not assign a functional value to participants who died.
A simple approach is to vary the probability of favorable function among living participants without assessments. This probability is not identified from their unobserved outcomes, so its chosen value is an assumption. It can differ by treatment group, and a range of values is usually more informative than presenting only one preferred value. When probabilities are used, multiplying each probability by the number of living participants with missing assessments gives an expected number of additional successes, which may be fractional.
Worked Example: Varying Success Probabilities Among the Living
For illustration, suppose an analyst assigns a 0.30 probability of favorable function to each living, unassessed treatment participant and a 0.20 probability to each living, unassessed control participant. These are hypothetical sensitivity assumptions, not facts learned from the observed data. Death remains a non-success under the composite definition.
In treatment, expected successes are \(54+9(0.30)=54+2.7=56.7\). The estimated composite success proportion is \(56.7/100=0.567\), or 56.7%. In control, expected successes are \(40+4(0.20)=40+0.8=40.8\), giving \(40.8/100=0.408\), or 40.8%. The estimated risk difference is \(0.567-0.408=0.159\), or 15.9 percentage points.
For comparison, the corresponding proportions among survivors are \(56.7/90=0.630\) in treatment and \(40.8/84=0.4857\) in control, a difference of 0.1443, or about 14.4 percentage points. This is a different summary, conditional on survival in each group. The two groups have different numbers of survivors, and their survivor populations may differ; the difference should not be presented as an effect in one common population of survivors.
The probability assumptions can be varied in a grid. For example, one can hold the control assumption at 0.20 and examine treatment assumptions from 0.10 to 0.70, then reverse the exercise or vary both groups. A tipping point is a value at which the chosen conclusion changes, but it is meaningful only relative to the specified estimand, outcome definition, and range of plausible assumptions. It does not reveal the true unobserved outcomes.
Bounds and a Short Justification
Even without assigning probabilities to living participants with missing assessments, the hypothetical counts give simple bounds for the composite success proportions. Keep all deaths as non-successes. In treatment, the 54 observed successes are certain, and at most all 9 living unassessed participants could be successes. Therefore the proportion is between \(54/100\) and \((54+9)/100\). In control, the corresponding range is from \(40/100\) to \((40+4)/100\).
The treatment-minus-control risk difference is at least the smallest treatment proportion minus the largest control proportion, and at most the largest treatment proportion minus the smallest control proportion:
These bounds are justified by considering the two extreme assignments of the unobserved living outcomes. For the lower bound, every unassessed treatment participant is a non-success and every unassessed control participant is a success. For the upper bound, reverse those assignments. The resulting range is a range of possible sample contrasts under the composite definition, not a confidence interval and not a probability statement about the true effect. Sampling uncertainty would require additional inferential calculations.
Diagnostics, Alternatives, and Limits
A useful analysis begins with an outcome-status table by treatment group: numbers randomized, deaths before assessment, survivors assessed, and survivors unassessed. Also report the timing of deaths and missing visits when available. Check that every participant belongs to exactly one status category at the assessment time. If the death counts or assessment status are wrong, sensitivity modeling cannot repair the data.
For missing assessments among living participants, examine whether recorded baseline characteristics and earlier functional measurements differ between assessed and unassessed participants. Such comparisons can reveal selection patterns, but they cannot establish the unobserved outcomes. Report which assumptions are used for the primary analysis and which are varied in sensitivity analyses. A model that produces implausibly favorable outcomes for a group with substantial death or dropout deserves scrutiny, as does an analysis that hides a differential death rate inside a single missing-data count.
The main approaches answer different questions:
| Approach | What is summarized | Key limitation |
|---|---|---|
| Composite outcome, such as alive with favorable function | All randomized participants, with death assigned an explicit outcome status | The death convention is part of the scientific definition; it is not a functional score. |
| Function among survivors | Functional outcomes conditional on being alive in each group | Treatment groups can have different survivor populations, so the contrast is not automatically a common-population causal effect. |
| Sensitivity analysis for living participants with missing assessments | How estimates change under specified assumptions about defined but unobserved outcomes | Results depend on assumptions that cannot be verified from those missing outcomes alone. |
| Rank or ordered composite including death | A prespecified ordering of death and levels of function | The ordering and interpretation must be clinically defensible and stated in advance. |
A hypothetical strategy that asks what function would have been if everyone had survived is another possible scientific question, but it concerns counterfactual outcomes under a different scenario. It cannot be obtained simply by filling in scores after death. Likewise, a survivor-only analysis does not solve the missingness problem among living participants; it may still require assumptions for their unrecorded assessments.
Common Mistakes
- Calling death an ordinary missing assessment. A living participant's unrecorded score is missing; a post-death score is undefined under the stated outcome. Specify an outcome that includes death rather than imputing a score.
- Counting every death as if it were a missing functional value. This leaves the estimand unclear. If death is a non-success, say that the analysis uses a composite success outcome and explain the definition.
- Reporting a survivor-only contrast as the all-randomized treatment effect. Survival can differ by treatment, so conditioning on survival changes the population being described.
- Treating an assumption as an observed result. A probability assigned to living participants with missing assessments is a sensitivity assumption. Label it and show how conclusions change under alternatives.
- Using a sensitivity range as a confidence interval. Bounds from possible missing outcomes describe uncertainty due to those unknown outcomes in the observed sample. They do not include sampling uncertainty.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A participant is alive at the scheduled stroke assessment, but no functional score was recorded. Is the score undefined or defined but unobserved?
- Under the composite definition “alive with favorable function,” calculate the treatment-group success proportion if all 9 living unassessed participants are assumed unfavorable.
- Using the hypothetical trial counts, give the lower and upper bounds for the control-group composite success proportion when deaths are non-successes but the 4 living unassessed outcomes are unknown.
- Why does a difference in functional outcomes among survivors not automatically represent a treatment effect in one common survivor population?
- If the analysis assigns a probability of favorable function to living participants without assessments, what kind of quantity is that probability?
Solutions. (1) It is defined but unobserved, because the participant was alive at the time the outcome was defined. (2) The 54 observed favorable outcomes are the only successes in this sensitivity case, so \(54/100=0.54\), or 54.0%. Deaths and the 9 assumed unfavorable outcomes are non-successes. (3) The 40 observed successes are certain and at most 4 more could be successes, so the bounds are \(40/100=0.40\) and \((40+4)/100=0.44\). (4) Treatment can affect who survives; the survivor sets can therefore differ between groups, even in a randomized trial. (5) It is an assumption about an unobserved outcome, not an observed probability estimated from those participants' missing assessments.