When an Event Changes What the Outcome Means
An intercurrent event can affect an endpoint in two different ways. It may prevent a particular event of interest from occurring first, as when death prevents a later hospitalization from being observed. Or it may make the endpoint itself undefined, as when a planned quality-of-life score at week 24 has no meaningful value for a participant who died before week 24. These situations are related, but they are not interchangeable.
The distinction matters because a participant with an undefined outcome is not simply a participant whose defined outcome was not recorded. Nor does excluding participants with an intercurrent event automatically answer the original treatment question. The event, endpoint, follow-up time, and target population must be specified together.
Whether an event is competing or makes an outcome undefined depends on the endpoint. Death can compete with a first hospitalization endpoint: after death, a later first hospitalization cannot occur. Death also truncates an endpoint such as a week-24 quality-of-life score if that score is defined only for people alive at week 24. In contrast, an event is not automatically a competing event for every endpoint, and the same intercurrent event may be handled differently for different estimands.
An estimand should state the population, treatment conditions, outcome variable and time, strategy for handling relevant intercurrent events, and population-level summary. Here the key discipline is to define the outcome and event handling before calculating a contrast. If the outcome has no defined value after an event, a numerical comparison of means requires an explicit target population or a revised outcome definition.
Competing Events: Keep the Full Event Categories in View
For a first-event endpoint, let \(T\) be the time to the first event and let \(J\) identify its type. For example, \(J=1\) might denote hospitalization and \(J=2\) death before hospitalization. At a fixed time \(t\), the risk of the event of interest is \(P(T\leq t, J=1)\). Participants who experience a competing event belong in the population denominator; they have experienced a different outcome category, not simply disappeared from the study.
This definition assumes that the event types and their ordering are specified consistently, that the follow-up period is stated, and that each participant can be classified according to the endpoint definition. If event types can overlap, the protocol must say how the first event is assigned. These risks describe what happened under each treatment condition. A comparison of them is not automatically a causal effect under a different hypothetical world in which competing events could not occur.
Worked Example: Hospitalization With Death as a Competing Event
Consider a hypothetical randomized study with 200 participants in each treatment group, followed for 24 weeks. In the new-treatment group, 30 participants are hospitalized first, 10 die before hospitalization, and 160 experience neither event by week 24. In the control group, the corresponding counts are 40, 5, and 155. The categories are mutually exclusive first-event outcomes.
Risk of first hospitalization in the new-treatment group. The denominator is all 200 randomized participants, including those who died before hospitalization: \(30/200=0.1500\), or 15.00%.
Risk of first hospitalization in the control group. \(40/200=0.2000\), or 20.00%. The treatment-minus-control risk difference is \(0.1500-0.2000=-0.0500\), or \(-5.00\) percentage points over 24 weeks. In this hypothetical sample, the observed first-hospitalization risk is 5 percentage points lower in the new-treatment group.
If the question instead concerns the proportion hospitalized among participants who did not die before hospitalization, the calculations change: \(30/(200-10)=30/190\approx0.1579\), or 15.79%, in the new-treatment group, and \(40/(200-5)=40/195\approx0.2051\), or 20.51%, in the control group. Those conditional proportions answer a different question. They exclude participants according to a post-randomization event and should not be presented as the all-participant risks of first hospitalization.
A frequent error is to treat a competing event as if it were ordinary loss to follow-up, then interpret a result based on the remaining participants as the original event risk. The first-event definition avoids that ambiguity: every participant is classified as having the event of interest first, a competing event first, or neither by the stated time. A different analysis can target a different quantity, but its estimand must say so.
Truncated Outcomes Require an Explicit Target
Suppose \(R^a(t)=1\) indicates that a participant assigned treatment condition \(a\) is alive and eligible for a week-\(t\) outcome, and \(R^a(t)=0\) indicates death before that time. If the outcome is defined only for those alive at \(t\), write \(Y^a(t)\) only when \(R^a(t)=1\). There is no value of \(Y^a(t)\) to recover for a participant with \(R^a(t)=0\) under that outcome definition.
Several distinct estimands are possible, but they answer different questions. A survivor-based summary under each treatment condition compares outcomes among those who would be alive under that condition. A principal-stratum contrast could instead target participants who would survive under either treatment. A hypothetical estimand could ask about the outcome under a clearly specified condition in which the truncating event did not occur. A composite strategy could define a new outcome that includes both the event and the original endpoint. None of these targets should be substituted silently for another.
Worked Example: A Week-24 Quality-of-Life Outcome Truncated by Death
In a hypothetical randomized study, 100 participants are assigned to each treatment condition. The week-24 quality-of-life score is defined only for participants alive at week 24. In the new-treatment group, 20 participants die before week 24; the 80 survivors have a mean score of 70 points. In the control group, 10 die; the 90 survivors have a mean score of 74 points.
The survivor mean contrast is \(70-74=-4\) points: the mean among survivors assigned the new treatment is 4 points lower than the mean among survivors assigned control. The calculation is valid as a descriptive comparison of these treatment-specific survivor groups, but it does not by itself estimate a treatment effect among one common population. The groups of survivors may differ because survival rates differ and because treatment can affect who survives.
For example, the principal-stratum question would compare scores among people who would survive under either treatment condition. That group is defined by the joint condition \(R^1(24)=R^0(24)=1\), not merely by observed survival in one assigned group. A hypothetical question would require a well-defined condition preventing death and an outcome under that condition. These are different targets; the survivor means alone do not establish either one.
If a survivor is alive at week 24 but the questionnaire was not returned, that score is missing, not truncated by death. The distinction is important: the outcome is defined for that survivor, even though it was not observed.
Making an Outcome Defined for Everyone
One option is to define a composite outcome that incorporates the intercurrent event. For example, a protocol could define success as being alive and not hospitalized by week 24. This binary outcome is defined for all participants, but it is not the same as the original quality-of-life score. The composite rule must be clinically interpretable and specified in advance; the components and their ordering can influence the treatment contrast.
Worked Example: A Composite of Death or Hospitalization
In a hypothetical study, 200 participants are assigned to each treatment condition. By week 24, 12 participants in the new-treatment group have died, 18 others have been hospitalized but are alive, and 170 have experienced neither. In the control group, 8 have died, 30 others have been hospitalized but are alive, and 162 have experienced neither. These categories are mutually exclusive: death is counted regardless of any earlier hospitalization, and the hospitalization category includes only those not counted as deaths.
Define the composite event as death by week 24 or hospitalization by week 24. The new-treatment risk is \((12+18)/200=30/200=0.1500\), or 15.00%. The control risk is \((8+30)/200=38/200=0.1900\), or 19.00%. The risk difference is \(0.1500-0.1900=-0.0400\), or \(-4.00\) percentage points.
This is an all-participant comparison of the defined composite, not a comparison of quality-of-life scores and not a claim that the two component events have equal clinical importance. Its interpretation depends on the stated rule that either event counts as a composite event.
Why the Choice of Estimand Matters
Different approaches can be appropriate for different scientific questions. Their results need not agree, because they summarize different outcomes or populations.
| Approach | What is defined or compared? | Key interpretive caution |
|---|---|---|
| Competing-event risk | Risk of a specified first event by a stated time, with competing event types retained as distinct outcomes. | Do not treat the competing category as if it were simply absent from the population. |
| Composite strategy | A new outcome combining the original endpoint and an intercurrent event according to a prespecified rule. | The rule determines how components contribute to the result. |
| Survivor-based summary | Outcome summaries among participants alive or otherwise eligible under each treatment condition. | The survivor groups can differ between treatment conditions. |
| Principal-stratum target | A contrast in a group defined by potential event status under multiple treatment conditions, such as survival under both. | Membership in that group is not directly observed for each participant. |
| Hypothetical target | An outcome contrast under a specified condition in which the event does not occur. | The hypothetical condition and assumptions needed to support it must be credible. |
The appropriate choice depends on the clinical question, not on which calculation is easiest. A composite is useful only if its event rule answers a relevant question. A survivor-based result is informative about outcomes among survivors in each treatment group, but it does not generally compare the same people. A principal-stratum or hypothetical question may be scientifically meaningful, but it requires assumptions beyond random assignment alone.
Derivation and Checks Before Interpretation
The denominator distinction in the first example follows directly from the event definition. Let \(H\) denote first hospitalization by time \(t\), and \(D\) denote death before hospitalization by \(t\). Under the mutually exclusive first-event classification, the full cohort is partitioned into \(H\), \(D\), and neither event. Therefore the all-participant risk is \(P(H)\), estimated by the count in category \(H\) divided by the full cohort size. By contrast, the proportion among those without a competing death estimates a conditional quantity:
Here the equality uses the fact that \(H\) and \(D\) are mutually exclusive, so \(H\subseteq D^c\). In the new-treatment group, this conditional proportion is \(30/190\), not \(30/200\); it answers a question restricted to those who avoided competing death. Thus changing the denominator changes the target summary, rather than merely correcting a calculation.
Before analysis, check that the endpoint definition remains meaningful after each event. If it does not, do not fill the undefined value with an ordinary missing-data imputation without first defining the target outcome. Then verify the event time, event categories, follow-up window, and denominator. For truncated outcomes, report the number experiencing the event and the number with an observed outcome among those for whom it is defined. These checks reveal whether a reported contrast is an all-participant event risk, a survivor summary, or another estimand.
Common Pitfalls and Limitations
- Calling every unobserved value missing. A questionnaire not returned by a living participant can be missing; a score defined only for survivors has no value after death under that definition.
- Dropping competing events without changing the interpretation. Dividing by the number remaining event-free gives a conditional proportion, not the original all-participant event risk.
- Calling survivor groups equivalent because the study was randomized. Randomization assigns treatment; it does not make the survivors under different treatments the same population.
- Combining outcomes without stating the rule. A composite must specify which events count, the time window, and how overlapping events are classified.
- Assuming a hypothetical outcome is observed data. An outcome under a condition that did not occur requires a clearly stated hypothetical target and assumptions; it is not supplied by the observed survivor mean.
These definitions do not by themselves identify every causal estimand or determine an estimator. In particular, observed data may not reveal joint potential survival status for a principal stratum, and a hypothetical estimand depends on its specified intervention and supporting assumptions. The goal at this stage is to make the target precise enough that later estimation and sensitivity analysis can be aligned with it.
Check Your Understanding
Answer each question by naming the target quantity, not just the arithmetic.
- In a 100-person group, 12 experience the event of interest first, 8 experience a competing event first, and 80 experience neither by the specified time. What is the all-participant risk of the event of interest?
- A week-12 score is defined only for participants alive at week 12. A living participant misses the questionnaire. Is that score truncated or missing?
- Why does comparing mean scores among survivors assigned to two treatments not necessarily compare outcomes in the same population?
- A study defines success as being alive and free of hospitalization at week 24. What kind of outcome is this, and why is it defined for all participants?
Solutions. (1) The all-participant risk is \(12/100=0.1200\), or 12.00%; the 8 competing events remain in the denominator. (2) It is missing: the score is defined for a living participant but was not observed. (3) Survival can differ by treatment, so each mean conditions on a treatment-specific survivor group; observed survivors need not represent the same population. (4) It is a composite binary outcome defined by a rule that counts death or hospitalization as failure, so every participant can be classified as success or failure by week 24.