Why Competing Events Need Prespecification
A protocol that asks whether a treatment prevents a nonfatal event must say what happens to people who die first. Death is not simply a missing observation: after death, the person cannot experience a later nonfatal event. Counting death as if it were the nonfatal event, ignoring it, or treating it as ordinary censoring can each correspond to a different question. The event definition and follow-up rules therefore help define the estimand, not just the analysis.
For example, a trial of two treatment strategies might ask about stroke within one year. A person who has a stroke and later dies experienced the outcome. A person who dies before having a stroke did not experience stroke within that person’s observed lifetime, and cannot go on to do so during the remainder of the year. The protocol should state how those two histories are classified.
Define the Event and Its Timing
Let \(a\) denote an assigned treatment strategy. Let \(T^a\) be the time from time zero to the first event that the protocol classifies as one of the specified event types under strategy \(a\), and let \(J^a\) identify that event type. For a nonfatal event of interest \(k\), its cumulative incidence by horizon \(t\) under strategy \(a\) is
This is the probability of experiencing event type \(k\) by \(t\), before any event that precludes it under the protocol. A causal contrast may compare \(F_k^1(t)\) and \(F_k^0(t)\), for example by a risk difference \(F_k^1(t)-F_k^0(t)\). The contrast must name the event, horizon, strategies, and scale. A statement such as “treatment reduces events” is not sufficiently specific.
Before analysis, the protocol should identify which event types are relevant, how each is ascertained, when follow-up starts and ends, and what happens if event dates tie or cannot be ordered. It should also specify whether the target outcome is the first qualifying event or whether later events count. If the outcome is “first stroke before death,” stroke after a prior death is impossible; if the outcome is “any stroke recorded during a defined period,” the protocol still needs a coherent rule for death before the end of that period.
The choice is outcome-specific. For all-cause mortality, death is the event of interest. For a composite of stroke or death, either event qualifies as the outcome. For nonfatal stroke, death before stroke is a competing event. A protocol should not use the label “competing risk” as a substitute for defining exactly what qualifies.
What the Competing-Event Risk Means
The event-specific cumulative incidence is an observed-world risk under a strategy: it includes the fact that some people may experience a competing event first. It does not answer the hypothetical question of what the stroke risk would have been if death could somehow be prevented. These are different estimands.
The distinction follows from partitioning histories by their first event. For a fixed horizon, experiencing event \(k\) by \(t\) requires that event \(k\) occurs first by \(t\). Histories with a competing event first are mutually exclusive from histories with event \(k\) first. Thus the event-specific risk is a joint probability involving both event time and event type, not a probability calculated as though competing events were merely absent.
In discrete time, if the possible first-event times are \(1,\ldots,t\), the same risk can be written as a sum over event times:
The terms are disjoint: a person cannot have the first qualifying event at two different times or have two different first-event types. Their sum therefore gives the probability of the specified event occurring first by the horizon. A competing event changes the probability of remaining able to experience event \(k\), so it can change this risk even if the treatment has no direct effect on the event process among people who remain alive and event-free.
Worked Example: Stroke Risk When Death Can Occur First
Consider a hypothetical randomized trial with 100 participants assigned to each of two strategies. Follow-up is one year. The protocol defines the outcome as first stroke before death; death before stroke is a competing event. By one year, the first-event counts are:
| Assigned strategy | Stroke first | Death before stroke | Neither by one year | Total |
|---|---|---|---|---|
| Strategy 1 | 12 | 8 | 80 | 100 |
| Strategy 0 | 8 | 20 | 72 | 100 |
The one-year stroke cumulative-incidence estimates from these hypothetical counts are \(12/100=0.12\), or 12%, under strategy 1 and \(8/100=0.08\), or 8%, under strategy 0. The risk difference is
so the estimated one-year risk of stroke before death is 4 percentage points higher under strategy 1 in this example. This is not, by itself, evidence that the strategy causes higher stroke risk; sampling uncertainty and the design or identification assumptions still matter. The result is specifically about stroke before death, not about a hypothetical risk with death removed.
If the protocol instead defined a composite outcome of stroke or death, the risks would be \((12+8)/100=20\%\) and \((8+20)/100=28\%\). The risk difference would then be \(0.20-0.28=-0.08\), or 8 percentage points lower under strategy 1. The opposite direction is possible because the composite and stroke-specific questions are different.
Why “Censor Death” Changes the Question
An ordinary censoring rule says that follow-up ends before the outcome status is known, for example because a record is unavailable or observation ends. Death before a nonfatal event is different: it is an observed event that prevents the later outcome. Treating it as ordinary censoring in a method that estimates risk as though those individuals could remain at risk does not generally recover the observed-world cumulative incidence.
A simple calculation illustrates the issue. Suppose, in a hypothetical group of 100, eight people die on day 30 before stroke and 12 others have stroke on day 60; all remaining people are alive and stroke-free through one year. The observed one-year stroke risk is \(12/100=12\%\). If deaths are treated as censored and a Kaplan–Meier calculation is used as if the censored people could represent those still at risk, the risk set just before the day-60 strokes is 92. The estimated survival after those strokes is \(1-12/92\), so the resulting estimated event risk is
or about 13.04%, not the observed 12%. More importantly, this calculation relies on treating the people who died as if their subsequent outcome experience could be represented by survivors. It does not turn death into a non-informative loss to follow-up. An analysis that removes the competing event targets a distinct hypothetical quantity and needs assumptions and a defensible interpretation for that quantity; simply censoring deaths does not establish it.
Worked Example: A Composite Can Reverse the Summary
Suppose a hypothetical study compares two strategies for people with chronic kidney disease. The protocol defines the event of interest as kidney failure of the native kidneys, and specifies that kidney transplantation before native-kidney failure makes that particular outcome impossible. At one year, the first-event counts among 80 people per group are:
| Strategy | Native-kidney failure first | Transplant before failure | Neither by one year | Total |
|---|---|---|---|---|
| A | 10 | 15 | 55 | 80 |
| B | 8 | 5 | 67 | 80 |
The one-year event-specific risks are \(10/80=0.125\), or 12.5%, for A and \(8/80=0.10\), or 10.0%, for B. The risk difference for native-kidney failure before transplantation is \(0.125-0.10=0.025\), or 2.5 percentage points higher under A.
If the protocol instead uses a composite of native-kidney failure or transplantation, the risks are \((10+15)/80=31.25\%\) for A and \((8+5)/80=16.25\%\) for B. The composite risk difference is \(0.3125-0.1625=0.15\), or 15 percentage points higher under A. Neither calculation is a correction of the other: one concerns native-kidney failure before transplant, and the other concerns either event. If the intended outcome included graft failure after transplant, transplant would not automatically make that broader outcome impossible, and the protocol would need a different definition.
Prespecifying the Protocol and Checking It
A useful protocol makes competing-event decisions explicit before outcomes are analyzed. These decisions should agree with the target-trial question, the endpoint definition, and the available event dates. For a causal comparison of strategies, the interpretation also depends on the usual design and identification conditions: the strategies and outcomes must be well-defined, treatment assignment must be randomized or adequately controlled for in an emulation, and relevant treatment options must be represented in the populations compared. A precise competing-risk estimand does not by itself guarantee an unbiased estimate.
State the clinical event, whether it is a first event, and the relevant horizon. Avoid labels that leave room for multiple endpoint definitions.
For each, state whether it is a competing event, a component of a composite, the outcome itself, or an event after which the outcome can still occur.
Specify time zero, the horizon, how event dates are compared, how ties are handled, and what determines the end of outcome ascertainment.
For example: “the one-year risk of nonfatal stroke before death under each assigned strategy, and the between-strategy risk difference.”
Confirm that the records distinguish event types and provide dates precise enough to determine their order. Flag ambiguous or incomplete event histories rather than silently assigning a convenient category.
Practical diagnostics include tabulating each first-event type by assigned strategy, checking the number and timing of ties, and comparing recorded event dates across data sources where possible. Large or unexpected differences in competing-event frequency can be clinically meaningful and can alter the event-specific risk. They also warrant checks for inconsistent definitions, differential outcome recording, and errors in event ordering. These checks do not establish causality, but can expose a protocol or data implementation that fails to match the intended question.
Choosing Among the Main Questions
| Protocol question | How the event is handled | Interpretation |
|---|---|---|
| Risk of a specific event before a competing event | Count the specified event if it occurs first; classify the precluding event separately. | Observed-world event-specific risk under the strategy. |
| Risk of either of two events | Count either component as the outcome. | Risk of the prespecified composite, not the risk of either component alone. |
| Risk if the competing event were eliminated | Define a hypothetical intervention or scenario that prevents the competing event. | A hypothetical estimand that is not obtained merely by censoring competing events. |
| Risk of the competing event itself | Define that event as the outcome. | A separate event-specific question, such as all-cause mortality. |
A common mistake is to say, “Death is a competing event, so censor deaths and estimate the same stroke risk.” That reasoning confuses an event that prevents stroke with loss of observation. The correction is to state whether the target is the observed-world risk of stroke before death, a composite, or a hypothetical risk under an explicitly defined intervention on death. The analysis must then match that stated question.
Another limitation is that a seemingly clinical label may not map cleanly to the data. If a database records only the month of stroke and death, same-month events may have unknown order. The protocol should specify a defensible tie rule and report the ambiguity; it should not imply that the true order is known. Detailed rules for genuinely missing or incomplete outcome information are a separate issue, but the competing-event definition must make clear which event history the protocol intends to classify.
Worked Example: Choosing a Defensible Endpoint for Hospitalization
A hypothetical protocol compares two strategies and asks about first unplanned hospitalization within 90 days. Death can occur before hospitalization. The protocol’s primary endpoint is hospitalization before death, not hospitalization or death. For 200 people assigned to each strategy, suppose 30 in strategy A and 24 in strategy B are hospitalized before death by day 90. The corresponding risks are \(30/200=0.15\) and \(24/200=0.12\); the risk difference is \(0.15-0.12=0.03\), or 3 percentage points higher under A.
If the clinical question is instead “unplanned hospitalization or death by day 90,” the protocol must count deaths as composite events. Suppose there are also 14 deaths before hospitalization under A and 20 under B, with no overlap because these are first-event categories. Composite risks are \((30+14)/200=22\%\) under A and \((24+20)/200=22\%\) under B, a difference of 0 percentage points. The equal composite risks do not contradict the different hospitalization-before-death risks. Each calculation answers its own prespecified question.
Check Your Understanding
Answer each question using the event definition and follow-up horizon stated.
- A protocol studies nonfatal myocardial infarction within one year, and a participant has an infarction at month 4 and dies at month 8. Does the participant experience the outcome?
- In the same protocol, what does death at month 4 before any infarction represent?
- In a hypothetical group of 150 people, 18 have the event of interest first by the horizon. What is the event-specific risk?
- Why does a composite of stroke or death answer a different question from stroke before death?
- A researcher proposes to censor deaths and call the result “the observed risk of stroke.” What is the error?
Solutions. (1) Yes. The infarction occurred before death and meets the stated endpoint. (2) It is a competing event because it precludes a later infarction during the remaining follow-up. (3) The risk is \(18/150=0.12\), or 12%. (4) The composite counts either event as an outcome, whereas the event-specific question counts stroke only when it occurs before death; their risks can differ, including in direction of a treatment contrast. (5) Censoring deaths treats a precluding event as though observation simply ended and does not generally estimate the observed-world stroke risk. The protocol must instead specify the stroke-before-death risk, a composite, or a separate hypothetical estimand.