Understanding Competing Risks

Time-to-event analysis sounds simple: start a clock, observe whether an event occurs, and compare groups. The difficulty appears when several mutually exclusive first events can end the same clock. A person may die from another cause before a cancer relapse, receive a transplant before dialysis, or die without ever having a first stroke.

Those are not merely missing observations. The competing event changes what can happen next. Once you see that, familiar survival curves, hazard ratios, and censoring rules all need more careful interpretation.

Start with a multistate picture#

Imagine that every participant begins alive and free of the event of interest. The first transition might be to event A, event B, or continued follow-up without either event. If event B makes event A impossible, B competes with A.

In an oncology study, relapse may be event A and death before relapse event B. In a kidney study, kidney failure may be event A and death before kidney failure event B. In a device study, revision may be event A and death with the original device still in place event B.

The definition depends on the research question. Death is not always a competing event. If the endpoint is all-cause death, no cause of death competes with it. If the endpoint is cardiovascular death, noncardiovascular death competes, and if the endpoint is a composite of cardiovascular death or heart attack, a cardiovascular death is part of the endpoint while a noncardiovascular death may compete. That is why a methods section should define its event types before it presents a model: statistical software cannot decide which transition represents the scientific question.

Competing events are not ordinary censoring#

Right censoring means the event status becomes unknown after a time. A participant may move away, withdraw, or reach the administrative end of follow-up without the event. Under standard assumptions, the remaining participants can stand in for those whose later outcomes are unobserved.

A competing event provides known, substantive information. A person who dies before relapse cannot later relapse. Treating that death as if it were ordinary loss to follow-up does not erase the fact. It changes the target toward a hypothetical setting where the person could somehow remain capable of relapse.

The distinction is conceptual before it is mathematical:

• Censoring says, “the later event status is not observed.”
• A competing event says, “the event of interest can no longer occur.”

Methods may still encode a competing event as censored for a particular cause-specific hazard calculation. That computational operation is valid for that estimand. It does not make the competing event noninformative or turn the resulting curve into real-world cumulative probability.

Why one minus Kaplan-Meier is usually wrong here#

The Kaplan-Meier estimator calculates survival by multiplying conditional probabilities across event times. When estimating time to relapse and treating deaths before relapse as censored, it removes those deaths from later risk sets. The complement, one minus Kaplan-Meier, then behaves as if people who died could have continued to face relapse risk like the people who remained alive.

The resulting quantity is not the observed probability of relapse in a world where death occurs. It is generally larger. The difference grows when competing events are common, follow-up is long, or the groups being compared have different competing-event rates.

Gooley and colleagues gave an intuitive account of why the cumulative-incidence estimator, not one minus Kaplan-Meier, represents event probability under competing risks. Andersen and colleagues likewise emphasize that cumulative incidence for one cause depends on both its own cause-specific hazard and the hazards of competing causes.

Kaplan-Meier itself is not defective. It estimates what it is designed to estimate. The error is labeling its complement as an observable cause-specific probability when a competing event has been censored.

Cumulative incidence answers the absolute-risk question#

For event type A, the cumulative incidence function is the probability of experiencing A by time t before experiencing a competing event; it accumulates the hazard of A among people still event-free, weighted by the probability of remaining free of every event up to each time.

That weighting is the correction. A contribution to relapse probability at year four can come only from people who were alive and relapse-free just before year four. Deaths before then reduce the population that can later relapse.

With several mutually exclusive first-event types, their cumulative incidences plus the probability of remaining event-free sum to one. This makes the curves interpretable as a partition of observed outcomes.

The nonparametric estimator is often described through the Aalen-Johansen framework. For a simple competing-risks setting, it produces event-specific cumulative-incidence curves. Gray's test can compare such curves between groups, although an absolute difference at a clinically useful time often tells you more than a P value. Useful reporting might state that by five years, 12 percent experienced relapse, 18 percent died without relapse, and 70 percent remained alive without relapse. Each percentage completes the others.

A cause-specific hazard asks about the current event rate#

The cause-specific hazard for event A is the instantaneous rate of A among people who, just before that moment, have experienced neither A nor any competing event, and when a competing event occurs, that person leaves the event-free risk set.

A cause-specific Cox model can estimate how a treatment or covariate is associated with that rate. It is often useful for etiologic or process-oriented questions: among people currently alive and relapse-free, is a marker associated with the next instantaneous relapse rate?

The cause-specific hazard does not by itself determine cumulative incidence. To derive the probability of event A, the analysis also needs the cause-specific hazards of all competing events. A treatment could leave the relapse hazard unchanged but reduce early death. More people then remain alive long enough to relapse, so observed relapse cumulative incidence can increase even though the treatment did not make relapse biology worse.

This is not a paradox once rate and probability are separated. A cause-specific hazard conditions on remaining in the starting state. Cumulative incidence reflects the full pathway by which people leave that state.

The Fine-Gray model targets a subdistribution#

Fine and Gray developed a proportional model for the subdistribution hazard, a quantity mathematically connected to the cumulative incidence function, and its risk set is constructed differently from the ordinary cause-specific risk set. People who have experienced a competing event are retained in a weighted or formal sense so that regression coefficients relate to changes in cumulative incidence.

That construction makes the model useful for prognostic questions and for comparing cumulative-incidence patterns. It also makes the subdistribution hazard ratio less intuitive. It is not the instantaneous event rate among people who are currently event-free. Calling it simply “the hazard ratio” invites a mistaken clinical reading.

A reported Fine-Gray result should therefore be labeled as a subdistribution hazard ratio. Its proportionality assumption should be considered, and it should be accompanied by predicted or observed cumulative-incidence values. A relative coefficient without absolute risks hides whether the event is common, rare, early, or late. Fine-Gray and cause-specific models can give different coefficients, or even different qualitative impressions, because they answer different questions, and picking the model after seeing which result is more favorable is not a principled solution.

Estimands should come before methods#

An estimand states the treatment or association effect to be estimated, including the population, outcome, handling of intercurrent events, and summary measure. For a competing-risks problem, useful questions include:

• What is the real-world probability of relapse by five years while death can occur?
• Among people still alive and relapse-free, how does treatment change the instantaneous relapse rate?
• What is the effect on being alive without relapse?
• What is the effect on relapse or death considered as a composite?
• How does treatment affect each transition separately?

These questions are related but not interchangeable. The cumulative incidence function directly addresses the first. A cause-specific hazard addresses part of the second. A composite endpoint addresses the fourth, but loses the distinction between its components. A multistate model can address the fifth and later transitions if the data support it.

The estimand also determines which events truly compete. A transplant might preclude observing dialysis under one protocol, but a model of treatment strategy could treat transplant as a transition rather than a terminal event. Clear science prevents a coding convention from silently defining the question.

Treatment can change the competing event#

Suppose a treatment substantially reduces noncancer death but has no direct effect on relapse. Treated participants survive longer and therefore have more time in which relapse can be observed. Their five-year relapse cumulative incidence might be higher than in control participants, even while overall survival is much better.

The reverse can also occur. A toxic treatment might increase early deaths, which mechanically reduces later observed relapse because fewer people remain capable of relapsing, and a lower relapse cumulative incidence alone could then look to you like success and be nothing of the kind.

Randomization protects treatment-group comparisons from baseline confounding, but it does not remove competing-risk structure. Austin and Fine found that many randomized-trial reports used conventional survival methods despite endpoints susceptible to competing events. Their recommendations emphasize cumulative incidence for absolute event probability and explicit attention to both the primary and competing outcomes. So ask what happened to every mutually exclusive outcome, not only whether one curve went down.

Composite endpoints solve one problem and create another#

Combining event A with the competing event can avoid treating the latter as censoring. “Relapse or death,” for example, measures failure of relapse-free survival regardless of which component occurs first.

The price is loss of specificity. If treatment reduces relapse but increases death, the composite may average effects that point in opposite directions, and a component that occurs more often or earlier can dominate the result even if it is less important to patients.

A composite is appropriate when its components form a coherent decision-relevant outcome and when each component is reported separately. It should not be used merely to avoid learning competing-risks methods. Event-specific cumulative incidence and an overall composite can be complementary.

Prediction brings calibration to the foreground#

A prognostic model is often intended to estimate an individual's absolute event probability. In older or medically complex populations, ignoring death before the outcome can overpredict the condition of interest. Wolbers and colleagues illustrated competing-risk methods in coronary-disease prediction, where noncoronary death becomes increasingly relevant with age and frailty.

Prediction performance must match the target. Calibration asks whether predicted probabilities agree with observed cumulative incidence at a stated horizon. Discrimination asks whether the model ranks people who experience different event pathways. Both require definitions adapted to competing outcomes and censoring.

A model can rank participants well yet systematically overstate their absolute risk. That matters when thresholds trigger treatment, screening, or referral. External validation should reproduce the same event definitions and account for different competing-event rates in the new population.

Cause-specific models can also generate valid cumulative-incidence predictions when all relevant cause-specific hazards are modeled and combined. Fine-Gray is not the only route to prediction. Model selection should reflect assumptions, flexibility, validation, and the intended use.

Causal questions require even more care#

Competing events complicate causal language because preventing death can make a later nonfatal event observable, and a contrast among people who would survive under either treatment refers to an unobservable subgroup defined by future potential outcomes. A hypothetical strategy that eliminates the competing event may be scientifically implausible.

Cause-specific hazards are descriptive conditional rates and can be affected by selection over time, even in a randomized trial, because the set of survivors can differ between treatment groups; subdistribution hazards are also not automatically causal effects. Randomization alone does not turn every post-baseline conditional contrast into a simple causal parameter. A causal analysis should specify the intervention, the intercurrent-event strategy, the target population, the time horizon, and the assumptions. Sometimes the clearest trial answer remains a set of intention-to-treat cumulative incidences for all outcomes plus an overall survival or event-free measure.

Common reading errors#

Several recurring mistakes are visible without recalculating anything:

1. The report calls death “censoring” without explaining whether death prevents the endpoint.
2. One minus Kaplan-Meier is labeled the probability of an event despite frequent competing deaths.
3. A Fine-Gray coefficient is described as the instantaneous risk among living event-free participants.
4. A cause-specific hazard ratio is translated directly into an absolute-risk reduction.
5. Only the event of interest is shown, with no counts or curves for competing events.
6. Groups are compared at different follow-up horizons or with unstable late curves.
7. A composite result is highlighted while its components move in opposite directions.

None of these automatically invalidates an entire study. Each changes what you can safely conclude.

A practical reporting set#

A transparent competing-risks report should define time zero, every first-event category, ordinary censoring, and the analysis horizon. It should provide event counts by type, cumulative-incidence curves with uncertainty, numbers at risk, and absolute probabilities at prespecified times.

Regression results should name the model and estimand. “Cause-specific hazard ratio” and “subdistribution hazard ratio” should not be shortened into an ambiguous single label. Proportionality and censoring assumptions deserve assessment, and sensitivity analyses should be prespecified when event definitions or missing causes are uncertain.

For a trial, show the effect on the event of interest, the competing event, and a patient-relevant overall outcome. For prediction, show calibration and discrimination at the intended horizon in an external population. For etiologic work, explain why the chosen conditional rate maps to the mechanism of interest. The goal is not to make you a survival statistician. It is to keep the scientific question visible through every transformation of the data.

References#

1. Introduction to survival analysis in the presence of competing risks
2. Fine and Gray proportional subdistribution hazards model
3. Estimation of failure probabilities with competing risks
4. Competing risks in epidemiology, possibilities and pitfalls
5. Prognostic models with competing risks
6. Accounting for competing risks in randomized trials