Competing risks
Estimate the probability of a particular event when other events can prevent it from occurring.
Concepts and reasoning
Some events change what can happen next
Suppose the endpoint is first relapse, but a patient can die before relapse. Death prevents that patient from subsequently experiencing relapse as the first event. It is therefore a competing event. The question “What is the chance of relapse by two years in this population?” must account for both the relapse process and the chance of dying before relapse.
Ordinary administrative censoring is different. A patient whose observation ends while still event-free could in principle experience the endpoint later. Treating death exactly like ordinary censoring when calculating one minus a Kaplan–Meier curve effectively removes an important part of the real-world pathway.
Rates and probabilities separate
The cause-specific hazard of relapse describes its instantaneous rate among people who have experienced no event yet. The cumulative incidence of relapse describes the probability of having experienced relapse first by a given time. The latter depends on the hazards of every competing cause, because all causes affect who remains event-free.
Consequently, a treatment can leave the cause-specific relapse hazard unchanged but alter relapse cumulative incidence by changing mortality. This is why a cause-specific hazard ratio alone does not describe the treatment’s effect on the real-world probability of relapse. The competing-risks tutorial explains the distinction between descriptive probabilities and regression targets.
Mathematical foundations
Build each cause’s probability from the event-free population
Let
The factor
At an observed event time
Cause-specific Cox models remove a person from the risk set after any first event and model each cause’s rate separately. Fine–Gray regression instead models a subdistribution hazard related to cumulative incidence. Its hazard ratio has a different interpretation; the two coefficients should not be treated as interchangeable measures.
Worked example
Calculate a small competing-risk example
Begin with 100 event-free people and no censoring. At time 1, 10 relapse and 20 die without relapse. The event-free survival becomes
At time 2, among the 70 remaining people, 7 relapse and 7 die first. The relapse increment is
| Time | Relapse cumulative incidence | Death cumulative incidence | Event-free survival |
|---|---|---|---|
| 1 | 0.100 | 0.200 | 0.700 |
| 2 | 0.170 | 0.270 | 0.560 |
The three columns sum to 1 at each time. If deaths were instead censored in a relapse Kaplan–Meier calculation, the estimated relapse-free curve would be
This illustration uses simple grouped times and no loss to follow-up. Real event-time data also require an explicit convention for ties and appropriately handled administrative censoring.
Interpreting the results
Interpret absolute risk in the presence of other events
A cumulative incidence function gives the probability of experiencing a particular cause by a given time while accounting for other causes that prevent it. For example, the probability of a nonfatal event is affected by deaths occurring first. Treating those deaths as ordinary censoring in one minus Kaplan–Meier can overstate the probability of the nonfatal event.
Event-free survival plus all cause-specific cumulative incidences equals one at each time, apart from numerical precision. This identity helps interpret the entire set of outcomes rather than one cause in isolation.
A cause-specific hazard ratio compares instantaneous rates among people still free of every event. It is not a cumulative incidence ratio or a subdistribution hazard ratio. A treatment can lower one cause-specific hazard yet have a complicated effect on that cause's cumulative incidence because it also changes the competing event process.
Assumptions and limitations
Define mutually exclusive first events
Each individual contributes follow-up to the first relevant event or censoring. Cause codes must consistently distinguish all events that prevent the event of interest. A later event following an earlier competing event is outside this first-event framework; recurrent or multi-state processes need additional methods.
A common time origin and appropriate independent right censoring are required. Competing events are observed outcomes, not missing event times. The distinction is substantive: death before recovery cannot be handled like a participant lost to contact if recovery is the endpoint.
Cause-specific regression adds the assumptions of the chosen hazard model, including proportional hazards and suitable covariate form. It does not require pretending that competing events could not occur. Causal claims still need a defensible treatment assignment or confounding strategy.
Diagnostics and sensitivity analysis
Reconcile causes and risk sets
Check counts and timing for every cause, including rare codes that might be data-entry errors. Verify that cumulative incidence never decreases and that its sum with event-free survival remains one. Inspect the numbers still under observation when interpreting late changes.
Compare cause-specific patterns rather than attributing a lower cumulative incidence automatically to a lower hazard for that cause. Differences in competing mortality or follow-up can change the observed risk. Report the follow-up horizon and cause definitions alongside every probability.
For regression, inspect event support and proportional-hazards diagnostics for the selected cause. Sparse causes may yield unstable coefficients even with many total participants. The Aalen–Johansen documentation explains why cumulative incidence differs from a Kaplan–Meier complement.
References and other material
Read the methodological references and supporting tutorials below. The examples in this lesson are original teaching calculations, not findings from these sources.
- lifelines: Aalen–Johansen fitter — Cumulative incidence concepts and competing-event data.