What are competing risks?

In survival analysis, a competing risk is an event that, if it happens first, prevents the event of interest from happening. Death from another cause is the classic example: a patient who dies of heart disease can no longer die of their cancer. Events that only change the chance of the event of interest, such as a relapse before death, are better handled with a multi-state model.

Competing risks are common. In studies of cause-specific mortality, every other cause of death competes with the one under study. After a stem-cell transplant, relapse competes with death without relapse. In older patients, death competes with almost everything: a hip replacement cannot be revised in a patient who has died.

Two questions, two quantities

With competing risks there are two different questions, and each has its own quantity:

• How fast does the event happen among those still free of every event? This is the cause-specific hazard. It describes the process behind the event, and it is the usual choice for aetiological questions: does this exposure change the rate of the event among those still event-free?
• What proportion will have the event by a given time? This is the cumulative incidence function: the probability of having the event by time $t$, in a world where the competing events can also happen. It is what matters for prognosis, for planning services, and for checking what a health economic model predicts.

The two can point in different directions. A treatment can lower the cause-specific hazard of an event and still raise its cumulative incidence, if it also keeps more people free of the competing event long enough to have it. Latouche et al. recommend reporting the cause-specific hazards and the cumulative incidence of every event, so that both questions are answered (J Clin Epidemiol 2013;66:648–653).

Why one minus Kaplan–Meier overstates the risk

A common mistake is to estimate the risk of the event as one minus the Kaplan–Meier estimate, with competing events treated as censored. Censoring assumes that the people removed could still have the event later, and a patient who has died cannot. So one minus Kaplan–Meier overstates the cumulative incidence, and the more common the competing event, the more it does. Only if the event and the competing events are independent, which the data cannot check, does it estimate the risk in a hypothetical world where the competing events were removed and nothing else changed, and that is rarely the question. Censoring a competing event: right for the hazard, wrong for the risk explains when censoring them is correct.

Methods

• The Aalen–Johansen estimator estimates the cumulative incidence of each event without a model, and Gray's test compares cumulative incidence between groups.
• Cause-specific Cox models fit one model for each event, treating the other events as censored. That is correct for the hazards; the cumulative incidence is then computed from all of them together.
• The Fine–Gray model models the cumulative incidence directly, through the subdistribution hazard. Its hazard ratio is not a ratio of event rates among patients who can still have the event, because the people it keeps at risk include those who have already had a competing event, so use it for the direction of the effect on the cumulative incidence, and show its size with predicted cumulative incidence curves. Separate Fine–Gray models for each event can give cumulative incidences that sum to more than one (Austin, Steyerberg and Putter, 2021).
• Flexible parametric models for the cause-specific hazards give smooth cumulative incidence curves for any covariate pattern, and can be carried beyond follow-up, though the data cannot confirm an extrapolation, so it needs external evidence and sensitivity analyses.
• Multi-state models generalise competing risks to events that can follow one another, such as relapse and then death.

Where the cause of death is unreliable or not recorded, as in many population cancer registries, relative survival estimates the excess mortality due to the cancer by comparing patients' survival with that expected in the general population. It gives net survival, the survival from the cancer if there were no other causes of death, which is a hypothetical measure, and the crude probabilities of dying of the cancer and of other causes, which are the real-world risks. Both rest on the assumption that the expected mortality, taken from population life tables, is right for the patients' mortality from other causes. Net survival also needs deaths from the cancer and from other causes to be independent, given the factors the life tables are matched on, which the data cannot check. See our relative survival tutorial.

In R, the survival package estimates cumulative incidence with the Aalen–Johansen estimator, fits cause-specific Cox models and, through finegray(), the Fine–Gray model; the cmprsk package also fits the Fine–Gray model and provides Gray's test. In Stata, stcox fits cause-specific models, stcrreg the Fine–Gray model, and the community-contributed stcompet estimates cumulative incidence. Our tutorial competing risks in merlin fits cause-specific models in Stata, and our multi-state and competing risks tool shows cumulative incidence interactively.

What to report

• The number of patients who had each event, the competing events included, and the number censored.
• The cumulative incidence of every event over time, from the Aalen–Johansen estimator or a model, in place of one minus Kaplan–Meier.
• Cause-specific hazard ratios for every event, as Latouche et al. recommend, so that a change in risk can be traced to the hazards behind it.
• For a Fine–Gray model, estimates labelled as subdistribution hazard ratios, with predicted cumulative incidence curves to show the size of the effect (Austin and Fine, Stat Med 2017).
• The time horizon of any risk you quote. A difference in cumulative incidence between groups can change size, and even direction, over time, as the constant-hazards example in censoring a competing event shows.

Want to learn more?

• Our competing risks course
• Censoring a competing event: right for the hazard, wrong for the risk
• Check out our Resources page

MethodCompeting risks