Censoring a competing event: right for the hazard, wrong for the risk

Treating a competing event as censored is right when you estimate a cause-specific hazard, and wrong when you turn that hazard into a risk. One minus the Kaplan–Meier estimate, with competing events censored, overstates the probability of the event of interest. The risk in a world where the competing events also happen is the cumulative incidence function, estimated with the Aalen–Johansen estimator.

What censoring assumes

Censoring a patient assumes that they could still have the event after their follow-up stops, and that the patients still being followed can stand in for them. A competing event breaks this. A patient who dies of cardiovascular disease cannot relapse afterwards, so censoring the death asks the remaining patients to stand in for someone whose risk of relapse is now zero.

One minus Kaplan–Meier then estimates the risk in a hypothetical world where the competing event could not happen and nothing else changed, and even that needs the two events to be independent, which the data cannot check (Andersen et al., 2012). The more common the competing event, the larger the overstatement. In a meta-analysis of 55 studies that reported both, the pooled ratio of one minus Kaplan–Meier to the cumulative incidence was 1.41, and 2.36 in studies with a high ratio of competing events to events of interest (Lacny et al., 2018).

A worked example

Take 100 hypothetical patients followed for three years, with nobody lost to follow-up. In the first year, 30 die of other causes without relapsing. In the second year, 20 relapse. The other 50 are alive and relapse-free at three years.

• Cumulative incidence. The risk of relapse by three years can be counted directly: 20 of 100, or 20%. The Aalen–Johansen estimator gives the same answer.
• One minus Kaplan–Meier. With the 30 deaths censored, 70 patients are at risk when the relapses happen, so Kaplan–Meier falls to 50/70, and one minus that is 20/70, or 28.6%.

The extra 8.6 percentage points are relapses credited to the 30 who died, as if 20 in 70 of them would otherwise have relapsed. Both analyses use the same relapses among the same 70 patients at risk, so they agree on the cause-specific hazard. They differ in what each relapse is weighted by. Aalen–Johansen uses the probability of still being free of every event, 70% at the first relapse. Kaplan–Meier uses its own estimate of remaining free of relapse with the deaths censored, 100% at the first relapse, because it treats the 30 who died as though they could still relapse. The ratio of the two weights stays at 0.7 at every relapse, so one minus Kaplan–Meier is 20% divided by 0.7.

When censoring the competing event is right

Censoring is right when the target is the cause-specific hazard, the rate of the event among patients still free of every event. A patient who has died is no longer at risk, so removing them from the risk set is what the definition asks for. A Cox or parametric model fitted with the competing events censored therefore estimates the cause-specific hazard and its hazard ratios, and needs no assumption that the events are independent. Patients lost to follow-up still need independent censoring.

The mistake comes at the next step, turning one hazard into a probability as though it were the only one. A patient has to be free of every event to have any one of them, so the cumulative incidence of each event depends on the cause-specific hazards of all of them, and the cumulative incidences add up to the probability of having had any event.

A higher hazard can mean a lower risk

It follows that a covariate can raise the cause-specific hazard of the event of interest and still lower its cumulative incidence, if it raises the hazard of the competing event enough that fewer patients stay free of both events long enough to have it. With constant hazards $h_1$ for the event and $h_2$ for the competing event, the cumulative incidence by time $t$ is $\{h_1/(h_1+h_2)\}\{1-e^{-(h_1+h_2)t}\}$. Take hypothetical rates of 0.10 per year for each event in one group, and in another 0.15 for the event of interest and 0.30 for the competing event. The second group has a cause-specific hazard ratio of 1.5 for the event of interest, yet its cumulative incidence by ten years is 33%, against 43%. By two years it is 20% against 16%, so the answer can also depend on the time horizon: here the two curves cross at about four years.

Both results are correct, and they answer different questions. The cause-specific hazard suits questions about aetiology, and the cumulative incidence, how many people will have the event, suits prognosis and planning. Decide which you are asking before you choose the model, and say which you reported.

The Fine–Gray alternative

The Fine–Gray model is a regression model for the cumulative incidence itself (Fine and Gray, 1999). Its hazard, the subdistribution hazard, keeps patients who have had a competing event in the risk set, so its hazard ratio moves with the cumulative incidence. If that ratio is constant over time, a ratio above 1 means a higher cumulative incidence at every time. Read it for the direction of the effect. A ratio of 2 does not mean twice the risk, and it is not a ratio of event rates among patients at risk (Austin and Fine, 2017). Show the size of the effect with predicted cumulative incidence curves. Separate Fine–Gray models for each event can give cumulative incidences that sum to more than 1 (Austin, Steyerberg and Putter, 2021).

What to report

Say, for each analysis, how the competing events were handled. Report the cumulative incidence from the Aalen–Johansen estimator or a model, in place of one minus Kaplan–Meier, and label every hazard ratio as cause-specific or subdistribution. Latouche et al. recommend reporting the cause-specific hazards and the cumulative incidence of every event, the competing ones included (Latouche et al., 2013).

Want to learn more?

• Our competing risks course
• What are competing risks?
• Competing risks in merlin, a tutorial fitting cause-specific models
• Our multi-state and competing risks tool

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