Competing Risks Analysis

Competing risks analysis is used when a person can experience different, mutually exclusive events and one event prevents the event of interest from occurring later. It estimates the probability of each event over time without treating the other events as ordinary censoring.

What are competing risks?

A competing event is an observed outcome that changes the possibility of observing the event of interest. The events are mutually exclusive for the first-event analysis: once one event occurs, the person is no longer at risk of experiencing another event first.

This differs from right censoring. A censored person has not experienced a known event by the last observed time, whereas a person with a competing event has experienced a known outcome that prevents the event of interest.

Why not use Kaplan-Meier?

A Kaplan-Meier curve treats observations that do not experience the event of interest as censored. If competing events are censored in the same way, the method effectively assumes that those people could still experience the event later. Because the competing event has made that impossible, the resulting failure probability can be too high.

With competing events, the cumulative incidence function is usually the appropriate descriptive estimate for the probability of a particular event. In particular, 1 minus the Kaplan-Meier estimate generally overestimates this probability when competing events occur.

Cumulative incidence function

The cumulative incidence function, abbreviated CIF, estimates the probability of experiencing a particular event by time t while accounting for all other event types. A separate CIF is calculated for each event type.

For example, a relapse CIF of 0.20 at 24 months means that the estimated probability of relapse by 24 months is 20%, considering that some patients may have experienced the competing event first. Confidence intervals describe the uncertainty around this estimate. At any time, the CIFs for all event types plus the probability of remaining event-free sum to one.

The CIF is a probability, whereas a hazard describes an instantaneous event rate among a defined risk set. A cause-specific hazard concerns people who have not yet experienced any event. The subdistribution hazard used by Fine-Gray regression uses a modified risk set so that covariate effects can be related to the CIF. These hazards answer different questions and their hazard ratios should not be interpreted as probabilities.

Gray's test

Gray's test compares cumulative incidence functions between independent groups. The null hypothesis is that the groups have the same cumulative incidence function for the event being tested. A small p-value provides evidence that at least one group differs.

Gray's test is the competing-risks counterpart to the log-rank test, but the two tests answer different questions. When other event types are treated as censored, the log-rank test concerns the cause-specific hazard, whereas Gray's test directly compares cumulative incidence in the presence of competing events.

Fine-Gray regression

Fine-Gray regression examines how predictors are associated with the cumulative incidence of the event of interest. It models the subdistribution hazard and reports a subdistribution hazard ratio, often abbreviated SHR.

An SHR above 1 indicates a higher subdistribution hazard and generally a higher cumulative incidence of the event of interest over time, given the model. An SHR below 1 indicates a lower subdistribution hazard. The SHR is not a probability, risk ratio, or ordinary Cox hazard ratio, so it should be reported with its 95% confidence interval and interpreted in terms of cumulative incidence.

Categorical predictors are interpreted relative to the displayed reference category. Metric predictors are interpreted per one-unit increase unless they were rescaled before the analysis. As with other observational regression analyses, an association in a Fine-Gray model does not by itself establish a causal effect.

Model diagnostics

The Fine-Gray model assumes proportional subdistribution hazards: the modeled predictor effect is constant over time. numiqo plots score residuals at the event-of-interest times. A roughly horizontal pattern around zero supports a constant effect, while a systematic trend or curvature can indicate a time-varying effect.

The moving average in the plot is a visual guide, not a formal hypothesis test. The regression output also reports convergence. Coefficients and SHRs should not be interpreted when the model has not converged.

Competing risks example

Suppose an oncology study records time from treatment to the first observed outcome. Relapse is coded as event 1, death without previous relapse as event 2, and the last follow-up without either event as 0. The analysis can then answer three related questions:

• What is the estimated probability of relapse or death over time?
• Does the relapse incidence differ between treatment groups?
• Which predictors are associated with the relapse incidence after adjustment?

The corresponding methods are the two CIFs, Gray's test for relapse, and Fine-Gray regression targeting relapse. Results for event 1 do not replace the CIF for event 2; both curves help describe how outcomes accumulate over follow-up.

Calculate competing risks online

Open the competing risks calculator, paste your data, and select the time and status variables. Use status 0 for censoring, 1 for the event of interest, and 2 for the competing event. Add a group to obtain Gray's test or add predictors for Fine-Gray regression.

numiqo displays CIF curves for both event types, pointwise confidence bands, numbers at risk, selected-time estimates, complete incidence tables, Gray's test, Fine-Gray coefficients and SHRs, convergence information, and residual diagnostics.

Methodological notes

The usual estimates assume that censoring is non-informative: conditional on variables included in the analysis, people who are censored should have comparable future event risks to those who remain under observation. Event times and causes must also be recorded correctly. Competing event types do not need to be statistically independent; each CIF accounts for the occurrence of all event types.

The CIF variance follows the Aalen estimator used by established competing-risks software, and confidence intervals use a log-log transformation. Fine-Gray censoring weights are estimated from the pooled complete-case sample without censoring strata. These choices are disclosed in the calculator output because different weighting or stratification choices can produce different results.

Automated reference tests compare the CIF estimates and variances, Gray's test, Fine-Gray coefficients, and score residuals with reproducible output from R's cmprsk package.