Competing Risks Analysis

Core Idea

Competing risks arise when subjects can experience more than one type of event, and the occurrence of one event precludes the others — a patient may die from cardiac causes, cancer, or other causes, but can only die once. Standard Kaplan-Meier and Cox methods treat competing events as censored observations, which overestimates the probability of the event of interest because it assumes censored subjects could still experience it. Two complementary approaches exist: cause-specific hazard models (separate Cox models for each event type, treating competing events as censored) and the Fine-Gray subdistribution hazard model (which directly models the cumulative incidence function, keeping subjects who experience competing events in the risk set). These approaches answer different questions and can lead to different conclusions.

Explainer

Standard survival analysis assumes a single event type and treats everything else as censoring. This works when censoring is truly non-informative — when a subject lost to follow-up has the same future risk as those remaining. But when a patient dies of cancer, they do not have the same future cardiac risk as a living patient. Cancer death is not non-informative censoring — it is a competing risk that permanently removes the patient from the possibility of experiencing the cardiac event. Treating it as censoring violates the independence assumption and inflates the estimated probability of the event of interest.

The cumulative incidence function (CIF) directly estimates the probability of experiencing a specific event type by time t, properly accounting for the fact that some subjects will be claimed by competing events first. Unlike the KM complement (1 - S(t)), which imagines a world without competing risks, the CIF represents the actual probability in the real world where multiple event types coexist. The CIF for cardiac death will always be less than or equal to the KM complement because it acknowledges that some subjects who would have eventually died of cardiac causes will instead die of cancer first.

Two regression frameworks address competing risks. The cause-specific hazard model fits a separate Cox model for each event type, treating competing events as censored observations. It estimates the instantaneous rate of each event among subjects still alive — the "etiological" quantity that tells you about the direct biological effect of a covariate on a specific cause of death. The Fine-Gray subdistribution hazard model takes a different approach: when a subject experiences a competing event, they remain in the risk set for the event of interest with zero probability of experiencing it. This produces a hazard that directly links to the cumulative incidence function, making it natural for prediction — if you want to estimate the 5-year probability of cardiac death for a patient with specific characteristics, the Fine-Gray model gives you that directly.

These two approaches can disagree. Consider a treatment that reduces cardiac death but increases cancer death. The cause-specific cardiac hazard will show a protective effect (lower cardiac death rate among the living). But the Fine-Gray subdistribution hazard may show no effect or even an adverse effect because the patients saved from cardiac death now live longer and accumulate more cancer deaths, altering the cumulative incidence. Neither approach is "correct" — they answer different questions. Most methodologists recommend reporting both, using the cause-specific model for understanding mechanisms and the Fine-Gray model for clinical prediction.