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Lesson 6 of 7

Competing Risks and Cumulative Incidence

Every lesson so far has tracked a single kind of exit: a patient either has the event, or is censored. Dr. Rao's newest study breaks that neat picture. She is following 200 leukemia patients after a bone-marrow transplant, and each one now has two different ways to leave the study, where only one can happen first.

The leukemia can come back, a relapse. Or the patient can die in remission from a transplant complication, an infection or graft-versus-host disease, before the leukemia ever returns. Take Anika: she dies of an infection in month 8, still in full remission. She can now never relapse. That one fact, that a patient claimed by one event is gone from the other forever, is what makes these events compete, and it quietly breaks the Kaplan-Meier estimator you have trusted for five lessons.

By the end of this lesson you will be able to:

  • Explain why a competing event is not the same as a censored patient
  • See why "1 minus Kaplan-Meier" overcounts a cause's incidence, and by how much
  • Estimate the cumulative incidence function for each cause in R, and fit both a cause-specific and a Fine-Gray model

Prerequisites: Lesson 1 (right-censoring, Surv(), the hazard), Lesson 2 (the Kaplan-Meier curve and 1 minus KM as incidence), and Lesson 3 (the Cox model and the hazard ratio). You can run R and read a coefficient table.

The idea

Two exits, and only one can be first

Line up what can happen to one of Dr. Rao's patients after the transplant. As the months pass they sit in remission, and then one of three things ends their follow-up: the leukemia relapses, or they die in remission, or the study simply ends while they are still healthy (that last one is ordinary right-censoring from Lesson 1). Relapse and death-in-remission are the two competing events: mutually exclusive endpoints racing each other, where whichever arrives first settles the patient's fate and rules the other one out.

Here is the crucial difference from everything so far. When a patient is censored, we stop watching, but the event could still be coming: censoring says "unknown, maybe later." When Anika dies in remission, relapse is not unknown, it is impossible. She has left the pool of people who could ever relapse, permanently. A competing event does not hide the outcome of interest, it forecloses it.

Key Insight
Censoring pauses the clock on an event that might still happen. A competing event stops that clock forever. Treating the second like the first is the mistake this whole lesson is about.

The stacked chart below is the honest bookkeeping. At any month it splits the cohort into three shares, still event-free on top, already relapsed, already dead, and the three always add up to the whole. Drag the time toggle and watch relapse and death both grow while the event-free slice shrinks: every patient one event takes is a patient the other can no longer claim.