11.2 Regression with competing risks

There are two regression models, answering different questions.

• The cause-specific Cox model fits a Cox model to \(h_k(t \mid x)\), treating failures from other causes as censored. This is legitimate, unlike the Kaplan–Meier estimate above, because the partial likelihood for \(h_k\) needs only the risk sets, and a subject who failed from another cause has left them. Its hazard ratios describe the effect of \(x\) on the rate of cause-\(k\) failure among those still event-free, the right target for questions about causes and mechanisms.
• The Fine–Gray model puts proportional hazards on the subdistribution hazard, \(\tilde h_k(t) = -\frac {\mathrm d}{\mathrm dt}\log \{1 - F_k(t)\}\), so that \(F_k(t \mid x) = 1 - \{1 - F_{k0}(t)\}^{\exp (\beta ^{\mathsf T}x)}\). Its risk set at \(t\) keeps subjects who have already failed from other causes, with weights that allow for censoring, and its coefficients describe the effect of \(x\) on the cumulative incidence itself, the right target for prediction and for questions such as “who is more likely to progress?”.

The two can differ in sign when \(x\) also affects the competing cause, and a full analysis reports the cause-specific models for every cause alongside the cumulative incidences.

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