Competing risks

Competing risks are mutually exclusive event types for which occurrence of one type prevents observation of the others as the first event.

A competing risks model assigns cause-specific hazards to mutually exclusive event types.

The cause-specific hazard $h_k(t)$ is the instantaneous rate of event type $k$ among individuals who have not yet experienced any competing event.

For event time $T$ and event type $J$, the cumulative incidence function is $F_k(t)=\mathbb{P}(T\le t,J=k)$. It is the absolute probability that cause $k$ occurs by time $t$ before any competing cause.