Cumulative incidence function ($F_k(t)$)

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.

For absolutely continuous first-event time $T$ with event type $J$, the event-free survivor function satisfies $S(t)=\exp (-{\int }_0^t{\sum }_j{h}_j(u)du)$. The cause-specific hazard definition gives unconditional event-type density $S(t)h_j(t)$, so $F_j(t)={\int }_0^{t}S(u)h_j(u)du$. Consequently ${\sum }_j{F}_j(t)=1-S(t)$ for exhaustive disjoint causes. Independent latent cause times are unnecessary. Substituting $1-e^{-\int h_j}$ instead estimates a hypothetical net risk and generally overstates observed incidence.