= Cause-specific hazard to cumulative incidence formula
{title2=$F_j(t)=\int_0^tS(u)h_j(u)\,du$}
For absolutely continuous first-event time $T$ with event type $J$, the event-free <survivor function> satisfies $S(t)=\exp(-\int_0^t\sum_jh_j(u)\,du)$. The <cause-specific hazard> definition gives unconditional event-type density $S(t)h_j(t)$, so $F_j(t)=\int_0^tS(u)h_j(u)\,du$. Consequently $\sum_jF_j(t)=1-S(t)$ 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.
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