Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 37 / 2

For the numerical competing risks update, the estimated probability of remaining event-free just before the tied time isThere were no intervening events, so the cumulative incidence functions did not jump between the two stated times. The Aalen–Johansen estimator uses the risk set immediately before the event time, including the subject censored at that same time under the usual events-before-censoring tie convention. Therefore the cause-$A$ increment is $0.474(2/20)$, andFor a consistency check, ${\hat{F}}_B(7.8)=0.241+0.474/20=0.2647$ and $\hat{S}(7.8)=0.474(1-3/20)=0.4029$; these sum to one. The censoring does not create an incidence jump and leaves $16$ subjects for subsequent follow-up. Dividing by $19$ would instead assume that the censored subject had already left before the event time, contrary to the stated immediately preceding risk set. This is a cumulative incidence update with tied censoring.

Interpret $h_A,h_B$ as cause-specific hazards: their conditioning population consists of individuals still free of both events. WriteIn a short interval, either event removes an individual from that population. Hence the survivor function satisfies $S^{\prime }(t)=-S(t)[h_A(t)+h_B(t)]$, with $S(0)=1$. Integrating givesNo independence assumption on hypothetical latent event times is needed: this follows directly from the specified cause-specific hazards.

The probability of a first event of type $A$ in $(u,u+du)$ is the probability $S(u)$ of still being at risk times $h_A(u)du$. Thus its cumulative incidence function isSimilarly $F_B(t)={\int }_0^t{h}_B(u)S(u)du$. In general $F_A(t)\ne 1-e^{-H_A(t)}$: that latter expression ignores the competing removal caused by $B$ and instead describes a net survival complement.

The two event types are mutually exclusive as observed first outcomes. Summing their cumulative incidence functions and using the survivor function givesTaking the increasing limit in time therefore yieldsThe expression is one when the sum of the two cumulative hazards diverges, with $e^{-\infty }=0$. Finite total cumulative hazard leaves a positive probability of never experiencing either event.