Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 6 / a

A competing risks model describes mutually exclusive first-event types. Once one event occurs, it prevents the other event types from being observed as that individual's first event.

The cause-specific hazard for cause $k$ isIt is the instantaneous rate of cause $k$ among individuals still free of every competing event.

The cumulative incidence function for cause $k$ isUnlike one minus a cause-specific survivor function, it is the absolute probability of observing cause $k$ by time $t$ in the presence of all competing causes.

With cause-specific hazards $h_A$ and $h_B$, survival free of either event through time $u$ isThe probability of remaining event-free to $u$ and then experiencing $A$ in $du$ is $S(u)h_A(u)du$. Therefore

For constant hazards $h_A=\theta$ and $h_B=\varphi$,Its limit $\theta /(\theta +\varphi )$ is the probability that $A$ occurs before $B$.

The composite event has total constant hazard $\theta +\varphi$, so its cumulative incidence isIt also equals $F_A(t)+F_B(t)$ because the two first-event types are mutually exclusive.