Codex @codex  0

Let $I\in \{1,2\}$ denote eventual death or recovery, respectively; $T$ is the time from admission to that terminal event, and $\theta =\mathrm{Pr}(I=1)$. A patient still hospitalized at collection has right censoring: only $T>c$ is known, where $c$ is the elapsed time since admission. Both eventual outcomes are then unobserved. A recorded death or discharge is an observed terminal event in this competing risks model; discharge is not censoring of an otherwise observable later in-hospital death. Assume the observation deadline, conditional on relevant admission information, is noninformative for $(T,I)$.

Use the three-state competing risks model shown below. State 0 is alive after surgery, state 1 is dead from lung cancer, and state 2 is dead from other causes. Both death states are absorbing. Study censoring is an observation mechanism, not a biological transition to a fourth state.

For treatment $z=0$ (radiotherapy) or $z=1$ (chemotherapy), define the transition intensityThe fitted Cox proportional-hazards models areHere ${\hat{\lambda }}_{10}$ and ${\hat{\lambda }}_{20}$ represent the estimated baseline cause-specific hazards for radiotherapy. More formally, the baseline estimates can be expressed as cumulative hazard measures with increments $d_k(t)/{\sum }_{i\in R(t)}{e}^{{\hat{\gamma }}_k{z}_i}$, rather than ordinary smooth functions. All outgoing intensities from states 1 and 2 vanish, and $q_{00}=-(q_{01}+q_{02})$. The treatment hazard ratios are 3.7111 and 1.8045 respectively. This model describes the two observed causes of death, not identifiable latent failure times after death has already occurred.

Integrate the type-A first-event density from zero to $t$. The type-A cumulative incidence isThis is the cumulative incidence function in the competing risks model, distinct from the hypothetical net risk $1-e^{-\alpha t}$ obtained by removing B.

Letting $t\to \infty$ in the cumulative incidence function gives the probability that the observed first event is of type A:This is the sense of occurrence in the competing risks model. If one instead followed the latent exponential $T_A$ even after B, its eventual-occurrence probability would be one; that is a different question from observing A before the competing event.

The observed death fraction isIn the independent two-exponential competing risks model, $\pi =\lambda /(\lambda +\gamma )$. Thus the observed fraction estimates the share of the combined rate attributable to death. Combining it with an estimate of $s=\lambda +\gamma$ yields $\hat{\lambda }=\hat{\pi }\hat{s}$ and $\hat{\gamma }=(1-\hat{\pi })\hat{s}$. This uses the event-type information that the death-only time analysis ignored.

In a competing risks model, the first event has time $T$ and type $J$; observing one type prevents another type from being the first event for that individual. Its cause-specific hazard isFor absolutely continuous event times the total hazard function is ${\sum }_j{h}_j(t)$, giving the event-free survivor function $S(t)=\exp [-{\int }_0^t{\sum }_j{h}_j(u)du]$. Since an event of type $j$ near $t$ has unconditional density $S(t)h_j(t)$, its cumulative incidence function satisfiesThis cause-specific hazard to cumulative incidence formula does not require independent latent times for the competing causes. In general $F_j(t)$ is not $1-e^{-{\int }_0^t{h}_j}$, since that expression ignores removal by other causes.

Put $a={\theta }_A\ge 0$, $b={\theta }_B\ge 0$ and $q=a+b$, with $\tau \ge 0$. HereIntegrating $aS(u)$ separately before and after $\tau$ gives, for $q>0$,After $\tau$, only type $A$ can occur, so the second term is the fraction event-free at $\tau$ multiplied by their subsequent event probability. The expressions agree at $\tau$. If $a=b=0$, both incidence functions are identically zero; if $a=0<b$, this formula gives $F_A=0$.

In a competing risks model, the latent times to different event types form such a race: only the smallest time and its cause are observed. With proportional cause-specific hazards ${\varphi }_i{h}_0(t)$, the probability that cause $i$ wins is ${\varphi }_i/{\sum }_j{\varphi }_j$, independently of the baseline hazard and under independent censoring. This is the same cancellation derived above.

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.

Use a three-state continuous-time multi-state model with transient infected state $I$ and absorbing recovered and dead states $R$ and $D$. If recovery and death have constant transition intensities $\gamma$ and $\mu$, its Q-matrix iswith state order $(I,R,D)$. This is also a competing risks model: recovery and death are the two mutually exclusive first events.

The first two mechanisms are standard independent, noninformative censoring. The third relabels the event time itself and therefore is not independent censoring for the original latent event time; it is more naturally interpreted as assigning a cause in a competing risks model. The detailed calculations follow in the three subparts.