Competing risks model

A competing risks model assigns cause-specific hazards to mutually exclusive event types.

Let eventual outcome $I=j$ have probability ${\pi }_j$, with $\log T\mid I=j\sim N({\mu }_j,{\sigma }_j^2)$. An observed outcome at time $t$ contributes ${\pi }_j{f}_j(t)$ to the likelihood, while right censoring at $c$ contributes ${\sum }_j{\pi }_j{S}_j(c)$. The expectation-maximization algorithm imputes both the censored outcome and its latent log event time. Its E-step uses the displayed conditional weights and the first two moments of a truncated normal distribution above $\log c$. With $z=(\log c-{\mu }_j)/{\sigma }_j$ and Inverse Mills ratio $\psi (z)$, these moments are $m={\mu }_j+{\sigma }_j\psi (z)$ and $s=m^2+{\sigma }_j^2[1-\psi (z)(\psi (z)-z)]$. Observed outcomes use deterministic class weights and moments $\log t,(\log t{)}^2$. The M-step updates each class probability to its average weight, its mean to the weighted first moment, and its variance to the weighted second moment minus the squared new mean. Outcome-specific cumulative probabilities are cumulative incidence functions, not the conditional distributions $F(t\mid I=j)$ themselves.

If one cause-specific hazard acts at constant rate $a$ only until time $\tau$, and another acts at constant rate $b>0$ indefinitely, overall survival is $S(t)=e^{-a\min (t,\tau )-bt}$. The first cause has ultimate cumulative incidence function $a(1-e^{-(a+b)\tau })/(a+b)$; the second accounts for all remaining eventual failures. This illustrates how a transient competing hazard modifies a persistent cause's cumulative risk.

For independent exponential competing times with rates $\lambda ,\gamma$, the observed first time has rate $s=\lambda +\gamma$, while death-type probability is $\lambda /s$. The joint density factors as $\lambda e^{-st}=(\lambda /s)s{e}^{-st}$. Thus the conditional observed time given type still has rate $s$. Independence of type and time generally fails for time-varying hazard ratios.

Under independent exponential death and censoring, a rate fit using only observed-death times estimates $s=\lambda +\gamma$. Multiplying by the observed death fraction consistently estimates $\lambda$. This plug-in correction generally differs in a finite sample from the full maximum-likelihood estimate, which uses observed exposure from both deaths and censored subjects.

For independent continuous event times, conditioning on $T_A<T_B$ weights the density of $T_A$ by the survivor function of $T_B$. Conditioning on $T_B<T_A$ instead weights it by one minus that survivor function. These are distributions of the latent $T_A$, not the distribution of the observed first event in every case.

The cause-specific hazard $h_k(t)$ is the instantaneous rate of event type $k$ among individuals who have not yet experienced any competing event.

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.