Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 35 / 7

Let $T=\min (T_A,T_B)$ be the first event time and $J$ its type. The cause-specific hazard for A isIt is a rate among people free of both event types, not the marginal probability of eventually experiencing A. Under the given independent latent event times, their constant hazard functions give exponential survivor functions. Therefore the first-event survivor isThe combined first-event rate is $s=\alpha +\beta >0$. Subsequent formulas concerning conditioning on type A require $\alpha >0$.

Multiply survival to $t$ by the conditional type-A event probability in the next infinitesimal interval. The unconditional type-A increment probability isEquivalently, its exact value over a positive interval of length $u$ is ${\int }_t^{t+u}\alpha e^{-(\alpha +\beta )v}dv$. The factor $e^{-(\alpha +\beta )t}$ is essential because a prior B event removes an individual from the relevant risk set.

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.

Divide the joint first-event/type-A density by the type probability. For $\alpha >0$,It integrates to one, so the conditional time is exponential with rate $\alpha +\beta$. Type-A events that win the competition tend to occur sooner than an unrestricted latent A time. In fact the same conditional density holds for B, so the observed first-event time and its type are independent in this constant-rate model.

A right-censored subject contributes survival up to the recorded censoring time, not no information. Under independent censoring, write $x_i=\min (T_i,C_i)$ and $v_i=1_{\{T_i\le C_i\}}$. For exponential death rate $\lambda$, the event-model survival likelihood isDiscarding censored subjects drops observed exposure and selects people whose death occurred before censoring. Their recorded event times are not an unselected sample of all death times. The death-only conditional distribution can differ from the marginal death-time distribution even when censoring is independent. A correct maximum-likelihood estimate uses the total observed person-time, including censored follow-up.

Let $\gamma$ be the exponential censoring rate and assume its time is independent of the exponential death time. Regard censoring as the competing event B. The preceding conditional-density calculation givesHence the reported death-only rate estimates the combined rate:It is not, in general, an estimate of the death cause-specific hazard alone. This interpretation assumes the stated two-exponential competition without an additional administrative end of observation that truncates the conditional time distribution.

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.

Applying death-only rate correction under exponential censoring to the death fraction and combined-rate estimate, the corrected plug-in death rate isThe corresponding censoring-rate estimate is $0.1{\text{year}}^{-1}$. This correction is consistent under the independent exponential competition model: the death-only conditional rate estimates $s$ and the observed fraction estimates $\lambda /s$.

There is an important distinction from the full-data maximum-likelihood estimate. If $T_D$ and $T_C$ are total observed follow-up for the 100 deaths and 50 censored subjects, the full likelihood for both rates isThe reported death-only rate determines $T_D=100/0.3$, but supplies no $T_C$, so the exact full-data estimate cannot be reconstructed from the numerical summaries alone. The value $0.2$ is the requested corrected rate using those summaries; it need not equal the realised full-data maximum-likelihood estimate. The full analysis should use the actual censored follow-up whenever available.