Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 6 (Analysis of Survival Data)

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.

Without administrative stopping, the probability that one patient experiences $A$ before $B$ is $\theta /(\theta +\varphi )$. For $n$ patients the expected event count is therefore

Time to the first of $A$ and $B$ is exponential with rate $\theta +\varphi$ and mean $1/(\theta +\varphi )$. The expected total person-time at risk is

The ratio of expected event count to expected person-time isThus incidence per unit person-time recovers the cause-specific hazard of interest despite independent competing censoring.

With administrative censoring at $c$, one patient is observed to experience $A$ with probabilitysoThe observed time is $\min (T_A,T_B,c)$, whose mean follows from the tail-sum formula for expectation:Hence

Ignoring factors that do not depend on $\theta$, each observed side-effect contributes $\theta e^{-\theta x_i}$ and each censored observation contributes $e^{-\theta x_i}$. ThusThe score equation $v_{+}/\theta -x_{+}=0$ givesThis event-count divided by person-time estimator is the sample analogue of the expectation ratio in part iv.

The second derivative isIt is negative when $v_{+}>0$, verifying a strict maximum. The Observed Fisher information is $v_{+}/{\theta }^2$: more observed side-effects sharpen the likelihood, while the curvature vanishes when no side-effect is observed and the maximum then lies at the boundary $\hat{\theta }=0$.

At ${\theta }_0$, the expected Fisher information isRequiring it to exceed $I_0$ directly controls the large-sample variance of the maximum-likelihood estimator, approximately $\mathcal{I}({\theta }_0{)}^{-1}$. Choose $I_0$ from the desired standard error, confidence-interval width, or power for clinically relevant alternatives, allowing for any planned significance level.

Part iv givesTherefore the information requirement isThe required integer sample size is