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

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