Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 207 / 4

At event time $t_j$, let $n_{Aj},n_{Bj}$ be the two risk set sizes, $n_j=n_{Aj}+n_{Bj}$, and let $d_{Aj},d_{Bj}$ be the event counts with $d_j=d_{Aj}+d_{Bj}$. Under the null hypothesis of equal hazards, conditioning on the risk set and total number of events gives a hypergeometric distribution, soandThe log-rank statistic and its estimated null variance areUnder the null, $U/\sqrt{V}$ is asymptotically standard normal, or $U^2/V$ is asymptotically chi-squared with one degree of freedom.

The expected Treatment A deaths are $1/2$ at month 1, $1/2$ at month 2, and $3(2/6)=1$ at the three tied deaths at month 5. ThusThis is not half of the five deaths because right censoring changes the treatment proportions in successive risk sets. For Treatment B, $O_B=2$ and $E_B=3$. One observed-to-expected relative-risk estimate is therefore

In a constant hazard survival model, the maximum-likelihood estimator is the number of observed events divided by total person-time at risk. Treatment A contributes $3$ deaths in $1+2+3+3+5=14$ months, while Treatment B contributes $2$ deaths in $1+5+5+7+10=28$ months. Hence

The Nelson–Aalen estimator isFor tied events, $d_j/r_j$ uses the number of events sharing time $t_j$ and the risk-set size just before that time.

At month 5,Their ratio is $29/10=2.9$.

The three estimates are $9/4=2.25$, $3$, and $2.9$. They use different weightings of follow-up time and event times, but all indicate a substantially greater mortality hazard under Treatment A; the exponential and Nelson–Aalen estimates are particularly close.