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

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.