Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 4 / c

Let $u_j^{(1)}=u_j$ and $u_j^{(0)}=1-u_j$. Applying the Nelson–Aalen estimator separately to group $k$ giveswith a zero contribution when the event at $a_{\ell }$ occurs in the other group.

The group-specific estimated cumulative hazard jumps only when that group experiences the event, so

Put $r_j=r_j^{(0)}+r_j^{(1)}$. The difference between the two Nelson–Aalen estimator increments isTherefore the log-rank weightsmake each summand of $T_W$ equal the corresponding summand of $T_0$, and hence $T_W=T_0$.

The variance of an estimated hazard increment is large when its group has few individuals in the risk set. The log-rank weights are near zero when either $r_j^{(0)}$ or $r_j^{(1)}$ is small and are largest when both groups retain substantial information. They therefore suppress noisy late-event comparisons and weight each observed-minus-expected event by its available information. Unit weights would instead give equal influence to unstable increments from depleted risk sets.