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

The risk set at event time $a_j$ contains individuals still under observation and event-free immediately before $a_j$. For group $k$ its size isThe use of $\ge$ keeps the individual who experiences the event at $a_j$ in the risk set just before that event.

Under the null hypothesis of equal event-time distributions, every member of the combined risk set has the same instantaneous chance of being the next event. Conditional on one event at $a_j$ and on the two risk-set sizes,so

The quantityis the observed-minus-expected group-1 event count at time $a_j$. A positive value is local evidence that group 1 has the greater hazard function; a negative value points toward group 0.

Summing the observed-minus-expected contributions gives the unstandardized log-rank statisticUnder the null it is centered at zero. A two-sided Log-rank test compares its magnitude with the square root of its null variance.

Let $Y(t)$ be the at-risk process and $N(t)$ the counting process for observed events. Over a short interval, the multiplicative-intensity model giveswhere $h$ is the hazard function and $H$ the cumulative hazard function. Solving this relation for the infinitesimal hazard increment suggests $d\hat{H}(t)=dN(t)/Y(t)$. Summing over distinct event times gives the Nelson–Aalen estimatorwhere $d_j$ events occur among $r_j$ individuals at risk. Here there are no ties, so $d_j=1$.

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.