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 Incoming links: Log-rank weight

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 4 / c / iii / Solution Created 2026-09-24 Updated 2026-09-25

Put

r_{j} = r_{j}^{(0)} + r_{j}^{(1)}

. The difference between the two Nelson–Aalen estimator increments is

\frac{u _{j}}{r _{j}^{(1)}} - \frac{1 - u _{j}}{r _{j}^{(0)}} = \frac{r _{j}}{r _{j}^{(0)} r _{j}^{(1)}} (u_{j} - \frac{r _{j}^{(1)}}{r _{j}}) .

Therefore the log-rank weights

w_{j} = \frac{r _{j}^{(0)} r _{j}^{(1)}}{r _{j}^{(0)} + r _{j}^{(1)}}

make each summand of

T_{W}

equal the corresponding summand of

T_{0}

, and hence

T_{W} = T_{0}

.

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 4 / c / iv / Solution Created 2026-09-24 Updated 2026-09-25

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.

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