Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 207 / 4

The Log-rank test compares observed failures in each group with the numbers expected under equal hazards, conditioning at every event time on the risk set and total failures. Its score sums $O_{1j}-Y_{1j}{d}_j/Y_j$ and is standardized by its null hypergeometric variance. It is most powerful for approximately proportional hazards. Strongly crossing survival curves can produce large positive and negative contributions that cancel, so very different distributions may yield a weak log-rank statistic.

Monthly assessment gives interval censoring, although recording failure at the visit treats it as exact. It also creates many tied times. Ordinary continuous-time log-rank calculations then require a tie convention or grouped-time method, and unequal or missed visits can induce informative observation.

At times 1 and 2, the risk sets immediately before failure contain respectively $(5,5)$ and $(4,4)$ low- and high-dose patients, with one failure each. The low-dose expected counts are $1/2$ and $1/2$. Immediately before time 5, one low-dose and four high-dose patients remain; three tied failures give low-dose expectation $3(1/5)=3/5$. Thus

No. The high-dose expectation isAlthough both groups start with five patients, censoring removes two low-dose patients at month 3 and one high-dose patient at month 1. Their later risk sets therefore differ; only the sum $E_A+E_B=5$ must equal the total observed failures.

The unstandardized low-dose log-rank statistic isIts positive sign means more low-dose failures than expected under equal survival, indicating a higher low-dose treatment-failure hazard and favoring the high dose.

An observed-to-expected estimate of the low-versus-high relative failure risk isThus the low-dose failure hazard is estimated to be roughly $3.2$ times the high-dose hazard, with great uncertainty in this tiny dataset.

A baseline-group log-rank test is inappropriate. Receiving surgery is determined at month 3, so classifying patients by that future decision gives the surgery group guaranteed survival without treatment failure to the decision time, creating immortal time bias. Fitness also strongly confounds surgery and prognosis. One can perform a month-3 landmark analysis among patients still at risk, or model surgery as a time-dependent covariate, while adjusting for the clinical variables driving the decision.