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

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.