Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 5 (Analysis of Survival Data)

At the month-48 analysis, the observed time from treatment start is:For example, patient D contributes $48-12+1=37$ months. This is right censoring at a common calendar data cutoff but with staggered treatment entry.

At duration seven there are six at risk and two deaths, at duration thirteen there are four at risk and one death, at duration twenty-six there are three at risk and one death, and at duration twenty-seven there are two at risk and one death. The Kaplan–Meier estimator is thereforeThe censoring of D at duration thirty-seven causes no multiplicative drop.

At month 24 the durations and statuses areAt duration seven, two of six patients die, giving $2/3$. Patient F is censored at duration nine. At duration thirteen, C dies while C, D, and A are at risk; treating an event before censoring at a tied time gives a factor $2/3$. Hence

At month 36, patients F and D are censored at treatment durations 21 and 25, after which patient A is the sole remaining member of the risk set and dies at duration 26. The corresponding Kaplan–Meier factor is $1-1/1=0$, sowithout needing the earlier factors.

At month 48, D remains at risk past duration 26 and is eventually censored at 37, so no event empties the risk set. Part ii gives

At month 12, patients C and D are censored at durations 6 and 1, respectively. When B dies at duration 7, only A and B remain at risk, so

At month 36, all six patients have at least twelve months of potential follow-up unless they die earlier. The only deaths by duration twelve are B and E, tied at duration seven, so

In the final data, the only deaths by treatment duration twelve are B and E. ThusThe month-24 estimate already has this numerical value, so month 24 is the first twelve-monthly analysis satisfying ${\hat{F}}_M(12)={\hat{F}}_{\infty }(12)$.

Equality is not yet knowable at month 24 because patient F has only nine months of follow-up and could still die before duration twelve. Patient F reaches twelve complete months at the end of month 27, so month 36 is the earliest scheduled analysis at which the final value at duration twelve is evaluable.

Every patient must either have an observed death or at least 48 months of follow-up. The unresolved patient is D, who starts in month 12 and completes 48 months at the end of month 59. Therefore the month-60 analysis is the earliest twelve-monthly analysis at which ${\hat{F}}_{\infty }(48)$ is evaluable.

The Kaplan–Meier estimator requires independent censoring: conditional on modeled covariates, censoring must carry no information about the future event time. Censoring at a common administrative data cutoff satisfies this condition when calendar entry time is independent of prognosis. Under that assumption, the month-24 analysis has valid administrative censoring despite unequal follow-up caused by staggered entry.

The researcher should not selectively add A's post-cutoff death to an analysis explicitly defined by the month-24 data cutoff. Doing so gives extra follow-up to a patient precisely because an event became known, creating outcome-dependent ascertainment. The defensible choices are to retain the locked month-24 analysis or update every patient's record to one common later cutoff and label it as a new analysis.

Yes, a selective update would invalidate the answer to part viii: A's censoring time would be extended because A died, while follow-up for other patients remained truncated at month 24. The resulting censoring mechanism is informative. A uniform update of every patient to the same later administrative cutoff would preserve independent censoring under the original entry-time assumption.