Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 46 / 4

Let $t_j$ be the distinct observed event times, with $r_j$ individuals at risk just before $t_j$ and $d_j$ events there. Parameterize the event-time distribution by the conditional failure probabilities $q_j=\mathbb{P}(T=t_j\mid T\ge t_j)$. Under independent censoring, the survival part of the nonparametric likelihood groups into factorsAn individual still under observation at $t_j$ contributes either a failure factor or a survival factor; later-censored individuals contribute the latter factors until censoring. Maximizing each factor independently gives ${\hat{q}}_j=d_j/r_j$, including the boundary solutions when $d_j=0$ or $d_j=r_j$. Multiplying the conditional survival probabilities yields the Kaplan–Meier estimatorThe notation $F$ here denotes the survivor function, as in the question. The estimator is constant between event times. Under the usual simultaneous-event convention, individuals censored at $t_j$ remain in its risk set while events are processed, and are then removed.

Let $r$ be the number at risk just after $a$, and suppose the $d$ events are grouped into tie sizes $d_1,\dots ,d_m$ at ordered times in $(a,b]$. With no intervening censoring or entry, their risk counts are $r$, $r-d_1$, $r-d_1-d_2$, and so on. Hence the Kaplan–Meier telescoping across a censor-free interval givesOnly the total $d$ appears. Consequently the order of the events and their grouping into ties do not affect the estimate at $b$. Censoring exactly at $b$ does not change this conclusion when it is processed after the events at that time, as in the standard convention. The calculation assumes $r>0$; no estimator beyond an exhausted risk set is being inferred from unobserved follow-up.

Use time since enrolment, and assume the cohorts share a common survival distribution with independent administrative censoring. The PDF starts cohort A in 2006, not the corrupted 2000 date in the TeX aid. During the first follow-up year the pooled sample begins with 200 subjects and has $28+31=59$ deaths, with no earlier censoring. By part (a),At one year, the 69 remaining cohort-B subjects are administratively censored. Cohort A supplies 72 subjects at risk for the second follow-up year, with 24 deaths and 48 survivors. ThereforeThe given grouped totals suffice because there is no censoring inside either follow-up interval; no assumption about the ordering or ties of its deaths is needed.

Cohort A alone would give ${\hat{F}}_A(2)=48/100=0.48$. But two-year survival is the product of first-year survival and conditional survival through the second year. Cohort B contributes a full year's information to the first factor even though it has no second-year follow-up. Omitting it discards useful early deaths and survivors. The pooled estimate $0.47$ combines the first-year information from both cohorts with the conditional second-year information from cohort A.

Thus lack of two-year follow-up is handled by censoring, not by removing the entire short-follow-up cohort. This is cohort pooling under administrative censoring. The criticism assumes comparable time-since-entry survival distributions: if enrolment cohort or calendar period genuinely changes the hazard, naive pooling is not justified. One should model or stratify that effect, and an estimand specifically restricted to cohort A may appropriately use its own survival distribution.