Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 207 / 4 / b

With no censoring or new entry within the interval, all $r$ individuals can be classified at its end: $d$ have experienced the event and $r-d$ remain event-free. Thus the empirical conditional survival probability isThis assumes $r>0$ and that the observed individuals represent the group whose conditional survival probability is being estimated.

An individual is right-censored at $t$ when follow-up establishes that the event has not occurred through $t$, but does not reveal the subsequent event time. The observed information is $T>t$ for the continuous-time convention.

Knowing only the total number censored inside an interval does not specify the risk sets at the event times. For example, start with two individuals, one event and one censoring. If censoring precedes the event, the Kaplan–Meier estimator contributes $1-1/1=0$; if the event precedes censoring, it contributes $1-1/2=1/2$. The totals $r=2,d=1,c=1$ are identical.

Therefore the timing of right censoring relative to the events is needed. Independent censoring is also required for these changing risk sets to yield a valid estimate of the underlying survivor function.

The estimate remains $(r-d)/r$. Individuals right-censored at $t_R$ are observed throughout the preceding interval, so they contribute to each risk set until the end. They must not be removed from the denominator at $t_L$.

If events and right censoring share the recorded time $t_R$, use the usual Kaplan–Meier estimator convention: count events first, using a risk set that still includes those censored at that time, and then remove the censored observations. With no earlier censoring, the product of event factors over the interval telescopes to $(r-d)/r$.