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

The survivor function of a nonnegative event time isHere the paper uses $F$ for the survival function, rather than for the cumulative distribution function; the latter is $1-F(t)$ for a continuous distribution. The survivor function is nonincreasing, starts at 1 when there is no atom at zero, and tends to zero for an almost surely finite event time.

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 $(r-d)/r$.

A potential event time is an allowed support point for the discrete event-time distribution used to derive the Kaplan–Meier estimator. A positive probability mass may be assigned there before fitting, even if no event was observed at that time.

This is a device for nonparametric maximum likelihood estimation from exact event times and right censoring; it does not assert that the underlying biological event time is genuinely discrete. Its estimated mass can be zero at an unobserved potential time.

The set must contain every observed event time. Otherwise a discrete survival likelihood would assign probability zero to at least one exact event observation.

The set can include additional times without observed events. Include time zero when events at zero are present, and allow remaining tail probability beyond the fitted range. Estimation of a conditional event mass requires a nonempty risk set; once follow-up ends, later probability masses are not identified by the data. The usual Kaplan–Meier estimator retains the final estimated survival probability as its tail convention.

Let $d_j$ be the number of events at potential time $a_j$ and let $r_j$ be the size of the risk set immediately before that time, including observations censored at $a_j$. Define the discrete conditional event probabilityUnder independent censoring, and ignoring the separate censoring distribution, the survival likelihood factorizes asThis follows by multiplying a survival factor for each individual at risk who does not fail at $a_j$, and an event factor for each failure. The log factor $d_j\log q_j+(r_j-d_j)\log (1-q_j)$ is maximized at ${\hat{q}}_j=d_j/r_j$, with the corresponding boundary values when $d_j=0$ or $d_j=r_j$.

The survival function is the product of successive conditional survival probabilities. ThereforeThis is the Kaplan–Meier estimator. A product over no event times is 1. If everyone remaining experiences an event, the product becomes zero and stays zero; subsequent empty risk sets supply no additional factor.

At an added potential event time with no event and a nonempty risk set, $d_j=0$, so maximum likelihood estimation gives ${\hat{q}}_j=0$. Its factor in the Kaplan–Meier estimator is therefore $1-d_j/r_j=1$.

Adding or deleting such times also leaves the risk sets at actual event times unchanged: those depend on the observed event and censoring histories. Consequently the fitted survivor function depends only on observed event times and their risk sets. Beyond the last observed follow-up, the tail is not identified; the standard flat continuation is a convention, rather than an additional consequence of likelihood maximization.

Choose the potential right-censoring times to include every observed right censoring time up to $t$, and also the endpoints 0 and $t$. Then no individual is censored strictly inside $(c_{k-1},c_k)$. Use an explicit endpoint convention: let $r_k$ count individuals still event-free after any events at $c_k$, but before censoring at $c_k$. If $d_k$ events occur in $(c_{k-1},c_k]$, the number at the beginning of that interval, after censoring at its left endpoint, is $r_k+d_k$.

The no-censoring interval estimate from the preceding parts is therefore $r_k/(r_k+d_k)$. The possible event mass at zero contributes $r_0/(r_0+d_0)$. Multiplying yieldsFor a continuous event-time distribution, normally $d_0=0$. Within interval $k$, put $b_k=r_k+d_k$ and let its successive event counts be $e_1,\dots ,e_m$. Its Kaplan–Meier estimator factors telescope:The same calculation at zero provesThe paper's phrase “at risk at $c_k$” does not specify its side of an event at that endpoint. If it denotes the conventional pre-event risk set size $r_k^{-}$ instead, and $e_k$ events occur exactly at $c_k$, substitute $r_k=r_k^{-}-e_k$. The interval factor is then $(r_k^{-}-e_k)/(r_k^{-}+d_k-e_k)$; at zero it is $(r_0^{-}-d_0)/r_0^{-}$. This keeps the events-before-censoring convention consistent. After an empty risk set, stop the product; if the fitted survival has reached zero, retain zero rather than creating a $0/0$ factor.