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

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.