Kaplan–Meier and Nelson–Aalen Estimation

Suppose we follow patients after a hospital discharge and record the time until readmission. Some patients are readmitted during follow-up; others move away, withdraw, or reach the end of the study without a readmission. We cannot simply average the observed times, because the censored patients have unknown event times. We need an estimator that uses exactly the information we have: each patient contributes to the risk set until the last time we know they were event-free.

In the previous chapter we separated event times, censoring times, survival functions, and hazard functions. We now turn those ideas into two nonparametric estimators. The Kaplan-Meier estimator estimates the survival function $S(t)=P(T>t)$ directly as a product of conditional survival probabilities. The Nelson-Aalen estimator estimates the cumulative hazard $H(t)$ by adding up estimated hazard increments at observed event times.

Both estimators are built from the same object: the risk set at each event time. That shared structure lets us derive them side by side. Kaplan-Meier answers “what fraction survived past this time?”, while Nelson-Aalen answers “how much hazard has accumulated by this time?”. We will see that the two answers are closely related but not identical, especially when events are tied or the risk sets are small.