KaplanMeier

The Kaplan-Meier estimator, also known as the product-limit estimator, is a nonparametric statistic used to estimate the survival function from time-to-event data, especially when some observations are censored. It is widely applied in medical research to describe the probability of surviving beyond a given time.

The estimator is constructed from observed event times. For each event time t_i, let n_i be the number of individuals at risk just before t_i and d_i be the number of events (for example, deaths) at t_i. The survival function S(t) is estimated by the product over all event times up to t of (1 − d_i / n_i):

S(t) = ∏_{t_i ≤ t} (1 − d_i / n_i).

The resulting curve is a step function that drops at event times and remains flat when censored observations occur.

Assumptions include noninformative censoring (the reason for censoring is independent of the survival process) and independence between individuals. The method can handle tied event times and right-censoring, which is common in clinical studies.

Variance and confidence intervals are typically derived using Greenwood’s formula or by transforming the standard error of the log of S(t). A common approach is to compute SE[log S(t)] = sqrt(∑ d_i / [n_i(n_i − d_i)]) and construct CIs as S(t) × exp(±zα/2 × SE[log S(t)]).

History and extensions: the method was introduced by Edward L. Kaplan and Paul Meier in 1958. It forms the basis for survival analysis in many fields, and is often used with the log-rank test for comparing groups or in stratified analyses.