KaplanMeierin

KaplanMeierin, commonly known as the Kaplan-Meier estimator or product-limit estimator, is a non-parametric statistic used to estimate the survival function S(t) of time-to-event data in the presence of right censoring. It was introduced by Edward L. Kaplan and Paul Meier in 1958 and remains a standard tool in medical research and reliability analysis. The survival function S(t) represents the probability of surviving beyond time t.

Computation and interpretation: At each observed event time t i, let d i be the number of events (such as deaths) and n i the number at risk just before t i. The estimator is S(t) = product over all t i ≤ t of (1 − d i / n i). The resulting curve is a step function that drops at event times and remains flat at censoring times. Censored observations reduce the number at risk in later intervals but do not contribute to d i.

Variance and confidence intervals: Variance is commonly estimated using Greenwood’s formula: Var[S(t)] = S(t)² sum over i with t i ≤ t of [ d i / ( n i ( n i − d i ) ) ]. Standard errors and confidence intervals can be constructed from this variance, often using a log-minus-log transformation for better coverage.

Assumptions and extensions: The method assumes non-informative censoring and independence between censoring and survival. It handles censored data without imputing missing times. Kaplan-Meier curves can be compared between groups with the log-rank test, and the approach serves as a foundation for more complex models, such as stratified analyses and Cox proportional hazards models that incorporate covariates. Applications span oncology, cardiology, epidemiology, and reliability engineering.