KaplanMeierestimation

Kaplan-Meier estimation, also known as the product-limit estimator, is a nonparametric method for estimating the survival function from time-to-event data, particularly in the presence of right censoring. It provides an estimate of the probability of surviving beyond a given time and yields a stepwise survival curve. The estimator uses observed event times t1 < t2 < ... and, at each event time i, di events (deaths) occur among ni individuals at risk just before time ti. The survival function is estimated by S(t) = product over i with ti ≤ t of (ni − di)/ni. The estimates update only at event times; censored observations contribute to the risk set up to their censoring time but do not count as events.

Key assumptions include non-informative censoring (censored individuals have the same future survival as those remaining under observation) and independence of survival times. The Kaplan-Meier curve starts at S(0) = 1 and decreases in a stepwise fashion, never increasing. The median survival time is the time at which S(t) = 0.5, if this point is observed within the data. Confidence intervals around the survival probability can be obtained using Greenwood’s formula. Variance estimates facilitate hypothesis testing and comparison across groups, commonly via the log-rank test. Kaplan-Meier estimation is widely used in clinical trials and epidemiology to summarize time-to-event data and to provide intuitive, visual representations of survival probabilities over time. Related methods include the Nelson-Aalen estimator for cumulative hazard and Cox proportional hazards models for regression analysis.