The Kaplan–Meier estimator, also known as the product limit estimator, is a non-parametric statistic used to estimate the survival function from lifetime data. In medical research, it is often used to measure the fraction of patients living for a certain amount of time after treatment. In other fields, Kaplan–Meier estimators may be used to measure the length of time people remain unemployed after a job loss, the time-to-failure of machine parts, or how long fleshy fruits remain on plants before they are removed by frugivores. The estimator is named after Edward L. Kaplan and Paul Meier, who each submitted similar manuscripts to the Journal of the American Statistical Association. The journal editor, John Tukey, convinced them to combine their work into one paper, which has been cited more than 34,000 times since its publication in 1958. The estimator of the survival function S ( t ) {\displaystyle S(t)} (the probability that life is longer than t {\displaystyle t} ) is given by:
S ^ ( t ) = ∏ i : t i ≤ t ( 1 − d i n i ) , {\displaystyle {\widehat {S}}(t)=\prod \limits _{i:\ t_{i}\leq t}\left(1-{\frac {d_{i}}{n_{i}}}\right),}
with t i {\displaystyle t_{i}} a time when at least one event happened, di the number of events (e.g., deaths) that happened at time t i {\displaystyle t_{i}} , and n i {\displaystyle n_{i}} the individuals known to have survived (have not yet had an event or been censored) up to time t i {\displaystyle t_{i}} .
Basic concepts A plot of the Kaplan–Meier estimator is a series of declining horizontal steps which, with a large enough sample size, approaches the true survival function for that population. The value of the survival function between successive distinct sampled observations ("clicks") is assumed to be constant. An important advantage of the Kaplan–Meier curve is that the method can take into account some types of censored data, particularly right-censoring, which occurs if a patient withdraws from a study, is lost to follow-up, or is alive without event occurrence at last follow-up. On the plot, small vertical tick-marks state individual patients whose survival times have been right-censored. When no truncation or censoring occurs, the Kaplan–Meier curve is the complement of the empirical distribution function. In medical statistics, a typical application might involve grouping patients into categories, for instance, those with Gene A profile and those with Gene B profile. In the graph, patients with Gene B die much quicker than those with Gene A. After two years, about 80% of the Gene A patients survive, but less than half of patients with Gene B. To generate a Kaplan–Meier estimator, at least two pieces of data are required for each patient (or each subject): the status at last observation (event occurrence or right-censored), and the time to event (or time to censoring). If the survival functions between two or more groups are to be compared, then a third piece of data is required: the group assignment of each subject.
Problem definition Let τ ≥ 0 {\displaystyle \tau \geq 0} be a random variable as the time that passes between the start of the possible exposure period, t 0 {\displaystyle t_{0}} , and the time that the event of interest takes place, t 1 {\displaystyle t_{1}} . As indicated above, the goal is to estimate the survival function S {\displaystyle S} underlying τ {\displaystyle \tau } . Recall that this function is defined as
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