# Kaplan–Meier estimator

File:Km plot.jpg
An example of a Kaplan–Meier plot for two conditions associated with patient survival.

The Kaplan–Meier estimator,[1][2] 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 convinced them to combine their work into one paper, which has been cited about 34,000 times since its publication.[3]

## 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 indicate 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 more quickly 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.

In order 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.[4]

## Formulation

Let S(t) be the probability that a member from a given population will have a lifetime exceeding time, t. For a sample of size N from this population, let the observed times until death of the N sample members be

$t_1 \le t_2 \le t_3 \le \cdots \le t_N.$

Corresponding to each ti is ni, the number "at risk" just prior to time ti, and di, the number of deaths at time ti.

Note that the intervals between events are typically not uniform. For example, a small data set might begin with 10 cases. Suppose subject 1 dies on day 3, subjects 2 and 3 die on day 11 and subject 4 is lost to follow-up (censored) at day 9. Data up to day 11 would be as follows.

$i$ 1 2
$t_i$ 3 11
$d_i$ 1 2
$n_i$ 10 8

The Kaplan–Meier estimator is the nonparametric maximum likelihood estimate of S(t), where the maximum is taken over the set of all piecewise constant survival curves with breakpoints at the event times ti. It is a product of the form

$\hat S(t) = \prod\limits_{t_i<t} \frac{n_i-d_i}{n_i}.$

When there is no censoring, ni is just the number of survivors just prior to time ti. With censoring, ni is the number of survivors minus the number of losses (censored cases). It is only those surviving cases that are still being observed (have not yet been censored) that are "at risk" of an (observed) death.[5]

There is an alternative definition that is sometimes used, namely

$\hat S(t) = \prod\limits_{t_i \le t} \frac{n_i-d_i}{n_i}.$

The two definitions differ only at the observed event times. The latter definition is right-continuous whereas the former definition is left-continuous.

Let T be the random variable that measures the time of failure and let F(t) be its cumulative distribution function. Note that

$S(t) = P[T>t] = 1-P[T \le t] = 1-F(t). \,$

Consequently, the right-continuous definition of $\scriptstyle\hat S(t)$ may be preferred in order to make the estimate compatible with a right-continuous estimate of F(t).

## Statistical considerations

The Kaplan–Meier estimator is a statistic, and several estimators are used to approximate its variance. One of the most common such estimators is Greenwood's formula:[6]

$\widehat{\operatorname{Var}}( \widehat S(t) ) = \widehat S(t)^2 \sum\limits_{t_i\le t} \frac{d_i}{n_i(n_i-d_i)}.$

In some cases, one may wish to compare different Kaplan–Meier curves. This may be done by several methods including:

## References

1. Kaplan, E. L.; Meier, P. (1958). "Nonparametric estimation from incomplete observations". J. Amer. Statist. Assn. 53 (282): 457–481. JSTOR 2281868.
2. Kaplan, E.L. in a retrospective on the seminal paper in "This week's citation classic". Current Contents 24, 14 (1983). Available from UPenn as PDF.
3. "Paul Meier, 1924–2011". Chicago Tribune. August 18, 2011.
4. Rich JT, Neely JG, Paniello RC, Voelker CC, Nussenbaum B, Wang EW (2010). "A practical guide to understanding Kaplan–Meier curves.". Otolaryngol Head Neck Surg 143 (3): 331–6. PMC 3932959. PMID 20723767. doi:10.1016/j.otohns.2010.05.007.
5. Costella, John P. (2010). "A simple alternative to Kaplan–Meier for survival curves" (PDF). Unpublished.
6. Greenwood, M. (1926). "The natural duration of cancer". Reports on Public Health and Medical Subjects (London: Her Majesty's Stationery Office) 33: 1–26.