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 nonparametric 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,^{[3]} the timetofailure 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 about 34,000 times since its publication.^{[4]}
Contents

Basic concepts 1

Formulation 2

Statistical considerations 3

Implementations in statistics packages 4

See also 5

References 6

Further reading 7

External links 8
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 rightcensoring, which occurs if a patient withdraws from a study, is lost to followup, or is alive without event occurrence at last followup. On the plot, small vertical tickmarks indicate individual patients whose survival times have been rightcensored. 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 rightcensored) 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.^{[5]}
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 t_{i} is n_{i}, the number "at risk" just prior to time t_{i}, and d_{i}, the number of deaths at time t_{i}.
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 followup (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 t_{i}. It is a product of the form

\hat S(t) = \prod\limits_{t_i
When there is no censoring, n_{i} is just the number of survivors just prior to time t_{i}. With censoring, n_{i} 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.^{[6]}
There is an alternative definition that is sometimes used, namely

\hat S(t) = \prod\limits_{t_i \le t} \frac{n_id_i}{n_i}.
The two definitions differ only at the observed event times. The latter definition is rightcontinuous whereas the former definition is leftcontinuous.
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] = 1P[T \le t] = 1F(t). \,
Consequently, the rightcontinuous definition of \scriptstyle\hat S(t) may be preferred in order to make the estimate compatible with a rightcontinuous 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:^{[7]}

\widehat{\operatorname{Var}}( \widehat S(t) ) = \widehat S(t)^2 \sum\limits_{t_i\le t} \frac{d_i}{n_i(n_id_i)}.
In some cases, one may wish to compare different Kaplan–Meier curves. This may be done by several methods including:
Implementations in statistics packages

R: the Kaplan–Meier estimator is available as part of the
survival
package.^{[8]}^{[9]}^{[10]}

Stata: the command
sts
returns the Kaplan–Meier estimator.^{[11]}

Python: The
lifelines
package includes the KaplanMeier estimator ^{[12]}
See also
References

^ Kaplan, E. L.; Meier, P. (1958). "Nonparametric estimation from incomplete observations".

^ 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.

^ Meyer, Bruce D. (1990). "Unemployment Insurance and Unemployment Spells".

^ "Paul Meier, 1924–2011". Chicago Tribune. August 18, 2011.

^ 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.

^ Costella, John P. (2010). "A simple alternative to Kaplan–Meier for survival curves" (PDF). Unpublished.

^

^ "survival: Survival Analysis". R Project.

^ Willekens, Frans (2014). Package"Survival"The . Multistate Analysis of Life Histories with R. Springer. pp. 135–153.

^ Chen, DingGeng; Peace, Karl E. (2014). Clinical Trial Data Analysis Using R. CRC Press. pp. 99–108.

^ "sts — Generate, graph, list, and test the survivor and cumulative hazard functions" (PDF). Stata Manual.

^ "lifelines". .
Further reading

Aalen, Odd; Borgan, Ornulf; Gjessing, Hakon (2008). Survival and Event History Analysis: A Process Point of View. Springer. pp. 90–104.


Jones, Andrew M.; Rice, Nigel; D'Uva, Teresa Bago; Balia, Silvia (2013). "Duration Data". Applied Health Economics. London: Routledge. pp. 139–181.
External links

Calculating Kaplan–Meier curves by Steve Dunn

Kaplan–Meier Survival Curves and the LogRank Test
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