The Kaplan–Meier estimator,^{[1]}^{[2]} also known as the product limit estimator, is an estimator for estimating 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 economics, it can be used to measure the length of time people remain unemployed after a job loss. In engineering, it can be used to measure the time until failure of machine parts. In ecology, it can be used to estimate how long fleshy fruits remain on plants before they are removed by frugivores. The estimator is named after Edward L. Kaplan and Paul Meier.
A plot of the Kaplan–Meier estimate of the survival function is a series of horizontal steps of declining magnitude which, when a large enough sample is taken, 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, i.e. is lost from the sample before the final outcome is observed. On the plot, small vertical tickmarks indicate losses, where a patient's survival time has 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.
Formulation
Let S(t) be the probability that a member from a given population will have a lifetime exceeding t. For a sample of size N from this population, let the observed times until death of the N sample members be
 $t\_1\; \backslash le\; t\_2\; \backslash le\; t\_3\; \backslash le\; \backslash cdots\; \backslash le\; t\_N.$
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). It is a product of the form
 $\backslash hat\; S(t)\; =\; \backslash prod\backslash limits\_\{t\_i\}\; \backslash frac\{n\_id\_i\}\{n\_i\}.\; math>$
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.^{[3]}
There is an alternative definition that is sometimes used, namely
 $\backslash hat\; S(t)\; =\; \backslash prod\backslash limits\_\{t\_i\; \backslash le\; t\}\; \backslash frac\{n\_id\_i\}\{n\_i\}.$
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
 $S(t)\; =\; P[T>t]\; =\; 1P[T\; \backslash le\; t]\; =\; 1F(t).\; \backslash ,$
Consequently, the rightcontinuous definition of $\backslash scriptstyle\backslash hat\; S(t)$ 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:^{[4]}
 $\backslash widehat\backslash mathrm\{Var\}(\; \backslash widehat\; S(t)\; )\; =\; \backslash widehat\; S(t)^2\; \backslash sum\backslash limits\_\{t\_i\}\; \{\backslash frac.\; math>$
In some cases, one may wish to compare different Kaplan–Meier curves. This may be done by several methods including:
See also
References
External links
 Calculating KaplanMeier curves by Steve Dunn
 KaplanMeier Survival Curves and the LogRank Test
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