Kaplan-Meier estimator(From Wikipedia, the free encyclopedia)
Kaplan-Meier estimator
From Wikipedia, the free encyclopedia
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The Kaplan-Meier estimator (also known as the product limit estimator) estimates the survival function from life-time data. In medical research, it might be used to measure the fraction of patients living for a certain amount of time after treatment. An economist might measure the length of time people remain unemployed after a job loss. An engineer might measure the time until failure of machine parts.
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 "censored" data — losses from the sample before the final outcome is observed (for instance, if a patient withdraws from a study). On the plot, small vertical tick-marks indicate losses, where patient data has been censored. When no truncation or censoring occurs, the Kaplan-Meier curve is equivalent to the empirical distribution.
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 still survive, but less than half of patients with Gene B.
[edit] Formulation
Let S(t) be the probability that an item from a given population will have a lifetime exceeding t. For a sample from this population of size N let the observed times until death of N sample members be
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 each time typically will not be uniform. For example, a small data set might begin with 10 cases, have a death at Day 3, a loss (censored case) at Day 9, and another death at Day 11. Then we have (t1 = 3,t2 = 11), (n1 = 10,n2 = 8), and (d1 = 1,d2 = 1).
The Kaplan-Meier estimator is the nonparametric maximum likelihood estimate of S(t). It is a product of the form
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 less 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.
There is an alternate definition that is sometimes used, namely
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
Consequently, the right-continuous definition of may be preferred in order to make the estimate compatible with a right-continuous estimate of F(t).
[edit] 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:
In some cases, one may wish to compare different Kaplan-Meier curves. This may be done by several methods including:
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