The
Kaplan–Meier estimator (also known as the
product limit estimator) estimates the
survival functionThe survival function, also known as a survivor function or reliability function, is a property of any random variable that maps a set of events, usually associated with mortality or failure of some system, onto time. It captures the probability that the system will survive beyond a specified time...
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
Kaplan–Meier estimator (also known as the
product limit estimator) estimates the
survival functionThe survival function, also known as a survivor function or reliability function, is a property of any random variable that maps a set of events, usually associated with mortality or failure of some system, onto time. It captures the probability that the system will survive beyond a specified time...
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 statisticsMedical statistics deals with applications of biostatistics to medicine and the health sciences, including epidemiology, public health, forensic medicine, and clinical research...
, 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.
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 alternative 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 functionIn probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X...
. 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).
Statistical considerations
The Kaplan–Meier estimator is a
statisticA statistic is the result of applying a function to a set of data.More formally, statistical theory defines a statistic as a function of a sample where the function itself is independent of the sample's distribution: the term is used both for the function and for the value of the function on a...
, and several estimators are used to approximate its
varianceIn probability theory and statistics, the variance of a random variable or distribution is the expected square deviation of that variable from its expected value or mean, or to put it another way: variance is the measure of the amount of variation of all the scores for a variable...
. 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:
- The log rank test
- The Cox proportional hazards test
-General:Proportional hazards models are a sub-class of survival models in statistics, in which the effect of a treatment under study has a multiplicative effect on the subject's hazard rate. For example, a drug may halve one's immediate probability of stroke...
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