Kaplan–Meier estimator

The Kaplan–Meier estimator,^[1][2] also known as the product limit estimator, is an estimator for estimating the survival function from life-time 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 example of a Kaplan–Meier plot for two conditions associated with patient survival

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, i.e. is lost from the sample before the final outcome is observed. On the plot, small vertical tick-marks indicate losses, where a patient's survival time has been right-censored. When no truncation or censoring occurs, the Kaplan–Meier curve is equivalent to 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 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

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

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 each time typically are not 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

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

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 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.^[3]

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^[4]:

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

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

• The log rank test
• The Cox proportional hazards test

See also

• Nelson–Aalen estimator

References

1. ^ Kaplan, E. L.; Meier, P.: Nonparametric estimation from incomplete observations. J. Amer. Statist. Assn. 53:457–481, 1958. 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. ^ John P. Costella, 2010, A simple alternative to Kaplan–Meier for survival curves.
4. ^ Greenwood M. The natural duration of cancer. Reports on Public Health and Medical Subjects. London: Her Majesty's Stationery Office 1926;33:1–26.

External links

• Calculating Kaplan-Meier curves by Steve Dunn