Kaplan–Meier estimator

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 non-parametric 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 time-to-failure 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]

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 right-censoring, which occurs if a patient withdraws from a study, is lost to follow-up, or is alive without event occurrence at last follow-up. On the plot, small vertical tick-marks indicate individual patients whose survival times have been right-censored. 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 right-censored) 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]

Benefits and limitations

Survival analysis is the evaluation of how long individuals, who are endangered of certain health risk, will survive. Kaplan–Meier estimator is one of the most frequently used survival analyses. It is based on a “time to event” model, meaning its endpoint is used to predict a time when certain positive events or negative events will occur. Benefits of the Kaplan-Meier include an estimate of recovery rates, the probability of death, and the effectiveness of medications. It can also measure multiple variables to determine if there is a correlation between more than one of the events. There are also disadvantages to the Kaplan–Meier estimator. When estimating the outcomes of numerous variables, researchers risk censoring the data. Censored data is when the variables being tested do not accurately display a real life scenario for the patient or majority of participants. Another potential error is if a competing event were to rise. For example, if a doctor was using the Kaplan–Meier estimator to follow a patient with a malignant brain tumor and the patient were to die, the estimator could no longer be used.^[6]

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 (or loss to follow-up) 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 follow-up (censored) at day 9. Data up to day 11 would be as follows.

$i$	$t_i$	$d_i$	$n_i$
1	3	1	10
2	11	2	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

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

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 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.^[7]

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:^[8]

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

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

Implementations in statistics packages

• SAS: The Kaplan–Meier estimator is implemented in the proc lifetest procedure.^[9]
• R: the Kaplan–Meier estimator is available as part of the survival package.^[10][11][12]
• Stata: the command sts returns the Kaplan–Meier estimator.^[13][14]
• Python: The lifelines package includes the Kaplan–Meier estimator ^[15]

See also

• Frequency of exceedance
• Median lethal dose
• Nelson–Aalen estimator

References

1. Kaplan, E. L.; Meier, P. (1958). "Nonparametric estimation from incomplete observations". J. Amer. Statist. Assn. 53 (282): 457–481. 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. Meyer, Bruce D. (1990). "Unemployment Insurance and Unemployment Spells". Econometrica 58 (4): 757–782. doi:10.2307/2938349.
4. "Paul Meier, 1924–2011". Chicago Tribune. August 18, 2011.
5. 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. doi:10.1016/j.otohns.2010.05.007. PMC 3932959. PMID 20723767.
6. Jager, K.; Dijk, P.; Zoccali, C.; Dekker, F. (2008). "The Analysis of Survival Data: The Kaplan–Meier Method". Kidney International 74 (5): 560–565. doi:10.1038/ki.2008.217.
7. Costella, John P. (2010). "A simple alternative to Kaplan–Meier for survival curves" (PDF). Unpublished.
8. Greenwood, M. (1926). "The natural duration of cancer". Reports on Public Health and Medical Subjects (London: Her Majesty's Stationery Office) 33: 1–26.
9. The LIFETEST Procedure
10. "survival: Survival Analysis". R Project.
11. Willekens, Frans (2014). "The Survival Package". Multistate Analysis of Life Histories with R. Springer. pp. 135–153. doi:10.1007/978-3-319-08383-4_6. ISBN 978-3-319-08383-4.
12. Chen, Ding-Geng; Peace, Karl E. (2014). Clinical Trial Data Analysis Using R. CRC Press. pp. 99–108.
13. "sts — Generate, graph, list, and test the survivor and cumulative hazard functions" (PDF). Stata Manual.
14. Cleves, Mario (2008). An Introduction to Survival Analysis Using Stata (Second ed.). College Station: Stata Press. pp. 93–107. ISBN 1-59718-041-6.
15. "lifelines"..

Further reading

• Aalen, Odd; Borgan, Ornulf; Gjessing, Hakon (2008). Survival and Event History Analysis: A Process Point of View. Springer. pp. 90–104. ISBN 978-0-387-68560-1.
• Greene, William H. (2012). "Nonparametric and Semiparametric Approaches". Econometric Analysis (Seventh ed.). Prentice-Hall. pp. 909–912. ISBN 978-0-273-75356-8.
• Jones, Andrew M.; Rice, Nigel; D'Uva, Teresa Bago; Balia, Silvia (2013). "Duration Data". Applied Health Economics. London: Routledge. pp. 139–181. ISBN 978-0-415-67682-3.
• Singer, Judith B.; Willett, John B. (2003). Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence. New York: Oxford University Press. pp. 483–487. ISBN 0-19-515296-4.

External links

• Calculating Kaplan–Meier curves by Steve Dunn
• Kaplan–Meier Survival Curves and the Log-Rank Test