The Kaplan–Meier estimator , 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, 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 50,000 times since its publication..
The estimator is given by:
with di the number of events and the total individuals at risk (have not yet had an event or been censored) at time i.
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.
Kaplan-Meier estimator can be derived from maximum likelihood estimation of hazard function . More specifically given as the number of events and the total individuals at risk at time , discrete hazard rate can be defined as the probability of an individual with an event at time . Then survival rate can be defined as:
and the likelihood function for the hazard function up to time is:
therefore the log likelihood will be:
finding the maximum of log likelihood with respect to yields:
where hat is used to denote maximum likelihood estimation. Given this result, we can write:
Benefits and limitations
The Kaplan–Meier estimator is one of the most frequently used methods of survival analysis. The estimate may be useful to examine recovery rates, the probability of death, and the effectiveness of treatment. It is limited in its ability to estimate survival adjusted for covariates; parametric survival models and the Cox proportional hazards model may be useful to estimate covariate-adjusted survival.
where is the number of cases and is the total number of observations, for .
- Mathematica: the built-in function
SurvivalModelFitcreates survival models.
- SAS: The Kaplan–Meier estimator is implemented in the
- R: the Kaplan–Meier estimator is available as part of the
- Stata: the command
stsreturns the Kaplan–Meier estimator.
- Python: the
lifelinespackage includes the Kaplan–Meier estimator.
- MATLAB: the
ecdffunction with the
'function','survivor'arguments can calculate or plot the Kaplan–Meier estimator.
- StatsDirect: The Kaplan–Meier estimator is implemented in the
- Kaplan, E. L.; Meier, P. (1958). "Nonparametric estimation from incomplete observations". J. Amer. Statist. Assn. 53 (282): 457–481. doi:10.2307/2281868. JSTOR 2281868.
- 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.
- Meyer, Bruce D. (1990). "Unemployment Insurance and Unemployment Spells". Econometrica. 58 (4): 757–782. doi:10.2307/2938349.
- "- Google Scholar". scholar.google.com. Retrieved 2017-03-04.
- "Paul Meier, 1924–2011". Chicago Tribune. August 18, 2011.
- 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 . PMID 20723767.
- (PDF) https://web.stanford.edu/~lutian/coursepdf/STAT331unit3.pdf. Missing or empty
- Greenwood, M. (1926). "The natural duration of cancer". Reports on Public Health and Medical Subjects. London: Her Majesty's Stationery Office. 33: 1–26.
- (PDF) https://www.math.wustl.edu/%7Esawyer/handouts/greenwood.pdf. Missing or empty
- Hall WJ and Wellner JA (1980) Confidence bands for a survival curve for censored data. Biometrika 69
- Nair VN (1984) Confidence bands for survival functions with censored data: A comparative study. Technometrics 26: 265–275
- "Survival Analysis - Mathematica SurvivalModelFit". wolfram.com. Retrieved 2017-08-14.
- The LIFETEST Procedure
- "survival: Survival Analysis". R Project.
- 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.
- Chen, Ding-Geng; Peace, Karl E. (2014). Clinical Trial Data Analysis Using R. CRC Press. pp. 99–108.
- "sts — Generate, graph, list, and test the survivor and cumulative hazard functions" (PDF). Stata Manual.
- Cleves, Mario (2008). An Introduction to Survival Analysis Using Stata (Second ed.). College Station: Stata Press. pp. 93–107. ISBN 1-59718-041-6.
- "Empirical cumulative distribution function - MATLAB ecdf". mathworks.com. Retrieved 2016-06-16.
- 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.
- Dunn, Steve (2002). "Survival Curves: Accrual and The Kaplan-Meier Estimate". Cancer Guide. Statistics.
- Staub, Linda; Gekenidis, Alexandros (Mar 7, 2011). "Kaplan–Meier Survival Curves and the Log-Rank Test" (PDF). Survival Analysis (pdf). Handout and presentation. Seminar for Statistics (SfS). Eidgenössische Technische Hochschule Zürich (ETH) [Swiss Federal Institute of Technology Zurich].
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