WordNet

a way of doing something, especially a systematic way; implies an orderly logical arrangement (usually in steps)

PrepTutorEJDIC

〈C〉(特に秩序だった)(…の)『方法』,方式《+『of』+『名』(do『ing』)》 / 〈C〉〈U〉(思考行・行為・行動の)きちょうめんさ;秩序,筋道

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/03/11 14:04:59」(JST)

wiki en

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 an estimator for estimating 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 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. Each submitted similar papers to the Journal of the American Statistical Association, but the editor then convinced them to combine their work into one paper, which has been cited about 34,000 times since its publication.^[3]

Basic concepts

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 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 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.

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 of the N sample members be

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.

	1	2
	3	11
	1	2
	10	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

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.^[4]

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 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).

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:^[5]

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

References

^ Kaplan, E. L.; Meier, P. (1958). "Nonparametric estimation from incomplete observations". J. Amer. Statist. Assn. 53 (282): 457–481. 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.
^ "Paul Meier, 1924 - 2011". Chicago Tribune. August 18, 2011.
^ Costella, John P. (2010). "A simple alternative to Kaplan–Meier for survival curves". Unpublished.
^ Greenwood, M. (1926). "The natural duration of cancer". Reports on Public Health and Medical Subjects (London: Her Majesty's Stationery Office) 33: 1–26.

External links

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

UpToDate Contents

全文を閲覧するには購読必要です。 To read the full text you will need to subscribe.

1. 生物統計学および疫学に関する一般用語集 glossary of common biostatistical and epidemiological terms
2. 大動脈解離のマネージメント management of aortic dissection
3. 小児における大動脈弁狭窄 valvar aortic stenosis in children
4. 腹部大動脈の血管内修復装置 endovascular devices for abdominal aortic repair
5. 大動脈内バルーンポンプカウンターパルセーション intraaortic balloon pump counterpulsation

English Journal

Renal Cell Carcinoma (RCC) in Patients With End-Stage Renal Disease Exhibits Many Favourable Clinical, Pathologic, and Outcome Features Compared With RCC in the General Population.

Neuzillet Y, Tillou X, Mathieu R, Long JA, Gigante M, Paparel P, Poissonnier L, Baumert H, Escudier B, Lang H, Rioux-Leclercq N, Bigot P, Bernhard JC, Albiges L, Bastien L, Petit J, Saint F, Bruyere F, Boutin JM, Brichart N, Karam G, Branchereau J, Ferriere JM, Wallerand H, Barbet S, Elkentaoui H, Hubert J, Feuillu B, Theveniaud PE, Villers A, Zini L, Descazeaux A, Roupret M, Barrou B, Fehri K, Lebret T, Tostain J, Terrier JE, Terrier N, Martin L, Dugardin F, Galliot I, Staerman F, Azemar MD, Irani J, Tisserand B, Timsit MO, Sallusto F, Rischmann P, Guy L, Valeri A, Deruelle C, Azzouzi AR, Chautard D, Mejean A, Salomon L, Rigaud J, Pfister C, Soulie M, Kleinclauss F, Badet L, Patard JJ; the members of the Comite de Transplantation de l’Association Francaise d’Urologie, the Comite de Cancerologie de l’Association Francaise d’Urologie.SourceDepartment of Urology, Foch Hospital, Suresnes, France.
European urology.Eur Urol.2011 Aug;60(2):366-373. Epub 2011 Mar 2.
BACKGROUND: Patients with end-stage renal disease (ESRD) are at risk of developing renal tumours.OBJECTIVE: Compare clinical, pathologic, and outcome features of renal cell carcinomas (RCCs) in ESRD patients and in patients from the general population.DESIGN, SETTING, AND PARTICIPANTS: Twenty-four F
PMID 21377780

Parafibromin expression in lung normal tissue and carcinoma: its comparison with clinicopathological parameters of carcinoma.

Xia P, Wang W, Xu XY, Wang JP, Takano Y, Zheng HC.SourceDepartment of Biochemistry and Molecular Biology, College of Basic Medicine, China Medical University, Shenyang, China.
Histology and histopathology.Histol Histopathol.2011 Aug;26(8):1039-47.
Parafibromin is a protein encoded by the hyperparathyroidism 2 oncosuppressor gene and its down-regulated expression is involved in the pathogenesis of parathyroid, gastric and colorectal carcinomas. To clarify the roles of parafibromin expression in lung carcinomas, it was examined by immunohistoch
PMID 21692036

Japanese Journal

An Approximate Likelihood Procedure for Competing Risks Data

Suzukawa Akio
Discussion Paper, Series A 231, 1-22, 2010-11
… Under censorship, the ordinary maximum likelihood method does not necessarily give consistent estimators. …
NAID 120002603322