|
|
| Line 1: |
Line 1: |
| {{lead too long|date=November 2013}}
| | I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Distributing manufacturing is exactly where her primary earnings comes from. To perform lacross is the factor I adore most of all. Kentucky is where I've usually been living.<br><br>Visit my blog: [http://koreanyelp.com/index.php?document_srl=1798&mid=SchoolNews free tarot readings] |
| [[File:Km plot.jpg|thumb|250px|An example of a Kaplan–Meier plot for two conditions associated with patient survival.]]
| |
| | |
| The '''Kaplan–Meier estimator''',<ref>{{cite journal |last=Kaplan |first=E. L. |last2=Meier |first2=P. |title=Nonparametric estimation from incomplete observations |journal=[[Journal of the American Statistical Association|J. Amer. Statist. Assn.]] |volume=53 |issue=282 |pages=457–481 |year=1958 |jstor=2281868}}</ref><ref>Kaplan, E.L. in a retrospective on the seminal paper in "This week's citation classic". ''Current Contents'' '''24''', 14 (1983). [http://www.garfield.library.upenn.edu/classics1983/A1983QS51100001.pdf Available from UPenn as PDF.]</ref> 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 (statistician)|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 important advantage of the Kaplan–Meier curve is that the method can take into account some types of [[Censoring (statistics)|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 ''t''. For a sample of size ''N'' from this population, let the observed times until death of the ''N'' sample members be
| |
| | |
| :<math>t_1 \le t_2 \le t_3 \le \cdots \le t_N. </math>
| |
| | |
| Corresponding to each ''t''<sub>''i''</sub> is ''n''<sub>''i''</sub>, the number "at risk" just prior to time ''t''<sub>''i''</sub>, and ''d''<sub>''i''</sub>, the number of deaths at time ''t''<sub>''i''</sub>.
| |
| | |
| 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.
| |
| | |
| {| class="wikitable" style="text-align: center; width:50%;margin:1em auto 1em auto;"
| |
| |-
| |
| ! scope="col" | <math>i </math>
| |
| ! scope="col" | 1
| |
| ! scope="col" | 2
| |
| |-
| |
| ! scope="row" | <math> t_i </math>
| |
| | 3 || 11
| |
| |-
| |
| ! scope="row" | <math> d_i </math>
| |
| | 1 || 2
| |
| |-
| |
| ! scope="row" | <math> n_i </math>
| |
| | 10 || 8
| |
| |}
| |
| | |
| The Kaplan–Meier estimator is the [[nonparametric]] maximum likelihood estimate of ''S''(''t''). It is a product of the form
| |
| | |
| :<math>\hat S(t) = \prod\limits_{t_i<t} \frac{n_i-d_i}{n_i}.</math>
| |
| | |
| When there is no censoring, ''n''<sub>''i''</sub> is just the number of survivors just prior to time ''t''<sub>''i''</sub>. With censoring, ''n''<sub>''i''</sub> 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.<ref name="costella">{{cite paper |first=John P. |last=Costella |year=2010 |url=http://assassinationscience.com/johncostella/physics/survival.pdf |title=A simple alternative to Kaplan–Meier for survival curves |work=Unpublished }}</ref>
| |
| | |
| There is an alternative definition that is sometimes used, namely
| |
| | |
| :<math>\hat S(t) = \prod\limits_{t_i \le t} \frac{n_i-d_i}{n_i}.</math>
| |
| | |
| The two definitions differ only at the observed event times. The latter definition is [[Right_continuous#Directional_continuity|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
| |
| | |
| :<math> S(t) = P[T>t] = 1-P[T \le t] = 1-F(t). \, </math>
| |
| | |
| Consequently, the right-continuous definition of <math>\scriptstyle\hat S(t)</math> 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:<ref>{{cite paper |authorlink=Major Greenwood |last=Greenwood |first=M. |title=The natural duration of cancer |work=Reports on Public Health and Medical Subjects |location=London |publisher=Her Majesty's Stationery Office |year=1926 |volume=33 |pages=1–26 }}</ref>
| |
| | |
| :<math> \widehat\mathrm{Var}( \widehat S(t) ) = \widehat S(t)^2 \sum\limits_{t_i<t} {\frac{{d_i}}{{n_i}({n_i-d_i})}}.</math>
| |
| | |
| 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 [[proportional hazards models|Cox proportional hazards test]]
| |
| | |
| ==See also==
| |
| * [[Nelson–Aalen estimator]]
| |
| * [[Median lethal dose]]
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| ==External links==
| |
| * [http://www.cancerguide.org/scurve_km.html Calculating Kaplan-Meier curves by Steve Dunn]
| |
| * [http://stat.ethz.ch/education/semesters/ss2011/seminar/contents/presentation_2.pdf Kaplan-Meier Survival Curves and the Log-Rank Test]
| |
| | |
| {{Statistics|analysis}}
| |
| | |
| {{DEFAULTSORT:Kaplan-Meier estimator}}
| |
| [[Category:Estimation theory]]
| |
| [[Category:Actuarial science]]
| |
| [[Category:Survival analysis]]
| |
| [[Category:Reliability engineering]]
| |
I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Distributing manufacturing is exactly where her primary earnings comes from. To perform lacross is the factor I adore most of all. Kentucky is where I've usually been living.
Visit my blog: free tarot readings