TheKaplan–Meier estimator,^[1][2] also known as theproduct limit estimator, is anon-parametricstatistic used to estimate thesurvival 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 byfrugivores. Theestimator is named afterEdward L. Kaplan andPaul Meier, who each submitted similar manuscripts to theJournal of the American Statistical Association.^[4] The journal editor,John Tukey, convinced them to combine their work into one paper, which has been cited more than 34,000 times since its publication in 1958.^[5][6]

Theestimator of thesurvival function${\displaystyle S(t)}$ (the probability that life is longer than${\displaystyle t}$) is given by:

${\displaystyle \hat{S}(t)=\prod _{i:t_i\le t}\left(1-\frac{d_i}{n_i}\right),}$

with${\displaystyle t_i}$ a time when at least one event happened,d_i thenumber of events (e.g., deaths) that happened at time${\displaystyle t_i}$, and${\displaystyle n_i}$ theindividuals known to have survived (have not yet had an event or been censored) up to time${\displaystyle t_i}$.

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 ofcensored data, particularlyright-censoring, which occurs if a patient withdraws from a study, islost to follow-up, or is alive without event occurrence at last follow-up. On the plot, small vertical tick-marks state individual patients whose survival times have been right-censored. When no truncation or censoring occurs, the Kaplan–Meier curve is thecomplement of theempirical distribution function.

Inmedical 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 quicker 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.

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.^[7]

Let${\displaystyle \tau \ge 0}$ be a random variable as the time that passes between the start of the possible exposure period,${\displaystyle t_0}$, and the time that the event of interest takes place,${\displaystyle t_1}$. As indicated above, the goal is to estimate thesurvival function${\displaystyle S}$ underlying${\displaystyle \tau }$. Recall that this function is defined as

${\displaystyle S(t)=\mathrm{Prob}(\tau >t)}$, where${\displaystyle t=0,1,\dots }$ is the time.

Let${\displaystyle {\tau }_1,\dots ,{\tau }_n\ge 0}$ be independent, identically distributed random variables, whose common distribution is that of${\displaystyle \tau }$:${\displaystyle {\tau }_j}$ is the random time when some event${\displaystyle j}$ happened. The data available for estimating${\displaystyle S}$ is not${\displaystyle ({\tau }_j{)}_{j=1,\dots ,n}}$, but the list of pairs${\displaystyle (({\tilde{\tau }}_j,c_j){)}_{j=1,\dots ,n}}$ where for${\displaystyle j\in [n]:=\{1,2,\dots ,n\}}$,${\displaystyle c_j\ge 0}$ is a fixed, deterministic integer, thecensoring time of event${\displaystyle j}$ and${\displaystyle {\tilde{\tau }}_j=min({\tau }_j,c_j)}$. In particular, the information available about the timing of event${\displaystyle j}$ is whether the event happened before the fixed time${\displaystyle c_j}$ and if so, then the actual time of the event is also available. The challenge is to estimate${\displaystyle S(t)}$ given this data.

Two derivations of the Kaplan–Meier estimator are shown. Both are based on rewriting the survival function in terms of what is sometimes calledhazard, ormortality rates. However, before doing this it is worthwhile to consider a naive estimator.

To understand the power of the Kaplan–Meier estimator, it is worthwhile to first describe a naive estimator of the survival function.

Fix${\displaystyle k\in [n]:=\{1,\dots ,n\}}$ and let${\displaystyle t>0}$. A basic argument shows that the following proposition holds:

Proposition 1: If the censoring time${\displaystyle c_k}$ of event${\displaystyle k}$ exceeds${\displaystyle t}$ (${\displaystyle c_k\ge t}$), then${\displaystyle {\tilde{\tau }}_k\ge t}$ if and only if${\displaystyle {\tau }_k\ge t}$.

Let${\displaystyle k}$ be such that${\displaystyle c_k\ge t}$. It follows from the above proposition that

${\displaystyle \mathrm{Prob}({\tau }_k\ge t)=\mathrm{Prob}({\tilde{\tau }}_k\ge t).}$

Let${\displaystyle X_k=\mathbb{I}({\tilde{\tau }}_k\ge t)}$ and consider only those${\displaystyle k\in C(t):=\{k:c_k\ge t\}}$, i.e. the events for which the outcome was not censored before time${\displaystyle t}$. Let${\displaystyle m(t)=|C(t)|}$ be the number of elements in${\displaystyle C(t)}$. Note that the set${\displaystyle C(t)}$ is not random and so neither is${\displaystyle m(t)}$. Furthermore,${\displaystyle (X_k{)}_{k\in C(t)}}$ is a sequence of independent, identically distributedBernoulli random variables with common parameter${\displaystyle S(t)=\mathrm{Prob}(\tau \ge t)}$. Assuming that${\displaystyle m(t)>0}$, this suggests to estimate${\displaystyle S(t)}$ using

${\displaystyle {\hat{S}}_{\text{naive}}(t)=\frac{1}{m(t)}\sum _{k:c_k\ge t}{X}_k=\frac{|\{1\le k\le n:{\tilde{\tau }}_k\ge t\}|}{|\{1\le k\le n:c_k\ge t\}|}=\frac{|\{1\le k\le n:{\tilde{\tau }}_k\ge t\}|}{m(t)},}$

where the second equality follows because${\displaystyle {\tilde{\tau }}_k\ge t}$ implies${\displaystyle c_k\ge t}$, while the last equality is simply a change of notation.

The quality of this estimate is governed by the size of${\displaystyle m(t)}$. This can be problematic when${\displaystyle m(t)}$ is small, which happens, by definition, when a lot of the events are censored. A particularly unpleasant property of this estimator, that suggests that perhaps it is not the "best" estimator, is that it ignores all the observations whose censoring time precedes${\displaystyle t}$. Intuitively, these observations still contain information about${\displaystyle S(t)}$: For example, when for many events with, also holds, we can infer that events often happen early, which implies that${\displaystyle \mathrm{Prob}(\tau \le t)}$ is large, which, through${\displaystyle S(t)=1-\mathrm{Prob}(\tau \le t)}$ means that${\displaystyle S(t)}$ must be small. However, this information is ignored by this naive estimator. The question is then whether there exists an estimator that makes a better use of all the data. This is what the Kaplan–Meier estimator accomplishes. Note that the naive estimator cannot be improved when censoring does not take place; so whether an improvement is possible critically hinges upon whether censoring is in place.

By elementary calculations,

where the second to last equality used that${\displaystyle \tau }$ is integer valued and for the last line we introduced

${\displaystyle q(t)=1-\mathrm{Prob}(\tau =t\mid \tau \ge t).}$

By a recursive expansion of the equality${\displaystyle S(t)=q(t)S(t-1)}$, we get

${\displaystyle S(t)=q(t)q(t-1)\cdots q(0).}$

Note that here${\displaystyle q(0)=1-\mathrm{Prob}(\tau =0\mid \tau >-1)=1-\mathrm{Prob}(\tau =0)}$.

The Kaplan–Meier estimator can be seen as a "plug-in estimator" where each${\displaystyle q(s)}$ is estimated based on the data and the estimator of${\displaystyle S(t)}$ is obtained as a product of these estimates.

It remains to specify how${\displaystyle q(s)=1-\mathrm{Prob}(\tau =s\mid \tau \ge s)}$ is to be estimated. By a similar reasoning that lead to the construction of the naive estimator above, for any${\displaystyle k\in [n]}$ such that${\displaystyle c_k>s}$,${\displaystyle \mathrm{Prob}(\tau =s)=\mathrm{Prob}({\tilde{\tau }}_k=s)}$ and${\displaystyle \mathrm{Prob}(\tau \ge s)=\mathrm{Prob}({\tilde{\tau }}_k\ge s)}$ both hold. Hence, for any${\displaystyle k\in [n]}$ such that${\displaystyle c_k>s}$,

${\displaystyle \mathrm{Prob}(\tau =s|\tau \ge s)=\mathrm{Prob}({\tilde{\tau }}_k=s)/\mathrm{Prob}({\tilde{\tau }}_k\ge s).}$

Hence we arrive at the estimator

${\displaystyle \hat{q}(s)=1-\frac{|\{1\le k\le n:c_k>s,{\tilde{\tau }}_k=s\}|}{|\{1\le k\le n:c_k>s,{\tilde{\tau }}_k\ge s\}|}}$

(think of estimating the numerator and denominator separately in the definition of the "hazard rate"${\displaystyle \mathrm{Prob}(\tau =s|\tau \ge s)}$). The Kaplan–Meier estimator is then given by

${\displaystyle \hat{S}(t)=\prod _{s=0}^t\hat{q}(s).}$

The form of the estimator stated at the beginning of the article can be obtained by some further algebra. For this, write${\displaystyle \hat{q}(s)=1-d(s)/n(s)}$ where, using the actuarial science terminology,${\displaystyle d(s)=|\{1\le k\le n:c_k>s,{\tilde{\tau }}_k=s\}|}$ is the number of known deaths at time${\displaystyle s}$, while${\displaystyle n(s)=|\{1\le k\le n:c_k>s,{\tilde{\tau }}_k\ge s\}|}$ is the number of those persons who are alive (and not being censored) at time${\displaystyle s}$.

Note that if${\displaystyle d(s)=0}$,${\displaystyle \hat{q}(s)=1}$. This implies that we can leave out from the product defining${\displaystyle \hat{S}(t)}$ all those terms where${\displaystyle d(s)=0}$. Then, letting be the times${\displaystyle s}$ when${\displaystyle d(s)>0}$,${\displaystyle d_i=d(t_i)}$ and${\displaystyle n_i=n(t_i)}$, we arrive at the form of the Kaplan–Meier estimator given at the beginning of the article:

${\displaystyle \hat{S}(t)=\prod _{i:t_i\le t}\left(1-\frac{d_i}{n_i}\right).}$

As opposed to the naive estimator, this estimator can be seen to use the available information more effectively: In the special case mentioned beforehand, when there are many early events recorded, the estimator will multiply many terms with a value below one and will thus take into account that the survival probability cannot be large.

Kaplan–Meier estimator can be derived frommaximum likelihood estimation of the discretehazard function.^[8] More specifically given${\displaystyle d_i}$ as the number of events and${\displaystyle n_i}$ the total individuals at risk at time ${\displaystyle t_i}$, discrete hazard rate${\displaystyle h_i}$ can be defined as the probability of an individual with an event at time ${\displaystyle t_i}$. Then survival rate can be defined as:

${\displaystyle S(t)=\prod _{i:t_i\le t}(1-h_i)}$

and the likelihood function for the hazard function up to time${\displaystyle t_i}$ is:

${\displaystyle \mathcal{L}(h_{j:j\le i}\mid d_{j:j\le i},n_{j:j\le i})=\prod _{j=1}^i{h}_j^{d_j}(1-h_j{)}^{n_j-d_j}(\genfrac{}{}{0ex}{}{n_j}{d_j})}$

therefore the log likelihood will be:

${\displaystyle \log (\mathcal{L})=\sum _{j=1}^i\left(d_j\log (h_j)+(n_j-d_j)\log (1-h_j)+\log (\genfrac{}{}{0ex}{}{n_j}{d_j})\right)}$

finding the maximum of log likelihood with respect to${\displaystyle h_i}$ yields:

${\displaystyle \frac{\mathrm{\partial }\log (\mathcal{L})}{\mathrm{\partial }{h}_i}=\frac{d_i}{{\hat{h}}_i}-\frac{n_i-d_i}{1-{\hat{h}}_i}=0\Rightarrow {\hat{h}}_i=\frac{d_i}{n_i}}$

where hat is used to denote maximum likelihood estimation. Given this result, we can write:

${\displaystyle \hat{S}(t)=\prod _{i:t_i\le t}\left(1-{\hat{h}}_i\right)=\prod _{i:t_i\le t}\left(1-\frac{d_i}{n_i}\right)}$

More generally (for continuous as well as discrete survival distributions), the Kaplan-Meier estimator may be interpreted as a nonparametric maximum likelihood estimator.^[9]

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 forcovariates; parametricsurvival models and the Coxproportional hazards model may be useful to estimate covariate-adjusted survival.

The Kaplan-Meier estimator is directly related to theNelson-Aalen estimator and both maximize theempirical likelihood.^[10]

The Kaplan–Meier estimator is astatistic, and several estimators are used to approximate itsvariance. One of the most common estimators is Greenwood's formula:^[11]

${\displaystyle \widehat{\mathrm{Var}}\left(\hat{S}(t)\right)=\hat{S}(t{)}^2\sum _{i:t_i\le t}\frac{d_i}{n_i(n_i-d_i)},}$

where${\displaystyle d_i}$ is the number of cases and${\displaystyle n_i}$ is the total number of observations, for.

In some cases, one may wish to compare different Kaplan–Meier curves. This can be done by thelog rank test, and theCox proportional hazards test.

Other statistics that may be of use with this estimator are pointwise confidence intervals,^[13] the Hall-Wellner band^[14] and the equal-precision band.^[15]

• Mathematica: the built-in functionSurvivalModelFit creates survival models.^[16]
• SAS: The Kaplan–Meier estimator is implemented in theproc lifetest procedure.^[17]
• R: the Kaplan–Meier estimator is available as part of thesurvival package.^[18][19][20]
• Stata: the commandsts returns the Kaplan–Meier estimator.^[21][22]
• Python: thelifelines andscikit-survival packages each include the Kaplan–Meier estimator.^[23][24]
• MATLAB: theecdf function with the'function','survivor' arguments can calculate or plot the Kaplan–Meier estimator.^[25]
• StatsDirect: The Kaplan–Meier estimator is implemented in theSurvival Analysis menu.^[26]
• SPSS: The Kaplan–Meier estimator is implemented in theAnalyze > Survival > Kaplan-Meier... menu.^[27]
• Julia: theSurvival.jl package includes the Kaplan–Meier estimator.^[28]
• Epi Info: Kaplan–Meier estimator survival curves and results for the log rank test are obtained with theKMSURVIVAL command.^[29]

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

• 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.
• Three evolving Kaplan–Meier curves onYouTube

• Estimator
• Actuarial science
• Survival analysis
• Reliability engineering

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