Kaplan–Meier estimator

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 state 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 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.{{cite journal |last1=Rich |first1=Jason T. |last2=Neely |first2=J. Gail |last3=Paniello |first3=Randal C. |last4=Voelker |first4=Courtney C. J. |last5=Nussenbaum |first5=Brian |last6=Wang |first6=Eric W. |title=A practical guide to understanding Kaplan-Meier curves |journal=Otolaryngology–Head and Neck Surgery |date=September 2010 |volume=143 |issue=3 |pages=331–336 |doi=10.1016/j.otohns.2010.05.007 |pmid=20723767 |pmc=3932959 }}

Let $\backslash tau\backslash ge\; 0$ be a random variable as the time that passes between the start of the possible exposure period, $t\_0$, and the time that the event of interest takes place, $t\_1$. As indicated above, the goal is to estimate the survival function $S$ underlying $\backslash tau$. Recall that this function is defined as

:$S(t)\; =\; \backslash operatorname\{Prob\}(\backslash tau\; >\; t\; )$, where $t=0,1,\backslash dots$ is the time.

Let $\backslash tau\_1,\backslash dots,\backslash tau\_n\backslash ge\; 0$ be independent, identically distributed random variables, whose common distribution is that of $\backslash tau$: $\backslash tau\_j$ is the random time when some event $j$ happened. The data available for estimating $S$ is not $(\backslash tau\_j)\_\{j=1,\backslash dots,n\}$, but the list of pairs $(\backslash ,\; (\; \backslash tilde\; \backslash tau\_j,\; c\_j\; )\backslash ,\; )\_\{j=1,\backslash dots,n\}$ where for $j\backslash in\; [n]\; :=\; \backslash \{1,2,\backslash dots,n\backslash \}$, $c\_j\backslash ge\; 0$ is a fixed, deterministic integer, the censoring time of event $j$ and $\backslash tilde\; \backslash tau\_j\; =\; \backslash min(\backslash tau\_j,c\_j)$. In particular, the information available about the timing of event $j$ is whether the event happened before the fixed time $c\_j$ and if so, then the actual time of the event is also available. The challenge is to estimate $S(t)$ 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 called hazard, or mortality 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 $k\backslash in\; [n]:=\backslash \{1,\backslash dots,n\backslash \}$ and let $t>0$. A basic argument shows that the following proposition holds:

:Proposition 1: If the censoring time $c\_k$ of event $k$ exceeds $t$ ($c\_k\backslash ge\; t$), then $\backslash tilde\; \backslash tau\_k\backslash ge\; t$ if and only if $\backslash tau\_k\backslash ge\; t$.

Let $k$ be such that $c\_k\backslash ge\; t$. It follows from the above proposition that

: $\backslash operatorname\{Prob\}(\backslash tau\_k\backslash ge\; t)\; =\; \backslash operatorname\{Prob\}(\backslash tilde\; \backslash tau\_k\backslash ge\; t).$

Let $$

X_k = \mathbb{I}(\tilde \tau_k\ge t)

and consider only those $$

k\in C(t) := \{ k \, :\, c_k \ge t\}

, i.e. the events for which the outcome was not censored before time $t$. Let $$

m(t)=|C(t)|

be the number of elements in $$

C(t)

. Note that the set $$

C(t)

is not random and so neither is $$

m(t)

. Furthermore, $$

(X_k)_{k\in C(t)}

is a sequence of independent, identically distributed Bernoulli random variables with common parameter $$

S(t)=\operatorname{Prob}(\tau\ge t)

. Assuming that $$

m(t)>0

, this suggests to estimate $S(t)$ using

: $$

\hat S_\text{naive}(t)

= \frac{1}{m(t)} \sum_{k:c_k\ge t} X_k

= \frac

= \frac

where the second equality follows because $\backslash tilde\; \backslash tau\_k\backslash ge\; t$ implies $c\_k\backslash ge\; t$, while the last equality is simply a change of notation.

The quality of this estimate is governed by the size of $m(t)$. This can be problematic when $m(t)$ 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 $t$. Intuitively, these observations still contain information about $S(t)$: For example, when for many events with ,

By elementary calculations,

: $\backslash begin\{align\}$

S(t) & = \operatorname{Prob}(\tau > t\mid\tau > t-1)\operatorname{Prob}(\tau > t-1) \\[4pt]

& = (1-\operatorname{Prob}(\tau\le t\mid\tau > t-1)) \operatorname{Prob}(\tau > t-1)\\[4pt]

& = (1-\operatorname{Prob}(\tau=t\mid\tau \ge t)) \operatorname{Prob}(\tau > t-1) \\[4pt]

& = q(t) S(t-1)\,,

\end{align}

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

: $q(t)\; =\; 1-\backslash operatorname\{Prob\}(\backslash tau=t\backslash mid\backslash tau\backslash ge\; t).$

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

: $S(t)\; =\; q(t)\; q(t-1)\; \backslash cdots\; q(0).$

Note that here $q(0)\; =\; 1-\backslash operatorname\{Prob\}(\backslash tau=0\backslash mid\backslash tau\; >\; -1)\; =\; 1-\backslash operatorname\{Prob\}(\backslash tau=0)$.

The Kaplan–Meier estimator can be seen as a "plug-in estimator" where each $q(s)$

is estimated based on the data and the estimator of $S(t)$

is obtained as a product of these estimates.

It remains to specify how $q(s)=1-\backslash operatorname\{Prob\}(\backslash tau=s\backslash mid\backslash tau\backslash ge\; s)$ is to be estimated. By Proposition 1, for any $k\backslash in\; [n]$ such that $c\_k\backslash ge\; s$, $\backslash operatorname\{Prob\}(\backslash tau=s)\; =\; \backslash operatorname\{Prob\}(\backslash tilde\; \backslash tau\_k=s)$ and $\backslash operatorname\{Prob\}(\backslash tau\backslash ge\; s)\; =\; \backslash operatorname\{Prob\}(\backslash tilde\; \backslash tau\_k\backslash ge\; s)$ both hold. Hence, for any $$

k\in [n] such that $c\_k\backslash ge\; s$,

: $\backslash operatorname\{Prob\}(\backslash tau=s|\backslash tau\backslash ge\; s)\; =\; \backslash operatorname\{Prob\}(\backslash tilde\; \backslash tau\_k=s)/\backslash operatorname\{Prob\}(\backslash tilde\; \backslash tau\_k\backslash ge\; s).$

By a similar reasoning that lead to the construction of the naive estimator above, we arrive at the estimator

: $\backslash hat\; q(s)$

= 1 - \frac

= 1 - \frac

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

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

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

Note that if $d(s)=0$, $\backslash hat\; q(s)=1$. This implies that we can leave out from the product defining $\backslash hat\; S(t)$ all those terms where $d(s)=0$. Then, letting

: $\backslash hat\; S(t)\; =\; \backslash prod\_\{i:t\_i\backslash le\; t\}\; \backslash left(1-\backslash frac\{d\_i\}\{n\_i\}\backslash right).$

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 from maximum likelihood estimation of the discrete hazard function.{{Cite web|url=https://web.stanford.edu/~lutian/coursepdf/STAT331unit3.pdf|title=STAT331 Unit 3, Kaplan-Meier (KM) Estimator|date=January 13, 2014|first=Lu|last=Tian|publisher=Stanford University|accessdate=12 May 2023}} More specifically given $d\_i$ as the number of events and $n\_i$ the total individuals at risk at time $t\_i$, discrete hazard rate $h\_i$ can be defined as the probability of an individual with an event at time $t\_i$. Then survival rate can be defined as:

: $S(t)\; =\; \backslash prod\backslash limits\_\{i:\backslash \; t\_i\backslash le\; t\}\; (1-h\_i)$

and the likelihood function for the hazard function up to time $t\_i$ is:

: $\backslash mathcal\{L\}(h\_\{j:\; j\backslash le\; i\}\backslash mid\; d\_\{j:\; j\backslash le\; i\},n\_\{j:\; j\backslash le\; i\})\; =\; \backslash prod\_\{j=1\}^i\; h\_j^\{d\_j\}(1-h\_j)^\{n\_j-d\_j\}\; \{n\_j\; \backslash choose\; d\_j\}$

therefore the log likelihood will be:

: $\backslash log(\backslash mathcal\{L\})\; =\; \backslash sum\_\{j=1\}^i\; \backslash left(d\_j\backslash log(h\_j)+(n\_j-d\_j)\backslash log(1-h\_j)\; +\; \backslash log\; \{n\_j\; \backslash choose\; d\_j\}\; \backslash right)$

finding the maximum of log likelihood with respect to $h\_i$ yields:

: $\backslash frac\{\backslash partial\; \backslash log(\backslash mathcal\{L\})\}\{\backslash partial\; h\_i\}\; =\; \backslash frac\{d\_i\}\{\backslash widehat\{h\}\_i\}-\backslash frac\{n\_i-d\_i\}\{1-\backslash widehat\{h\}\_i\}\; =\; 0\; \backslash Rightarrow\; \backslash widehat\{h\}\_i=\backslash frac\{d\_i\}\{n\_i\}$

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

: $\backslash widehat\; S(t)\; =\; \backslash prod\backslash limits\_\{i:\backslash \; t\_i\backslash le\; t\}\; \backslash left(1\; -\; \backslash widehat\{h\}\_i\backslash right)\; =\; \backslash prod\backslash limits\_\{i:\backslash \; t\_i\backslash le\; t\}\; \backslash left(1\; -\; \backslash frac\{d\_i\}\{n\_i\}\backslash right)$

More generally (for continuous as well as discrete survival distributions), the Kaplan-Meier estimator may be interpreted as a nonparametric maximum likelihood estimator.{{cite book |last1=Andersen |first1=Per Kragh |last2=Borgan |first2=Ornulf |last3=Gill |first3=Richard D. |last4=Keiding |first4=Niels |title=Statistical models based on counting processes |date=1993 |publisher=Springer-Verlag |location=New York |isbn=0-387-97872-0}}

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.

The Kaplan-Meier estimator is directly related to the Nelson-Aalen estimator and both maximize the empirical likelihood.Zhou, M. (2015). Empirical Likelihood Method in Survival Analysis (1st ed.). Chapman and Hall/CRC. https://doi.org/10.1201/b18598, https://books.google.com/books?id=9-b5CQAAQBAJ&dq=Does+the+Nelson%E2%80%93Aalen+estimator+construct+an+empirical+likelihood%3F&pg=PA7

The Kaplan–Meier estimator is a statistic, and several estimators are used to approximate its variance. One of the most common estimators is Greenwood's formula:{{cite book |last1=Greenwood |first1=Major |date=1926 |title=A report on the natural duration of cancer |series=Issue 33 of Reports on public health and medical subjects |publisher=HMSO |oclc=14713088 }}

:$\backslash widehat\{\backslash operatorname\{Var\}\}\; \backslash left(\; \backslash widehat\; S(t)\; \backslash right)\; =\; \backslash widehat\; S(t)^2\; \backslash sum\_\{i:\backslash \; t\_i\backslash le\; t\}\; \backslash frac\{d\_i\}\{n\_i(n\_i-d\_i)\},$

where $d\_i$ is the number of cases and $n\_i$ is the total number of observations, for .

{{Hidden|titlestyle=background-color:#f2dfce; color:black;|contentstyle=border:1px #C4C3D0 solid;|header=For a 'sketch' of the mathematical derivation of the equation above, click on "show" to reveal|content=

Greenwood's formula is derived{{Cite web|url=https://www.math.wustl.edu/%7Esawyer/handouts/greenwood.pdf|title=The Greenwood and Exponential Greenwood Confidence Intervals in Survival Analysis|accessdate=12 May 2023}}{{self-published inline|date=November 2021}} by noting that probability of getting $d\_i$ failures out of $n\_i$ cases follows a binomial distribution with failure probability $h\_i$. As a result for maximum likelihood hazard rate $\backslash widehat\{h\}\_i\; =\; d\_i/n\_i$ we have $E\backslash left(\backslash widehat\{h\}\_i\backslash right)\; =\; h\_i$

and $\backslash operatorname\{Var\}\backslash left(\backslash widehat\{h\}\_i\backslash right)\; =\; h\_i(1-h\_i)/n\_i$. To avoid dealing with multiplicative probabilities we compute variance of logarithm of $\backslash widehat\; S(t)$ and will use the delta method to convert it back to the original variance:

: $\backslash begin\{align\}$

\operatorname{Var}\left(\log \widehat{S}(t)\right) &\sim \frac{1}{{\widehat{S}(t)}^2} \operatorname{Var} \left(\widehat{S}(t)\right) \Rightarrow \\

\operatorname{Var}\left( \widehat{S}(t)\right) &\sim {{\widehat{S}(t)}^2}\operatorname{Var}\left(\log\widehat{S}(t) \right)

\end{align}

using martingale central limit theorem, it can be shown that the variance of the sum in the following equation is equal to the sum of variances:

: $\backslash log\backslash widehat\; S(t)\; =\; \backslash sum\backslash limits\_\{i:\backslash \; t\_i\backslash le\; t\}\; \backslash log\backslash left(1\; -\; \backslash widehat\{h\}\_i\backslash right)$

as a result we can write:

: $\backslash begin\{align\}$

\operatorname{Var}( \widehat{S}(t)) &\sim

{{\widehat{S}(t)}^2}\operatorname{Var}\left(\sum_{i:\ t_i\le t} \log\left(1 - \widehat{h}_i\right)\right) \\

&\sim

{{\widehat{S}(t)}^2}\sum\limits_{i:\ t_i\le t} \operatorname{Var}\left(\log\left(1 - \widehat{h}_i\right)\right)

\end{align}

using the delta method once more:

: $\backslash begin\{align\}$

\operatorname{Var}( \widehat{S}(t)) &\sim

{{\widehat{S}(t)}^2}\sum_{i:\ t_i\le t}\left(\frac{\partial \log\left(1 - \widehat{h}_i\right)}{\partial \widehat{h}_i}\right)^2 \operatorname{Var}\left(\widehat{h}_i\right)\\

&={{\widehat{S}(t)}^2}\sum_{i:\ t_i\le t}\left(\frac{1}{1-\widehat{h}_i}\right)^2\frac{\widehat{h}_i \left( 1-\widehat{h}_i \right)}{n_i} \\

&= {{\widehat{S}(t)}^2}\sum_{i:\ t_i\le t} \frac{\widehat{h}_i}{n_i\left(1-\widehat{h}_i\right)} \\

&= {{\widehat{S}(t)}^2}\sum_{i:\ t_i\le t} \frac{d_i}{n_i(n_i-d_i)}

\end{align}

as desired.

----

}}

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

Other statistics that may be of use with this estimator are pointwise confidence intervals,{{cite journal |last1=Fay |first1=Michael P. |last2=Brittain |first2=Erica H.|author2-link=Erica Brittain |last3=Proschan |first3=Michael A. |title=Pointwise confidence intervals for a survival distribution with small samples or heavy censoring |journal=Biostatistics |date=1 September 2013 |volume=14 |issue=4 |pages=723–736 |doi=10.1093/biostatistics/kxt016 |pmid=23632624 |pmc=3769999 }} the Hall-Wellner band{{cite journal |last1=Hall |first1=W. J. |last2=Wellner |first2=Jon A. |title=Confidence bands for a survival curve from censored data |journal=Biometrika |date=1980 |volume=67 |issue=1 |pages=133–143 |doi=10.1093/biomet/67.1.133 }} and the equal-precision band.{{cite journal |last1=Nair |first1=Vijayan N. |title=Confidence Bands for Survival Functions With Censored Data: A Comparative Study |journal=Technometrics |date=August 1984 |volume=26 |issue=3 |pages=265–275 |doi=10.1080/00401706.1984.10487964 }}

• Mathematica: the built-in function SurvivalModelFit creates survival models.{{Cite web|url=http://reference.wolfram.com/language/ref/SurvivalModelFit.html|title=Survival Analysis – Mathematica SurvivalModelFit|website=wolfram.com|access-date=2017-08-14}}
• SAS: The Kaplan–Meier estimator is implemented in the proc lifetest procedure.{{Cite web|url=https://support.sas.com/documentation/cdl/en/statug/68162/HTML/default/viewer.htm#statug_lifetest_overview.htm|title=SAS/STAT(R) 14.1 User's Guide|website=support.sas.com|accessdate=12 May 2023}}
• R: the Kaplan–Meier estimator is available as part of the survival package.{{cite web |last=Therneau |first=Terry M. |date=2022-08-09 |title=survival: Survival Analysis |url=https://cran.r-project.org/web/packages/survival/index.html |access-date=2022-11-30 |work=The Comprehensive R Archive Network}}{{cite book |first=Frans |last=Willekens |chapter=Statistical Packages for Multistate Life History Analysis |series=Use R! |title=Multistate Analysis of Life Histories with R |publisher=Springer |year=2014 |isbn=978-3-319-08383-4 |pages=135–153 |chapter-url=https://books.google.com/books?id=Cd2CBAAAQBAJ&pg=PA135 |doi=10.1007/978-3-319-08383-4_6 }}{{cite book |first1=Ding-Geng |last1=Chen |first2=Karl E. |last2=Peace |title=Clinical Trial Data Analysis Using R |publisher=CRC Press |year=2014 |isbn= 9781439840214|pages=99–108 |url=https://books.google.com/books?id=fGnRBQAAQBAJ&pg=PA99 }}
• Stata: the command sts returns the Kaplan–Meier estimator.{{cite web |title=sts — Generate, graph, list, and test the survivor and cumulative hazard functions |work=Stata Manual |url=https://www.stata.com/manuals15/ststs.pdf }}{{cite book |first=Mario |last=Cleves |title=An Introduction to Survival Analysis Using Stata |location=College Station |publisher=Stata Press |edition=Second |year=2008 |isbn=978-1-59718-041-2 |pages=93–107 |url=https://books.google.com/books?id=xttbn0a-QR8C&pg=PA93 }}
• Python: the lifelines and scikit-survival packages each include the Kaplan–Meier estimator.{{Cite web|url=https://lifelines.readthedocs.io/en/latest/|title=lifelines — lifelines 0.27.7 documentation|website=lifelines.readthedocs.io|accessdate=12 May 2023}}{{Cite web|url=https://scikit-survival.readthedocs.io/en/stable/api/generated/sksurv.nonparametric.kaplan_meier_estimator.html|title=sksurv.nonparametric.kaplan_meier_estimator — scikit-survival 0.20.0|website=scikit-survival.readthedocs.io|accessdate=12 May 2023}}
• MATLAB: the ecdf function with the 'function','survivor' arguments can calculate or plot the Kaplan–Meier estimator.{{Cite web|url=http://mathworks.com/help/stats/ecdf.html|title=Empirical cumulative distribution function – MATLAB ecdf|website=mathworks.com|access-date=2016-06-16}}
• StatsDirect: The Kaplan–Meier estimator is implemented in the Survival Analysis menu.{{cite web|website=statsdirect.co.uk

|url=https://www.statsdirect.co.uk/help/Default.htm#survival_analysis/kaplan_meier.htm |title=Kaplan-Meier Survival Estimates|access-date=12 May 2023}}

• SPSS: The Kaplan–Meier estimator is implemented in the Analyze > Survival > Kaplan-Meier... menu.{{Cite web|url=https://statistics.laerd.com/spss-tutorials/kaplan-meier-using-spss-statistics.php|title = Kaplan-Meier method in SPSS Statistics {{pipe}} Laerd Statistics}}
• Julia: the Survival.jl package includes the Kaplan–Meier estimator.{{Cite web|url=https://juliastats.org/Survival.jl/latest/km/|title = Kaplan-Meier · Survival.jl}}
• Epi Info: Kaplan–Meier estimator survival curves and results for the log rank test are obtained with the KMSURVIVAL command.{{Cite web|url=https://www.cdc.gov/epiinfo/user-guide/command-reference/analysis-commands-kmsurvival.html|title =Epi Info™ User Guide - Command Reference - Analysis Commands: KMSURVIVAL|accessdate=30 Oct 2023}}

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

{{Reflist|30em}}

• {{cite book |last1=Aalen |first1=Odd |first2=Ornulf |last2=Borgan |first3=Hakon |last3=Gjessing |title=Survival and Event History Analysis: A Process Point of View |publisher=Springer |year=2008 |isbn=978-0-387-68560-1 |pages=90–104 }}
• {{cite book |last=Greene |first=William H. |author-link=William Greene (economist) |chapter=Nonparametric and Semiparametric Approaches |title=Econometric Analysis |publisher=Prentice-Hall |edition=Seventh |year=2012 |isbn=978-0-273-75356-8 |pages=909–912 |chapter-url=https://books.google.com/books?id=-WFPYgEACAAJ&pg=PA909 }}
• {{cite book |last1=Jones |first1=Andrew M. |first2=Nigel |last2=Rice |first3=Teresa Bago |last3=D'Uva |first4=Silvia |last4=Balia |chapter=Duration Data |title=Applied Health Economics |location=London |publisher=Routledge |year=2013 |isbn=978-0-415-67682-3 |pages=139–181 |chapter-url=https://books.google.com/books?id=7tdcCol9mNEC&pg=PA141 }}
• {{cite book |first1=Judith B. |last1=Singer |first2=John B. |last2=Willett |title=Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence |location=New York |publisher=Oxford University Press |year=2003 |isbn=0-19-515296-4 |pages=483–487 |url=https://books.google.com/books?id=PpnA1M8VwR8C&pg=PA483 }}

• {{cite web |url= http://www.cancerguide.org/scurve_km.html |title= Survival Curves: Accrual and The Kaplan–Meier Estimate |first= Steve |last= Dunn |website= Cancer Guide |series= Statistics |date= 2002 }}
• {{youTube|5C_zzD1pOAg|Three evolving Kaplan–Meier curves}}

{{Statistics|analysis}}

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