2015-09-12

Kaplan-Meier estimator

The Kaplan-Meier estimator,^{1 2} 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,³ 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 34,000 times since its publication.⁴

Formulation

Let $S(t)$S(t) be the probability that a member from a given population will have a lifetime exceeding time, $t$t. For a sample of size $N$N from this population, let the observed times until death of the $N$N sample members be

$t_1 t_2 t_3 t_N.$t1​≤t2​≤t3​≤⋯≤tN​.

Corresponding to each $t_i$ti​ is $n_i$ni​, the number “at risk” just prior to time $t_i$ti​, and $d_i$di​, the number of deaths at time $t_i$ti​.

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.

$i$i	1	2
$t_i$ti​	3	11
$d_i$di​	1	2
$n_i$ni​	10	8

The Kaplan-Meier estimator is the nonparametric maximum likelihood estimate of $S(t)$S(t), where the maximum is taken over the set of all piecewise constant survival curves with breakpoints at the event times $t_i$ti​. It is a product of the form

$S(t) = _{i=1}^{t} .$S^(t)=∏i=1t​ni​ni​−di​​.

When there is no censoring, $n_i$ni​ is just the number of survivors just prior to time $t_i$ti​. With censoring, $n_i$ni​ 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.⁵

There is an alternative definition that is sometimes used, namely

$(t) = _{i=1}^{t} .$S^(t)=∏i=1t​ni​ni​−di​​.

The two definitions differ only at the observed event times. The latter definition is right-continuous whereas the former definition is left-continuous.

Let $T$T be the random variable that measures the time of failure and let $F(t)$F(t) be its cumulative distribution function. Note that

$S(t) = P[T>t] = 1-P[T t] = 1-F(t). ,$S(t)=P[T>t]=1−P[T≤t]=1−F(t).

Consequently, the right-continuous definition of $(t)$S^(t) may be preferred in order to make the estimate compatible with a right-continuous estimate of $F(t)$F(t).

---

1. Kaplan, E. L.; Meier, P. (1958). “Nonparametric estimation from incomplete observations”. J. Amer. Statist. Assn. 53 (282): 457-481. JSTOR 2281868.↩

2. 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.↩

3. Meyer, Bruce D. (1990). “Unemployment Insurance and Unemployment Spells”. Econometrica 58 (4): 757-782. doi:10.2307/2938349.↩

4. “Paul Meier, 1924-2011”. Chicago Tribune. August 18, 2011.↩

5. Costella, John P. (2010). “A simple alternative to Kaplan-Meier for survival curves” (PDF). Unpublished.↩

mathematics science statistics

---

Previous post

hello, world! hello, world! hello, world! hello, world! hello, world! hello, world!1 hello, world! https://vimeo.com/56267591 A “Hello

Next post

Picture of the day Utah, USA. A woman looks at a damaged vehicle swept away during a flash flood in Hildale. (Rick Bowmer/AP)