Kaplan–Meier estimator

Builds on Survival function Censoring Estimator Empirical distribution function

Definition

Let $(\tilde t_1, \delta_1), \dots, (\tilde t_n, \delta_n)$ be right-censored observations of a lifetime. Here $\tilde t_i \ge 0$ is the observed time, and $\delta_i = 1$ if it is an event and $\delta_i = 0$ if it is censored. Let $t_{(1)} < \dots < t_{(m)}$ be the distinct times $\tilde t_i$ with $\delta_i = 1$ . For each $j$ , let $d_j = \#\set{i : \tilde t_i = t_{(j)}, \delta_i = 1}$ be the number of events at $t_{(j)}$ and $n_j = \#\set{i : \tilde t_i \ge t_{(j)}}$ the number at risk just before $t_{(j)}$ . The Kaplan–Meier estimator of the survival function is $\hat S(t) = \prod_{j \,:\, t_{(j)} \le t} \left(1 - \frac{d_j}{n_j}\right), \qquad t \ge 0,$ with the empty product equal to $1$ .

Remarks

It is also called the product-limit estimator. Its variance is usually estimated by Greenwood's formula $\hat S(t)^2 \sum_{j : t_{(j)} \le t} d_j / (n_j (n_j - d_j))$ . If no observation is censored, $\hat S(t) = 1 - \hat F_n(t)$ , where $\hat F_n$ is the empirical distribution function.

Example: take the observations $2, 3^+, 5, 7$ , where $3^+$ is censored. Then $(n_j, d_j)$ is $(4, 1)$ at $2$ , $(2, 1)$ at $5$ and $(1, 1)$ at $7$ . So $\hat S = 1$ on $[0, 2)$ , $3/4$ on $[2, 5)$ , $3/8$ on $[5, 7)$ and $0$ from $7$ on.

Non-example: dropping the censored observation and using $1 - \hat F_3$ on $2, 5, 7$ gives $1/3$ on $[5, 7)$ instead of $3/8$ . This ignores the information that the censored subject survived past $3$ , and it is biased downwards in general.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`W94W93RHF5`
URI	`m2://W94W93RHF5`
Kind	definition
Writing	W0 stub
MSC	62N02, 62G05, 62P10
Defines	Kaplan–Meier estimator
Uses	Survival function, Censoring, Estimator, Empirical distribution function

Kaplan–Meier estimator

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here