# Kaplan-Meier Estimator

The Kaplan-Meier estimator is a nonparametric estimator of the survivor function, i.e. the probability of survival beyond a given time.

The estimate is given by

\[\hat{S}(t) = \prod_{i: t_i < t} \left( 1 - \frac{d_i}{n_i} \right)\]

where $d_i$ is the number of observed events at time $t_i$ and $n_i$ is the number of subjects at risk for the event just before time $t_i$.

The pointwise standard error of the log of the survivor function can be computed using Greenwood's formula:

\[\text{SE}(\log \hat{S}(t)) = \sqrt{\sum_{i: t_i < t} \frac{d_i}{n_i (n_i - d_i)}}\]

## API

`Survival.KaplanMeier`

— Type`KaplanMeier{S,T}`

An immutable type containing survivor function estimates computed using the Kaplan-Meier method. The type has the following fields:

`events`

: An`EventTable`

summarizing the times and events used to compute the estimates. The time values are of type`T`

.`survival`

: Estimate of the survival probability at each time. Values are of type`S`

.`stderr`

: Standard error of the log survivor function at each time. Values are of type`S`

.

Use `fit(KaplanMeier, ...)`

to compute the estimates as `Float64`

values and construct this type. Alternatively, `fit(KaplanMeier{S}, ...)`

may be used to request a particular value type `S`

for the estimates.

`StatsAPI.fit`

— Method`fit(KaplanMeier, times, status) -> KaplanMeier`

Given a vector of times to events and a corresponding vector of indicators that denote whether each time is an observed event or is right censored, compute the Kaplan-Meier estimate of the survivor function.

`StatsAPI.confint`

— Method`confint(km::KaplanMeier; level=0.05)`

Compute the pointwise log-log transformed confidence intervals for the survivor function as a vector of tuples.

## References

Kaplan, E. L., and Meier, P. (1958).

*Nonparametric Estimation from Incomplete Observations*. Journal of the American Statistical Association, 53(282), 457-481. doi:10.2307/2281868Greenwood, M. (1926).

*A Report on the Natural Duration of Cancer*. Reports on Public Health and Medical Subjects. London: Her Majesty's Stationery Office. 33, 1-26.