# kaplan.meier

##### Kaplan-Meier Estimator using Histogram Data

Compute the Kaplan-Meier estimator of a survival time distribution function, from histogram data

- Keywords
- spatial, nonparametric

##### Usage

`kaplan.meier(obs, nco, breaks, upperobs=0)`

##### Arguments

- obs
- vector of $n$ integers giving the histogram of all observations (censored or uncensored survival times)
- nco
- vector of $n$ integers giving the histogram of uncensored observations (those survival times that are less than or equal to the censoring time)
- breaks
- Vector of $n+1$ breakpoints which were used to form both histograms.
- upperobs
- Number of observations beyond the rightmost breakpoint, if any.

##### Details

This function is needed mainly for internal use in

Suppose $T_i$ are the survival times of individuals $i=1,\ldots,M$ with unknown distribution function $F(t)$ which we wish to estimate. Suppose these times are right-censored by random censoring times $C_i$. Thus the observations consist of right-censored survival times $\tilde T_i = \min(T_i,C_i)$ and non-censoring indicators $D_i = 1{T_i \le C_i}$ for each $i$.

If the number of observations $M$ is large, it is efficient to
use histograms.
Form the histogram `obs`

of all observed times $\tilde T_i$.
That is, `obs[k]`

counts the number of values
$\tilde T_i$ in the interval
`(breaks[k],breaks[k+1]]`

for $k > 1$
and `[breaks[1],breaks[2]]`

for $k = 1$.
Also form the histogram `nco`

of all uncensored times,
i.e. those $\tilde T_i$ such that $D_i=1$.
These two histograms are the arguments passed to `kaplan.meier`

.
The vectors `km`

and `lambda`

returned by `kaplan.meier`

are (histogram approximations to) the Kaplan-Meier estimator
of $F(t)$ and its hazard rate $\lambda(t)$.
Specifically, `km[k]`

is an estimate of
`F(breaks[k+1])`

, and `lambda[k]`

is an estimate of
the average of $\lambda(t)$ over the interval
`(breaks[k],breaks[k+1])`

.

The histogram breaks must include $0$.
If the histogram breaks do not span the range of the observations,
it is important to count how many survival times
$\tilde T_i$ exceed the rightmost breakpoint,
and give this as the value `upperobs`

.

##### Value

- A list with two elements:
km Kaplan-Meier estimate of the survival time c.d.f. $F(t)$ lambda corresponding Nelson-Aalen estimate of the hazard rate $\lambda(t)$ - These are numeric vectors of length $n$.

##### See Also

*Documentation reproduced from package spatstat, version 1.11-8, License: GPL version 2 or newer*