# km.rs

##### Kaplan-Meier and Reduced Sample Estimator using Histograms

Compute the Kaplan-Meier and Reduced Sample estimators of a survival time distribution function, using histogram techniques

- Keywords
- spatial, nonparametric

##### Usage

`km.rs(o, cc, d, breaks)`

##### Arguments

- o
- vector of observed survival times
- cc
- vector of censoring times
- d
- vector of non-censoring indicators
- breaks
- Vector of breakpoints to be used to form histograms.

##### 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 $i$.

The arguments to this function are
vectors `o`

, `cc`

, `d`

of observed values of $\tilde T_i$, $C_i$
and $D_i$ respectively.
The function computes histograms and forms the reduced-sample
and Kaplan-Meier estimates of $F(t)$ by
invoking the functions `kaplan.meier`

and `reduced.sample`

.
This is efficient if the lengths of `o`

, `cc`

, `d`

(i.e. the number of observations) is large.

The vectors `km`

and `hazard`

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])`

. This approximation is exact only if the
survival times are discrete and the
histogram breaks are fine enough to ensure that each interval
`(breaks[k],breaks[k+1])`

contains only one possible value of
the survival time.

The vector `rs`

is the reduced-sample estimator,
`rs[k]`

being the reduced sample estimate of `F(breaks[k+1])`

.
This value is exact, i.e. the use of histograms does not introduce any
approximation error in the reduced-sample estimator.

##### Value

- A list with five elements
rs Reduced-sample estimate of the survival time c.d.f. $F(t)$ km Kaplan-Meier estimate of the survival time c.d.f. $F(t)$ hazard corresponding Nelson-Aalen estimate of the hazard rate $\lambda(t)$ r values of $t$ for which $F(t)$ is estimated breaks the breakpoints vector

##### See Also

*Documentation reproduced from package spatstat, version 1.41-1, License: GPL (>= 2)*