R: Kaplan-Meier and Reduced Sample Estimator using Histograms

km.rs {spatstat} | R Documentation

## Kaplan-Meier and Reduced Sample Estimator using Histograms

### Description

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

### 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 spatstat, but may be useful in other applications where you want to form the Kaplan-Meier estimator from a huge dataset.

Suppose T[i] are the survival times of individuals i=1,…,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 T*[i] = min(T[i],C[i]) and non-censoring indicators D[i] = 1(T[i] <= C[i]) for each i.

The arguments to this function are vectors `o`, `cc`, `d` of observed values of 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

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au

and Rolf Turner r.turner@auckland.ac.nz

### See Also

`reduced.sample`, `kaplan.meier`

[Package spatstat version 1.54-0 Index]