Compute the Reduced Sample estimator of a survival time distribution function, from histogram data

`reduced.sample(nco, cen, ncc, show=FALSE, uppercen=0)`

nco

vector of counts giving the histogram of uncensored observations (those survival times that are less than or equal to the censoring time)

cen

vector of counts giving the histogram of censoring times

ncc

vector of counts giving the histogram of censoring times for the uncensored observations only

uppercen

number of censoring times greater than the rightmost histogram breakpoint (if there are any)

show

Logical value controlling the amount of detail returned by the function value (see below)

If `show = FALSE`

, a numeric vector giving the values of
the reduced sample estimator.
If `show=TRUE`

, a list with three components which are
vectors of equal length,

Reduced sample estimate of the survival time c.d.f. \(F(t)\)

numerator of the reduced sample estimator

denominator of the reduced sample estimator

This function is needed mainly for internal use in spatstat, but may be useful in other applications where you want to form the reduced sample estimator from a huge dataset.

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 `cen`

of all censoring times \(C_i\).
That is, `obs[k]`

counts the number of values
\(C_i\) 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 \(D_i=1\),
and the histogram of all censoring times for which the survival time
is uncensored,
i.e. those \(C_i\) such that \(D_i=1\).
These three histograms are the arguments passed to `kaplan.meier`

.

The return value `rs`

is the reduced-sample estimator
of the distribution function \(F(t)\). Specifically,
`rs[k]`

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

.
The value is exact, i.e. the use of histograms does not introduce any
approximation error.

Note that, for the results to be valid, either the histogram breaks
must span the censoring times, or the number of censoring times
that do not fall in a histogram cell must have been counted in
`uppercen`

.