km.rs.Rd
\name{km.rs}
\alias{km.rs}
\title{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{
\item{o}{vector of observed survival times
}
\item{cc}{vector of censoring times
}
\item{d}{vector of non-censoring indicators
}
\item{breaks}{Vector of breakpoints to be used to form histograms.
}
}
\value{
A list with five elements
\item{rs}{Reduced-sample estimate of the survival time c.d.f. \eqn{F(t)}
}
\item{km}{Kaplan-Meier estimate of the survival time c.d.f. \eqn{F(t)}
}
\item{hazard}{corresponding Nelson-Aalen estimate of the
hazard rate \eqn{\lambda(t)}{lambda(t)}
}
\item{r}{values of \eqn{t} for which \eqn{F(t)} is estimated
}
\item{breaks}{the breakpoints vector
}
}
\details{
This function is needed mainly for internal use in \pkg{spatstat},
but may be useful in other applications where you want to form the
Kaplan-Meier estimator from a huge dataset.
Suppose \eqn{T_i}{T[i]} are the survival times of individuals
\eqn{i=1,\ldots,M} with unknown distribution function \eqn{F(t)}
which we wish to estimate. Suppose these times are right-censored
by random censoring times \eqn{C_i}{C[i]}.
Thus the observations consist of right-censored survival times
\eqn{\tilde T_i = \min(T_i,C_i)}{T*[i] = min(T[i],C[i])}
and non-censoring indicators
\eqn{D_i = 1\{T_i \le C_i\}}{D[i] = 1(T[i] <= C[i])}
for each \eqn{i}.
The arguments to this function are
vectors \code{o}, \code{cc}, \code{d}
of observed values of \eqn{\tilde T_i}{T*[i]}, \eqn{C_i}{C[i]}
and \eqn{D_i}{D[i]} respectively.
The function computes histograms and forms the reduced-sample
and Kaplan-Meier estimates of \eqn{F(t)} by
invoking the functions \code{\link{kaplan.meier}}
and \code{\link{reduced.sample}}.
This is efficient if the lengths of \code{o}, \code{cc}, \code{d}
(i.e. the number of observations) is large.
The vectors \code{km} and \code{hazard} returned by \code{kaplan.meier}
are (histogram approximations to) the Kaplan-Meier estimator
of \eqn{F(t)} and its hazard rate \eqn{\lambda(t)}{lambda(t)}.
Specifically, \code{km[k]} is an estimate of
\code{F(breaks[k+1])}, and \code{lambda[k]} is an estimate of
the average of \eqn{\lambda(t)}{lambda(t)} over the interval
\code{(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
\code{(breaks[k],breaks[k+1])} contains only one possible value of
the survival time.
The vector \code{rs} is the reduced-sample estimator,
\code{rs[k]} being the reduced sample estimate of \code{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.
}
\seealso{
\code{\link{reduced.sample}},
\code{\link{kaplan.meier}}
}
\author{\adrian
and \rolf
}
\keyword{spatial}
\keyword{nonparametric}