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