Compute the Kaplan-Meier and Reduced Sample estimators of a survival time distribution function, using histogram techniques
km.rs(o, cc, d, breaks)A list with five elements
Reduced-sample estimate of the survival time c.d.f. \(F(t)\)
Kaplan-Meier estimate of the survival time c.d.f. \(F(t)\)
corresponding Nelson-Aalen estimate of the hazard rate \(\lambda(t)\)
values of \(t\) for which \(F(t)\) is estimated
the breakpoints vector
vector of observed survival times
vector of censoring times
vector of non-censoring indicators
Vector of breakpoints to be used to form histograms.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au
and Rolf Turner r.turner@auckland.ac.nz
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 \(i=1,\ldots,M\) 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.
reduced.sample,
kaplan.meier