spatstat (version 1.64-1) 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

Usage, cc, d, breaks)



vector of observed survival times


vector of censoring times


vector of non-censoring indicators


Vector of breakpoints to be used to form histograms.


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


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.

See Also

reduced.sample, kaplan.meier