| km.rs {spatstat} | R Documentation |
Compute the Kaplan-Meier and Reduced Sample estimators of a survival time distribution function, using histogram techniques
km.rs(o, cc, d, breaks)
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. |
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,...,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 T*[i] = min(T[i],C[i]) and non-censoring indicators D[i] = 1(T[i] <= C[i]) for each i.
The arguments to this function are
vectors o, cc, d
of observed values of 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.
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 |
Adrian Baddeley adrian@maths.uwa.edu.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner rolf@math.unb.ca http://www.math.unb.ca/~rolf