Kaplan-Meier Estimator using Histogram Data
Compute the Kaplan-Meier estimator of a survival time distribution function, from histogram data
kaplan.meier(obs, nco, breaks, upperobs=0)
obs |
vector of n integers giving the histogram of all observations (censored or uncensored survival times) |
nco |
vector of n integers giving the histogram of uncensored observations (those survival times that are less than or equal to the censoring time) |
breaks |
Vector of n+1 breakpoints which were used to form both histograms. |
upperobs |
Number of observations beyond the rightmost breakpoint, if any. |
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.
If the number of observations M is large, it is efficient to
use histograms.
Form the histogram obs
of all observed times T*[i].
That is, obs[k]
counts the number of values
T*[i] in the interval
(breaks[k],breaks[k+1]]
for k > 1
and [breaks[1],breaks[2]]
for k = 1.
Also form the histogram nco
of all uncensored times,
i.e. those T*[i] such that D[i]=1.
These two histograms are the arguments passed to kaplan.meier
.
The vectors km
and lambda
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])
.
The histogram breaks must include 0.
If the histogram breaks do not span the range of the observations,
it is important to count how many survival times
T*[i] exceed the rightmost breakpoint,
and give this as the value upperobs
.
A list with two elements:
km |
Kaplan-Meier estimate of the survival time c.d.f. F(t) |
lambda |
corresponding Nelson-Aalen estimate of the hazard rate lambda(t) |
These are numeric vectors of length n.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au
and Rolf Turner r.turner@auckland.ac.nz
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