kaplan.meier
Kaplan-Meier Estimator using Histogram Data
Compute the Kaplan-Meier estimator of a survival time distribution function, from histogram data
- Keywords
- spatial, nonparametric
Usage
kaplan.meier(obs, nco, breaks, upperobs=0)Arguments
- 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.
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 $i$.
If the number of observations $M$ is large, it is efficient to
use histograms.
Form the histogram obs of all observed times $\tilde T_i$.
That is, obs[k] counts the number of values
$\tilde T_i$ 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 $\tilde T_i$ 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
$\tilde T_i$ exceed the rightmost breakpoint,
and give this as the value upperobs.
Value
- 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 $n$.