Survival Distribution Estimates for the Cox Model
Two estimators of the survivor function
are available: one is
the productlimit estimate and the other is based on
the empirical cumulative hazard function.
ProductLimit Estimates
Let C_{i} denote the
set of individuals censored in the halfopen interval [t_{i} , t_{i+1}),
where t_{0}=0 and .Let denote
the censoring times in
[t_{i} , t_{i+1}); l ranges over
C_{i} .
The likelihood function for all individuals is given by
where D_{0} is empty.
The likelihood L is maximized by taking
S_{0}(t)=S_{0}(t_{i}+0) for
and allowing the
probability mass to fall only on the observed event times
t_{1}, , t_{k}.
By considering
a discrete model with hazard contribution
at t_{i}, you take
, where
. Substitution into the likelihood function produces
If you replace with estimated from the partial
likelihood function and then
maximize with respect to , , ,
the maximum likelihood estimate
of becomes a solution of
When only a single failure occurs at t_{i}, can be found explicitly. Otherwise,
an iterative solution is obtained by the Newton method.
The estimated baseline cumulative hazard function is
where is the estimated baseline
survivor function given by
For details, refer to Kalbfleisch and Prentice (1980).
For a given realization of the
explanatory variables ,the productlimit estimate of the survival function at is
Empirical Cumulative Hazards Function Estimates
Let be a given realization of the
explanatory variables.
The empirical cumulative hazard function estimate
at is
The variance estimator of is given by
the following (Tsiatis 1981):
where is the estimated covariance matrix of and
The empirical cumulative hazard function (CH) estimate of the
survivor function for is
Confidence Intervals for the Survivor Function
Let and correspond to the productlimit (PL) and
empirical cumulative hazard function (CH) estimates of the survivor function
for , respectively.
Both the standard error of log() and the standard error of
log() are approximated by , which is the
square root of the variance estimate of
; refer to Kalbfleish and
Prentice (1980, p. 116). By the
delta method, the standard errors of and are given by
respectively. The standard errors of log[log()] and
log[log()] are given by
respectively.
Let be the upper percentile point of the standard normal distribution.
A confidence interval for the survivor function
is given in the following table.
Method

CLTYPE

Confidence Limits

LOG  PL  
LOG  CH  
LOGLOG  PL  
LOGLOG  CH  
NORMAL  PL  
NORMAL  CH  
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.