Survival Distribution Estimates for the Cox Model
Two estimators of the survivor function
are available: one is
the product-limit estimate and the other is based on
the empirical cumulative hazard function.
Product-Limit Estimates
Let Ci denote the
set of individuals censored in the half-open interval [ti , ti+1),
where t0=0 and
.Let
denote
the censoring times in
[ti , ti+1); l ranges over
Ci .
The likelihood function for all individuals is given by
![{\cal L}=\prod_{i=0}^k
\{ \prod_{l \in {\cal D}_i}
( [S_{0}(t_{i})]^{ {\rm exp...
... )
\prod_{l \in {\cal C}_i} [S_{0}(\gamma_{l}+0)]^{{\rm exp}(z'_{l}{\beta})}
\}](images/phreq136.gif)
where D0 is empty.
The likelihood L is maximized by taking
S0(t)=S0(ti+0) for
and allowing the
probability mass to fall only on the observed event times
t1, , tk.
By considering
a discrete model with hazard contribution
at ti, you take
, where
. Substitution into the likelihood function produces
![{\cal L}=\prod_{i=0}^k \{ \prod_{j \in {\cal D}_i }
( 1-\alpha_{i}^{{\rm exp}(z...
...d_{ l \in {\cal R}_{i}-{\cal D}_{i} }
\alpha_{i}^{ {\rm exp}(z'_{l}{\beta})} \}](images/phreq141.gif)
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
![\sum_{ j \in {\cal D}_i }
\frac { {\rm exp}(z'_{j}\hat{{\beta}}) }
{ 1-\hat{\a...
..._{j}\hat{{\beta}}) } }
=\sum_{l \in {\cal R}_i } {\rm exp}(z'_{l}\hat{{\beta}})](images/phreq146.gif)
When only a single failure occurs at ti,
can be found explicitly. Otherwise,
an iterative solution is obtained by the Newton method.
The estimated baseline cumulative hazard function is
![\hat{H}_{0}(t)=-{\rm log}(\hat{S}_{0}(t))](images/phreq147.gif)
where
is the estimated baseline
survivor function given by
![\hat{S}_{0}(t)=\hat{S}_{0}(t_{i-1}+0)
=\prod_{j=0}^{i-1} \hat{\alpha}_{j} ,
t_{i-1} \lt t \leq t_{i}](images/phreq149.gif)
For details, refer to Kalbfleisch and Prentice (1980).
For a given realization of the
explanatory variables
,the product-limit estimate of the survival function at
is
![\hat{S}(t,{\xi})= [\hat{S}_{0}(t)]^{\exp({\beta}'{\xi})}](images/phreq152.gif)
Empirical Cumulative Hazards Function Estimates
Let
be a given realization of the
explanatory variables.
The empirical cumulative hazard function estimate
at
is
![\hat{\Lambda}(t,{\xi}) = \sum_{i=1}^n\int_{0}^t
\frac{dN_{i}(s)}{\sum_{j=1}^nY_{j}(s)\exp(\hat{{\beta}}'(z_{j} - {\xi}))}](images/phreq153.gif)
The variance estimator of
is given by
the following (Tsiatis 1981):
![& &\hat{var}\{n^{\frac{1}2}(\hat{\Lambda}(t,{\xi}) -
\Lambda(t,{\xi}))\} \ & = ...
...i}))]^2} + \biggr.
\biggl. H'(t,{\xi})\hat{V}(\hat{{\beta}})H(t,{\xi}) \biggr\}](images/phreq155.gif)
where
is the estimated covariance matrix of
and
![H(t,{\xi}) = \sum_{i=1}^n\int_{0}^t
\frac{\sum_{l=1}^n Y_{l}(s)(Z_{l}-{\xi})\ex...
...{\xi}))}
{[\sum_{j=1}^n Y_{j}(s)\exp(\hat{{\beta}}'(z_{j} - {\xi}))]^2} dN_i(s)](images/phreq157.gif)
The empirical cumulative hazard function (CH) estimate of the
survivor function for
is
![\tilde{S}(t,{\xi})= \exp(-\hat{\Lambda}(t,{\xi}))](images/phreq158.gif)
Confidence Intervals for the Survivor Function
Let
and
correspond to the product-limit (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
![\hat{\sigma}_{1}(t,{\xi}) \dot{=} \hat{S}(t,{\xi})\tilde{\sigma}_{0}(t,{\xi})
...
...tilde{\sigma}_{1}(t,{\xi}) \dot{=} \tilde{S}(t,{\xi})\tilde{\sigma}_{0}(t,{\xi})](images/phreq162.gif)
respectively. The standard errors of log[-log(
)] and
log[-log(
)] are given by
![\hat{\sigma}_{2}(t,{\xi}) \dot{=} \frac{- \tilde{\sigma}_{0}(t,{\xi})}{\log(\hat...
..._{2}(t,{\xi}) \dot{=} \frac{\tilde{\sigma}_{0}(t,{\xi})}{\hat{\Lambda}(t,{\xi})}](images/phreq163.gif)
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 | ![\exp[\log(\hat{S}(t,{\xi})) +- z_{\frac{\alpha}2}\tilde{\sigma}_{0}(t,{\xi})]](images/phreq166.gif) |
LOG | CH | ![\exp[\log(\tilde{S}(t,{\xi})) +- z_{\frac{\alpha}2}\tilde{\sigma}_{0}(t,{\xi})]](images/phreq167.gif) |
LOGLOG | PL | ![\exp\{-\exp[\log(-\log(\hat{S}(t,{\xi}))) +- z_{\frac{\alpha}2}\hat{\sigma}_{2}(t,{\xi})]\}](images/phreq168.gif) |
LOGLOG | CH | ![\exp\{-\exp[\log(-\log(\tilde{S}(t,{\xi}))) +- z_{\frac{\alpha}2}\tilde{\sigma}_{2}(t,{\xi})]\}](images/phreq169.gif) |
NORMAL | PL | ![\hat{S}(t,{\xi})+- z_{\frac{\alpha}2}\hat{\sigma}_{1}(t,{\xi})](images/phreq170.gif) |
NORMAL | CH | ![\tilde{S}(t,{\xi})+- z_{\frac{\alpha}2}\tilde{\sigma}_{1}(t,{\xi})](images/phreq171.gif) |
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.