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The PHREG Procedure

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 t_{k+1}=\infty.Let \gamma_l 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})}
where D0 is empty. The likelihood L is maximized by taking S0(t)=S0(ti+0) for t_{i}\lt t \leq t_{i+1} and allowing the probability mass to fall only on the observed event times t1, , tk. By considering a discrete model with hazard contribution 1-\alpha_{i} at ti, you take S_{0}(t_{i})=S_{0}(t_{i-1}+0)=\prod_{j=0}^{i-1} \alpha_j, where \alpha_{0}=1. 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})} \}
If you replace {\beta} with \hat{{\beta}} estimated from the partial likelihood function and then maximize with respect to \alpha_{1} , ,\alpha_{k} , the maximum likelihood estimate \hat{\alpha}_i of \alpha_i 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}})
When only a single failure occurs at ti, \hat{\alpha}_ican 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))
where \hat{S}_{0}(t) is the estimated baseline survivor function given by
 =\prod_{j=0}^{i-1} \hat{\alpha}_{j} ,
  t_{i-1} \lt t \leq t_{i}

For details, refer to Kalbfleisch and Prentice (1980). For a given realization of the explanatory variables {\xi},the product-limit estimate of the survival function at Z={\xi} is

\hat{S}(t,{\xi})= [\hat{S}_{0}(t)]^{\exp({\beta}'{\xi})}

Empirical Cumulative Hazards Function Estimates

Let {\xi} be a given realization of the explanatory variables. The empirical cumulative hazard function estimate at Z={\xi} 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}))}
The variance estimator of \hat{\Lambda}(t,{\xi}) 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\}
where \hat{V}(\hat{{\beta}}) is the estimated covariance matrix of \hat{{\beta}} and
H(t,{\xi}) = \sum_{i=1}^n\int_{0}^t
 \frac{\sum_{l=1}^n Y_{l}(s)(Z_{l}-{\xi})\ex...
 {[\sum_{j=1}^n Y_{j}(s)\exp(\hat{{\beta}}'(z_{j} - {\xi}))]^2} dN_i(s)

The empirical cumulative hazard function (CH) estimate of the survivor function for Z={\xi} is

\tilde{S}(t,{\xi})= \exp(-\hat{\Lambda}(t,{\xi}))

Confidence Intervals for the Survivor Function

Let \hat{S}(t,{\xi}) and \tilde{S}(t,{\xi}) correspond to the product-limit (PL) and empirical cumulative hazard function (CH) estimates of the survivor function for Z={\xi}, respectively. Both the standard error of log(\hat{S}(t,{\xi})) and the standard error of log(\tilde{S}(t,{\xi})) are approximated by \tilde{\sigma}_{0}(t,{\xi}), which is the square root of the variance estimate of \hat{\Lambda}(t,{\xi}); refer to Kalbfleish and Prentice (1980, p. 116). By the delta method, the standard errors of \hat{S}(t,{\xi}) and \tilde{S}(t,{\xi})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})
respectively. The standard errors of log[-log(\hat{S}(t,{\xi}))] and log[-log(\tilde{S}(t,{\xi}))] 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})}

Let z_{\alpha /2} be the upper 100(1-\frac{\alpha}2)percentile point of the standard normal distribution. A 100(1-{\alpha})\% confidence interval for the survivor function S(t,{\xi}) is given in the following table.
Method CLTYPE Confidence Limits
LOGPL\exp[\log(\hat{S}(t,{\xi})) +- z_{\frac{\alpha}2}\tilde{\sigma}_{0}(t,{\xi})]
LOGCH\exp[\log(\tilde{S}(t,{\xi})) +- z_{\frac{\alpha}2}\tilde{\sigma}_{0}(t,{\xi})]
LOGLOGPL\exp\{-\exp[\log(-\log(\hat{S}(t,{\xi}))) +- z_{\frac{\alpha}2}\hat{\sigma}_{2}(t,{\xi})]\}
LOGLOGCH\exp\{-\exp[\log(-\log(\tilde{S}(t,{\xi}))) +- z_{\frac{\alpha}2}\tilde{\sigma}_{2}(t,{\xi})]\}
NORMALPL\hat{S}(t,{\xi})+- z_{\frac{\alpha}2}\hat{\sigma}_{1}(t,{\xi})
NORMALCH\tilde{S}(t,{\xi})+- z_{\frac{\alpha}2}\tilde{\sigma}_{1}(t,{\xi})

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Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.