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The PHREG Procedure |
Two estimators of the survivor function are available: one is the product-limit estimator (Kalbfleisch and Prentice; 1980, pp. 84–86) and the other is the Breslow (1972) estimator based on the empirical cumulative hazard function.
Let denote the set of individuals censored in the half-open interval
, where
and
. Let
denote the censoring times in
; l ranges over
.
The likelihood function for all individuals is given by
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where is empty. The likelihood
is maximized by taking
for
and allowing the probability mass to fall only on the observed event times
,
,
. By considering a discrete model with hazard contribution
at
, you take
, where
. Substitution into the likelihood function produces
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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
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When only a single failure occurs at ,
can be found explicitly. Otherwise, an iterative solution is obtained by the Newton method.
The estimated baseline cumulative hazard function is
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where is the estimated baseline survivor function given by
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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
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Let be a given realization of the explanatory variables. The empirical cumulative hazard function estimate at
is
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The variance estimator of is given by the following (Tsiatis; 1981):
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where is the estimated covariance matrix of
and
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For the marginal model, the variance estimator computation follows Spiekerman and Lin (1998).
The empirical cumulative hazard function (CH) estimate of the survivor function for is
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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 Kalbfleisch and Prentice (1980, p. 116). By the delta method, the standard errors of
and
are given by
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respectively. The standard errors of log[–log()] and log[–log(
)] are given by
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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.
CLTYPE |
Method |
Confidence Limits |
---|---|---|
LOG |
PL |
|
LOG |
CH |
|
LOGLOG |
PL |
|
LOGLOG |
CH |
|
NORMAL |
PL |
|
NORMAL |
CH |
|
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Copyright © 2009 by SAS Institute Inc., Cary, NC, USA. All rights reserved.