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The LIFETEST Procedure |
Pointwise confidence limits are computed for the survivor function, and for the density function and hazard function when the life-table method is used. Let be specified by the ALPHA= option. Let
be the critical value for the standard normal distribution. That is,
, where
is the cumulative distribution function of the standard normal random variable.
When the computation of confidence limits for the survivor function is based on the asymptotic normality of the survival estimator
, the approximate confidence interval might include impossible values outside the range [0,1] at extreme values of
. This problem can be avoided by applying the asymptotic normality to a transformation of
for which the range is unrestricted. In addition, certain transformed confidence intervals for
perform better than the usual linear confidence intervals (Borgan and Liestøl; 1990). The CONFTYPE= option enables you to pick one of the following transformations: the log-log function (Kalbfleisch and Prentice; 1980), the arcsine-square root function (Nair; 1984), the logit function (Meeker and Escobar; 1998), the log function, and the linear function.
Let be the transformation that is being applied to the survivor function
. By the delta method, the standard error of
is estimated by
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where is the first derivative of the function
. The 100(1–
)% confidence interval for
is given by
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where is the inverse function of
.
The estimated variance of is
The 100(1–
)% confidence interval for
is given by
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This is the same as having no transformation in which is the identity. The 100(1–
)% confidence interval for
is given by
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For the life-table method, a 100(1–)% confidence interval for hazard function or density function at time
is computed as
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where is the estimate of either the hazard function or the density function at time
, and
is the corresponding standard error estimate.
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Copyright © 2009 by SAS Institute Inc., Cary, NC, USA. All rights reserved.