PROC LIFETEST: Pointwise Confidence Limits in the OUTSURV= Data Set :: SAS/STAT(R) 9.2 User's Guide, Second Edition

The LIFETEST Procedure

Pointwise Confidence Limits in the OUTSURV= Data Set

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 $\text{[math]}$ be specified by the ALPHA= option. Let $\text{[math]}$ be the critical value for the standard normal distribution. That is, $\text{[math]}$ , where $\text{[math]}$ is the cumulative distribution function of the standard normal random variable.

Survivor Function

When the computation of confidence limits for the survivor function $\text{[math]}$ is based on the asymptotic normality of the survival estimator $\text{[math]}$ , the approximate confidence interval might include impossible values outside the range [0,1] at extreme values of $\text{[math]}$ . This problem can be avoided by applying the asymptotic normality to a transformation of $\text{[math]}$ for which the range is unrestricted. In addition, certain transformed confidence intervals for $\text{[math]}$ 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 $\text{[math]}$ be the transformation that is being applied to the survivor function $\text{[math]}$ . By the delta method, the standard error of $\text{[math]}$ is estimated by

$\text{[math]}$

where $\text{[math]}$ is the first derivative of the function $\text{[math]}$ . The 100(1– $\text{[math]}$ )% confidence interval for $\text{[math]}$ is given by

$\text{[math]}$

where $\text{[math]}$ is the inverse function of $\text{[math]}$ .

Arcsine-Square Root Transformation

The estimated variance of $\text{[math]}$ is $\text{[math]}$ The 100(1– $\text{[math]}$ )% confidence interval for $\text{[math]}$ is given by

$\text{[math]}$

Linear Transformation

This is the same as having no transformation in which $\text{[math]}$ is the identity. The 100(1– $\text{[math]}$ )% confidence interval for $\text{[math]}$ is given by

$\text{[math]}$

Log Transformation

The estimated variance of $\text{[math]}$ is $\text{[math]}$ The 100(1– $\text{[math]}$ )% confidence interval for $\text{[math]}$ is given by

$\text{[math]}$

Log-Log Transformation

The estimated variance of $\text{[math]}$ is $\text{[math]}$ The 100(1– $\text{[math]}$ )% confidence interval for $\text{[math]}$ is given by

$\text{[math]}$

Logit Transformation

The estimated variance of $\text{[math]}$ is $\text{[math]}$ The 100(1– $\text{[math]}$ )% confidence limits for $\text{[math]}$ are given by

$\text{[math]}$

Density and Hazard Functions

For the life-table method, a 100(1– $\text{[math]}$ )% confidence interval for hazard function or density function at time $\text{[math]}$ is computed as

$\text{[math]}$

where $\text{[math]}$ is the estimate of either the hazard function or the density function at time $\text{[math]}$ , and $\text{[math]}$ is the corresponding standard error estimate.

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