# 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 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.

##### Survivor Function

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 t. 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 g be the transformation that is being applied to the survivor function . By the delta method, the standard error of is estimated by

where is the first derivative of the function g. The 100(1–)% confidence interval for is given by

where is the inverse function of g. That choices of the transformation g are as follows:

• arcsine-square root transformation: The estimated variance of is The 100(1–)% confidence interval for is given by

• linear transformation: This is the same as having no transformation in which g is the identity. The 100(1–)% confidence interval for is given by

• log transformation: The estimated variance of is The 100(1–)% confidence interval for is given by

• log-log transformation: The estimated variance of is The 100(1–)% confidence interval for is given by

• logit transformation: The estimated variance of is

The 100(1–)% confidence limits for are given by

##### Density and Hazard Functions

For the life-table method, a 100(1–)% confidence interval for hazard function or density function at time t is computed as

where is the estimate of either the hazard function or the density function at time t, and is the corresponding standard error estimate.