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 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
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.