The QUANTLIFE Procedure

Relationship of Quantile Function and Survival Function

Both quantile function and survival function are useful in characterizing a lifetime distribution.

By the definition of the quantile function upper Q Subscript upper T Baseline left-parenthesis tau vertical-bar x right-parenthesis,

upper F left-parenthesis upper Q Subscript upper T Baseline left-parenthesis tau vertical-bar x right-parenthesis right-parenthesis equals upper P left-parenthesis upper T less-than-or-equal-to upper Q Subscript upper T Baseline left-parenthesis tau vertical-bar x right-parenthesis right-parenthesis equals tau

In other words, the cumulative distribution function upper F Subscript upper T Baseline left-parenthesis t vertical-bar x right-parenthesis maps upper Q Subscript upper T Baseline left-parenthesis tau vertical-bar x right-parenthesis to tau, and thus the corresponding survival function upper S Subscript upper T Baseline left-parenthesis t vertical-bar x right-parenthesis maps upper Q Subscript upper T Baseline left-parenthesis tau vertical-bar x right-parenthesis to 1 minus tau.

When you specify the LOG option, the QUANTLIFE procedure fits a linear quantile regression model for a log transformation of the lifetime as

upper Q Subscript normal l normal o normal g left-parenthesis upper T right-parenthesis Baseline left-parenthesis tau vertical-bar x right-parenthesis equals x prime bold-italic beta left-parenthesis tau right-parenthesis

where upper Q Subscript normal l normal o normal g left-parenthesis upper T right-parenthesis Baseline left-parenthesis tau vertical-bar x right-parenthesis is the tauth quantile of log left-parenthesis upper T right-parenthesis at x. The estimated quantile function for T given x is ModifyingAbove upper Q With caret Subscript upper T Baseline left-parenthesis tau vertical-bar x right-parenthesis equals e Superscript x prime ModifyingAbove bold-italic beta With caret left-parenthesis tau right-parenthesis, because the quantile function is invariant under a monotone transformation.

You can specify the covariates x in the COVARIATES= data set of the BASELINE statement and the PLOTS=(QUANTILE SURVIVAL) option in the PROC statement. Then the conditional quantile function at x is plotted as ModifyingAbove upper Q With caret Subscript upper T Baseline left-parenthesis tau vertical-bar x right-parenthesis against tau, and the conditional survival function at x is plotted as 1 minus tau against ModifyingAbove upper Q With caret Subscript upper T Baseline left-parenthesis tau vertical-bar x right-parenthesis.

Last updated: December 09, 2022