The QUANTLIFE Procedure

Kaplan-Meier-Type Estimator for Censored Quantile Regression

Portnoy (2003) proposes the use of weighted quantile regression to sequentially estimate beta left-parenthesis tau Subscript k Baseline right-parenthesis along the equally spaced grid 0 less-than tau 1 less-than midline-horizontal-ellipsis less-than tau Subscript upper M Baseline less-than 1. You can request this method by specifying the METHOD=KM option in the PROC QUANTLIFE statement. The grid points 0 less-than tau 1 less-than midline-horizontal-ellipsis less-than tau Subscript upper M Baseline less-than 1 are equally spaced, with tau 1 specified by the INITTAU= option and the step between adjacent grid points specified by the GRIDSIZE=option.

This method uses a weight function w Subscript i Baseline left-parenthesis tau right-parenthesis for each censored observation. The weight function is constructed as follows: Let ModifyingAbove tau With caret Subscript i be the first grid point at which x prime Subscript i Baseline ModifyingAbove beta With caret left-parenthesis tau Subscript i Baseline right-parenthesis greater-than-or-equal-to upper C Subscript i and x prime Subscript i Baseline ModifyingAbove beta With caret left-parenthesis tau Subscript i plus 1 Baseline right-parenthesis less-than upper C Subscript i; otherwise let ModifyingAbove tau With caret Subscript i Baseline equals 1. When computing the tauth quantile, assign weight w Subscript i Baseline left-parenthesis tau right-parenthesis equals StartFraction tau minus ModifyingAbove tau With caret Subscript i Baseline Over 1 minus ModifyingAbove tau With caret Subscript i Baseline EndFraction to the censored observation upper Y Subscript i if tau greater-than ModifyingAbove tau With caret Subscript i; otherwise assign w Subscript i Baseline left-parenthesis tau right-parenthesis equals 1. The algorithm for computing ModifyingAbove beta With caret left-parenthesis tau Subscript k Baseline right-parenthesis comma k equals 1 comma ellipsis comma upper M comma is as follows:

  1. Compute ModifyingAbove beta With caret left-parenthesis tau 1 right-parenthesis by using the standard quantile regression method.

  2. For k equals 2 comma ellipsis comma upper M, obtain ModifyingAbove beta With caret left-parenthesis tau Subscript k Baseline right-parenthesis sequentially by minimizing the following weighted quantile regression objective function:

    StartLayout 1st Row 1st Column r Subscript w Baseline left-parenthesis b right-parenthesis 2nd Column equals sigma-summation Underscript normal upper Delta Subscript i Baseline equals 1 Endscripts rho Subscript tau Sub Subscript k Baseline left-parenthesis upper Y Subscript i Baseline minus x prime Subscript i Baseline b right-parenthesis 2nd Row 1st Column Blank 2nd Column plus sigma-summation Underscript normal upper Delta Subscript i Baseline equals 0 Endscripts StartSet w Subscript i Baseline left-parenthesis tau Subscript k Baseline right-parenthesis rho Subscript tau Sub Subscript k Subscript Baseline left-parenthesis upper Y Subscript i Baseline minus x prime Subscript i Baseline b right-parenthesis plus left-parenthesis 1 minus w Subscript i Baseline left-parenthesis tau Subscript k Baseline right-parenthesis right-parenthesis rho Subscript tau Sub Subscript k Subscript Baseline left-parenthesis upper Y Superscript asterisk Baseline minus x prime Subscript i Baseline b right-parenthesis EndSet EndLayout

    where w Subscript i Baseline left-parenthesis tau Subscript k Baseline right-parenthesis is the weight for the right-censored observation upper Y Subscript i at computing ModifyingAbove beta With caret left-parenthesis tau Subscript k Baseline right-parenthesis, and the complementary weight 1 minus w Subscript i Baseline left-parenthesis tau Subscript k Baseline right-parenthesis is for upper Y Superscript asterisk, a large constant that is greater than all x prime Subscript i Baseline ModifyingAbove beta With caret left-parenthesis tau right-parenthesis.

The weighted quantile regression method is similar to Efron’s redistribution-of-mass idea (Efron 1967) for the Kaplan-Meier estimator.

Note that if all observations are uncensored, ModifyingAbove beta With caret left-parenthesis tau Subscript k Baseline right-parenthesis is the same as the standard quantile regression estimator.

Last updated: December 09, 2022