The QUANTLIFE Procedure

Notation for Censored Quantile Regression

Let T be a dependent variable, such as a survival time, and let x be a p times 1 covariate vector. Quantile regression methods focus on modeling the conditional quantile function, upper Q Subscript upper T Baseline left-parenthesis tau vertical-bar x right-parenthesis, which is defined as

upper Q Subscript upper T Baseline left-parenthesis tau vertical-bar x right-parenthesis equals inf StartSet t colon upper P left-parenthesis upper T less-than-or-equal-to t vertical-bar x right-parenthesis equals tau EndSet comma 0 less-than tau less-than 1

For example, upper Q Subscript upper T Baseline left-parenthesis 0.5 vertical-bar x right-parenthesis is the conditional median quantile, and upper Q Subscript upper T Baseline left-parenthesis 0.95 vertical-bar x right-parenthesis is the conditional quantile function that corresponds to the 95th percentile.

A linear quantile regression model for upper Q Subscript upper T Baseline left-parenthesis tau vertical-bar x right-parenthesis has the form x prime bold-italic beta left-parenthesis tau right-parenthesis. One of the advantages of quantile regression analysis is that the covariate effect bold-italic beta left-parenthesis tau right-parenthesis can change with tau. Unlike ordinary least squares regression, which estimates the conditional expectation function upper E left-parenthesis upper T vertical-bar x right-parenthesis, quantile regression offers the flexibility to model the entire conditional distribution.

Given observations StartSet left-parenthesis upper T Subscript i Baseline comma x Subscript i Baseline right-parenthesis comma i equals 1 comma ellipsis comma n EndSet, standard quantile regression estimates the regression coefficients beta left-parenthesis tau right-parenthesis by minimizing the following objective function over b:

r left-parenthesis b right-parenthesis equals sigma-summation Underscript i equals 1 Overscript n Endscripts rho Subscript tau Baseline left-parenthesis upper T Subscript i Baseline minus x prime Subscript i Baseline b right-parenthesis

where rho Subscript tau Baseline left-parenthesis u right-parenthesis equals u left-parenthesis tau minus upper I left-parenthesis u less-than 0 right-parenthesis right-parenthesis period

However, in many applications, the responses upper T Subscript i are subject to censoring. For example, in a biomedical study, censoring occurs when patients withdraw from the study or die from a cause that is unrelated to the disease being studied.

Let upper C Subscript i denote the censoring variable. In the case of right-censoring, the triples left-parenthesis x Subscript i Baseline comma upper Y Subscript i Baseline comma normal upper Delta Subscript i Baseline right-parenthesis are observed, where upper Y Subscript i Baseline equals min left-parenthesis upper T Subscript i Baseline comma upper C Subscript i Baseline right-parenthesis and normal upper Delta Subscript i Baseline equals upper I left-parenthesis upper T Subscript i Baseline less-than-or-equal-to upper C Subscript i Baseline right-parenthesis are the observed response variable and the censoring indicator, respectively. Standard quantile regression can lead to a biased estimator of the regression parameters beta left-parenthesis tau right-parenthesis when censoring occurs.

The following sections describe two methods for estimating the quantile coefficient beta left-parenthesis tau right-parenthesis in the presence of right-censoring.

Last updated: December 09, 2022