QUANTILE CALCULUS AND CENSORED REGRESSION.

Quantile regression has been advocated in survival analysis to assess evolving covariate effects. However, challenges arise when the censoring time is not always observed and may be covariate-dependent, particularly in the presence of continuously-distributed covariates. In spite of several recent advances, existing methods either involve algorithmic complications or impose a probability grid. The former leads to difficulties in the implementation and asymptotics, whereas the latter introduces undesirable grid dependence. To resolve these issues, we develop fundamental and general quantile calculus on cumulative probability scale in this article, upon recognizing that probability and time scales do not always have a one-to-one mapping given a survival distribution. These results give rise to a novel estimation procedure for censored quantile regression, based on estimating integral equations. A numerically reliable and efficient Progressive Localized Minimization (PLMIN) algorithm is proposed for the computation. This procedure reduces exactly to the Kaplan-Meier method in the k-sample problem, and to standard uncensored quantile regression in the absence of censoring. Under regularity conditions, the proposed quantile coefficient estimator is uniformly consistent and converges weakly to a Gaussian process. Simulations show good statistical and algorithmic performance. The proposal is illustrated in the application to a clinical study.

[1]  Lan Wang,et al.  Locally Weighted Censored Quantile Regression , 2009 .

[2]  Bo E. Honoré,et al.  Quantile regression under random censoring , 2002 .

[3]  J. Powell,et al.  Least absolute deviations estimation for the censored regression model , 1984 .

[4]  I. James,et al.  Linear regression with censored data , 1979 .

[5]  M. Kosorok.  Introduction to Empirical Processes and Semiparametric Inference , 2008 .

[6]  R. Koenker,et al.  Computing regression quantiles , 1987 .

[7]  Philip E. Gill,et al.  Numerical Linear Algebra and Optimization , 1991 .

[8]  L. J. Wei,et al.  On the Cox Model With Time-Varying Regression Coefficients , 2005 .

[9]  V. Chernozhukov.  Extremal quantile regression , 2005, math/0505639.

[10]  A. Tsiatis.  Estimating Regression Parameters Using Linear Rank Tests for Censored Data , 1990 .

[11]  R. Koenker,et al.  Reappraising Medfly Longevity , 2001 .

[12]  Odd Aalen,et al.  A Model for Nonparametric Regression Analysis of Counting Processes , 1980 .

[13]  Limin Peng,et al.  Survival analysis with temporal covariate effects , 2007 .

[14]  Z. Ying,et al.  A simple resampling method by perturbing the minimand , 2001 .

[15]  A. Owen.  Empirical Likelihood Ratio Confidence Regions , 1990 .

[16]  B. Efron.  The two sample problem with censored data , 1967 .

[17]  C. Gutenbrunner,et al.  Regression Rank Scores and Regression Quantiles , 1992 .

[18]  J. Robins,et al.  Toward a curse of dimensionality appropriate (CODA) asymptotic theory for semi-parametric models. , 1997, Statistics in medicine.

[19]  R. Koenker,et al.  Regression Quantiles , 2007 .

[20]  Limin Peng,et al.  Survival Analysis With Quantile Regression Models , 2008 .

[21]  Zhiliang Ying,et al.  Survival analysis with median regression models , 1995 .

[22]  Stephen Portnoy,et al.  Correction to Censored Regression Quantiles by S. Portnoy, 98 (2003), 1001–1012 , 2006 .

[23]  R. Gill,et al.  A Survey of Product-Integration with a View Toward Application in Survival Analysis , 1990 .

[24]  S. Portnoy,et al.  On Extreme Regression Quantiles , 1999 .

[25]  R. Koenker.  Censored Quantile Regression Redux , 2008 .

[26]  D. Harrington,et al.  Counting Processes and Survival Analysis , 1991 .

[27]  Stephen Portnoy,et al.  Censored Regression Quantiles , 2003 .