Lifetime Data Analysis

, Volume 18, Issue 2, pp 177–194

Censored quantile regression for residual lifetimes

Article

Abstract

We propose a regression method that studies covariate effects on the conditional quantiles of residual lifetimes at a certain followup time point. This can be particularly useful in cancer studies, where more patients survive cancers initially and a patient’s residual life expectancy is used to compare the efficacy of secondary or adjuvant therapies. The new method provides a consistent estimator that often exhibits smaller standard error in real and simulated examples, compared to the existing method of Jung et al. (2009). It also provides a simple empirical likelihood inference method that does not require estimating the covariance matrix of the estimator or resampling. We apply the new method to a breast cancer study (NSABP Protocol B-04, Fisher et al. (2002)) and estimate median residual lifetimes at various followup time points, adjusting for important prognostic factors.

Keywords

Cancer Empirical likelihood Quantile regression Residual lifetime regression Survival analysis Wilks’ theorem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Akritas M (2000) The central limit theorem under censoring. Bernoulli 6: 1109–1120MathSciNetMATHCrossRefGoogle Scholar
  2. Coombes RC, Hall E, Gibson LJ, Paridaens R, Jassem J, Delozier T, Jones S, Alvarez I, Bertelli G, Ortmann O, Coates AS, Bajetta E, Dodwell D, Coleman RE, Fallowfield LJ, Mickiewicz E, Andersen J, Lønning PE, Cocconi G, Stewart A, Stuart N, Snowdon CF, Carpentieri M, Massimini G, Bliss JM, fortheIntergroup Exemestane Study (2004) A randomized trial of exemestane after two to three years of tamoxifen therapy in postmenopausal women with primary breast cancer. N Engl J Med 350(11): 1081–1092CrossRefGoogle Scholar
  3. Efron B, Johnstone IM (1990) Fisher’s information in terms of the hazard rate. Ann Statist 18: 38–62MathSciNetMATHCrossRefGoogle Scholar
  4. Fisher B, Jeong J, Anderson S, Bryant J, Fisher E, Wolmark N (2002) Twenty-five year findings from a randomized clinical trial comparing radical mastectomy with total mastectomy and with total mastectomy followed by radiation therapy. N Engl J Med 347(8): 567–575CrossRefGoogle Scholar
  5. Gelfand A, Kottas A (2003) Bayesian semiparametric regression for median residual life. Scand J Stat 30: 651–665MathSciNetMATHCrossRefGoogle Scholar
  6. Goss PE, Ingle JN, Martino S, Robert NJ, Muss HB, Piccart MJ, Castiglione M, Tu D, Shepherd LE, Pritchard KI, Livingston RB, Davidson NE, Norton L, Perez EA, Abrams JS, Therasse P, Palmer MJ, Pater JL (2003) A randomized trial of letrozole in postmenopausal women after five years of tamoxifen therapy for early-stage breast cancer. N Engl J Med 349(19): 1793–1802CrossRefGoogle Scholar
  7. Honoré B, Khan S, Powell J (2002) Quantile regression under random censoring. J Econom 109: 67–105MATHCrossRefGoogle Scholar
  8. Huang J, Ma S, Xie H (2007) Least absolute deviations estimation for the accelerated failure time model. Statistica Sinica 17: 1533–1548MathSciNetMATHGoogle Scholar
  9. Jeong JH, Jung SH, Costantino JP (2008) Nonparametric inference on median residual life function. Biometrics 64: 157–163MathSciNetMATHCrossRefGoogle Scholar
  10. Jung SH, Jeong JH, Bandos H (2009) Regression on quantile residual life. Biometrics 65: 1203–1212MathSciNetMATHCrossRefGoogle Scholar
  11. Kaplan ED, Meier P (1958) Nonparametric estimation from incomplete observations. J Amer Statist Assoc 53: 457–481MathSciNetMATHCrossRefGoogle Scholar
  12. Koenker R (2005) Quantile Regression Cambridge University PressGoogle Scholar
  13. Koenker R, Bassett G (1978) Regression quantiles. Econometrica 46: 33–50MathSciNetMATHCrossRefGoogle Scholar
  14. Murphy S, van der Vaart A (1997) Semi-parametric likelihood ratio inference. Ann Stat 25: 1471–1509MATHCrossRefGoogle Scholar
  15. Owen A (2001) Empirical likelihood. Chapman & Hall, LondonMATHCrossRefGoogle Scholar
  16. Pan X, Zhou M (1999) Using 1-parameter sub-family of distributions in empirical likelihood ratio with censored data. J Stat Plan Inf 75: 379–392MathSciNetMATHCrossRefGoogle Scholar
  17. Peng L, Huang Y (2008) Survival analysis with quantile regression models. J Amer Statist Assoc 103: 637–649MathSciNetMATHCrossRefGoogle Scholar
  18. Pollard D (1991) Asymptotics of least absolute deviation regression estimators. Econometric Theory 7: 186–199MathSciNetCrossRefGoogle Scholar
  19. Portnoy S (2003) Censored regression quantiles. J Amer Statist Assoc 98: 1001–1012MathSciNetMATHCrossRefGoogle Scholar
  20. Powell J (1986) Censored regression quantiles. J Econom 32: 143–155MATHCrossRefGoogle Scholar
  21. R Development Core Team (2008) R: A language and environment for statistical computing. R Foundation for Statistical Computing, http://www.R-project.org
  22. Rotnitzky A, Robins JM (2005) Inverse probability weighted estimation in survival analysis 2nd edn. WileyGoogle Scholar
  23. Stute W (1993) Consistent estimation under random censorship when covariables are present. J Multivariate Anal 45: 89–103MathSciNetMATHCrossRefGoogle Scholar
  24. Stute W (1995) The central limit theorem under random censorship. Ann Statist 23: 422–439MathSciNetMATHCrossRefGoogle Scholar
  25. Stute W (1996) Distributional convergence under random censorship when covariables are present. Scand J Stat 23: 461–471MathSciNetMATHGoogle Scholar
  26. van der Laan M, Robins JM (2003) Unified methods for censored longitudinal data and causality. Springer, New YorkMATHGoogle Scholar
  27. Wang JH, Wang L (2009) Locally weighted censored quantile regression. J Am Stat Assoc 104: 1117–1128CrossRefGoogle Scholar
  28. Ying Z, Jung S, Wei L (1995) Survival analysis with median regression models. J Am Stat Assoc 90: 178–184MathSciNetMATHCrossRefGoogle Scholar
  29. Zhou M (2011) A wilks thorem for teh censored empirical liklihood of means. Unpublished Manuscript: http://www.ms.uky.edu/~mai/research/Note3.pdf
  30. Zhou M, Kim M, Bathke A (2011) Empirical likelihood analysis for the heteroscedastic accelerated failure time model. Statistica Sinica (in press)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Cincinnati Children’s Medical CenterCincinnatiUSA
  2. 2.University of KentuckyLexingtonUSA
  3. 3.University of PittsburghPittsburghUSA

Personalised recommendations