Lifetime Data Analysis

, Volume 22, Issue 1, pp 100–121 | Cite as

Non-crossing weighted kernel quantile regression with right censored data

  • Sungwan Bang
  • Soo-Heang Eo
  • Yong Mee Cho
  • Myoungshic Jhun
  • HyungJun Cho
Article
First Online:
Received:
Accepted:

Abstract

Regarding survival data analysis in regression modeling, multiple conditional quantiles are useful summary statistics to assess covariate effects on survival times. In this study, we consider an estimation problem of multiple nonlinear quantile functions with right censored survival data. To account for censoring in estimating a nonlinear quantile function, weighted kernel quantile regression (WKQR) has been developed by using the kernel trick and inverse-censoring-probability weights. However, the individually estimated quantile functions based on the WKQR often cross each other and consequently violate the basic properties of quantiles. To avoid this problem of quantile crossing, we propose the non-crossing weighted kernel quantile regression (NWKQR), which estimates multiple nonlinear conditional quantile functions simultaneously by enforcing the non-crossing constraints on kernel coefficients. The numerical results are presented to demonstrate the competitive performance of the proposed NWKQR over the WKQR.

Keywords

Kernel Multiple quantiles regression Non-crossing  Right censored data 
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Notes

Acknowledgments

The authors are grateful to the editor, the associate editor, and the reviewers for their constructive and insightful comments and suggestions, which helped to dramatically improve the quality of this paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by (1) the Ministry of Science, ICT and Future Planning (NRF-2013R1A1A1007536) for S. Bang, (2) the Ministry of Education (NRF-2013R1A1A2A10007545) for M. Jhun, and (3) the Ministry of Education, Science and Technology (2010-0007936) for H. Cho.

References

  1. Andersen ED, Roos C, Terlaky T (2003) On implementing a primal-dual interior-point method for conic quadratic optimization. Math Programm 95:2MathSciNetCrossRefMATHGoogle Scholar
  2. Bang H, Tsiatis AA (2002) Median regression with censored cost data. Biometrics 55:643–649MathSciNetCrossRefMATHGoogle Scholar
  3. Cai T, Huang J, Tian L (2009) Regularized estimation for the accelerated failure time model. Biometrics 65:394–404MathSciNetCrossRefMATHGoogle Scholar
  4. Cheong CW (2010) Estimating the Hurst parameter in financial time series via heuristic approaches. J Appl Stat 37:201–214MathSciNetCrossRefGoogle Scholar
  5. Cristianini N, Shawe-Taylor J (2000) An introduction to support vector machines (and other kernel-based learning methods). Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  6. Friberg HA (2013) Rmosek: the R-to-MOSEK optimization interface. R package version 7.0.1. http://rmosek.r-forge.r-project.org/, http://www.mosek.com/
  7. Gelius-Dietrich G (2013) cplexAPI: R interfact to C API of IBM ILOG CPLEX. R package version 1.2.9. http://cran.r-project.org/web/packages/cplexAPI
  8. Hendricks W, Koenker R (1992) Hierarchical spline models for conditional quantiles and the demand for electricity. J Am Stat Assoc 87:58–68CrossRefGoogle Scholar
  9. Huang H, Haaland P, Lu X, Liu Y, Marron JS (2013) DWD: DWD implementation based on A IPM SOCP solver. R package version 0.11. http://CRAN.R-project.org/package=DWD
  10. Huang J, Ma S, Xie H (2007) Least absolute deviations estimation for the accelerated failure time model. Stat Sin 17:1533–1548MathSciNetMATHGoogle Scholar
  11. Kaplan EL, Meier P (1958) Nonparametric estimation from incomplete observations. J Am Stat Assoc 53:457–481MathSciNetCrossRefMATHGoogle Scholar
  12. Karatzoglou A, Smola A, Hornik K, Zeileis A (2004) kernlab—an S4 Package for Kernel Methods in R. J Stat Softw 11(9):1–20. http://www.jstatsoft.org/v11/i09/
  13. Kimeldorf G, Wahba G (1971) Some results on Tchebycheffian spline functions. J Math Anal Appl 33:82–95MathSciNetCrossRefMATHGoogle Scholar
  14. Koenker R, Bassett G (1978) Regression quantiles. Econometrica 4:33–50MathSciNetCrossRefMATHGoogle Scholar
  15. Koenker R, Geling R (2001) Reappraising medfly longevity: a quantile regression survival analysis. J Am Stat Assoc 96:458–468MathSciNetCrossRefMATHGoogle Scholar
  16. Koenker R, Hallock K (2001) Quantile regression. J Econ Perspect 15:143–156CrossRefGoogle Scholar
  17. Koenker R, Ng P, Portnoy S (1994) Quantile smoothing splines. Biometrika 81:673–680Google Scholar
  18. Koul H, Susarla V, Van Ryzin J (1981) Regression analysis with randomly right censored data. Ann Stat 9:1276–1288CrossRefMathSciNetMATHGoogle Scholar
  19. León LF, Cai T, Wei LJ (2009) Robust inferences for covariate effects on survival time with censored linear regression models. Stat Biosci 1:50–64CrossRefGoogle Scholar
  20. Li Y, Liu Y, Zhu J (2007) Quantile regression in reproducing kernel hilbert spaces. J Am Stat Assoc 102:255–268MathSciNetCrossRefMATHGoogle Scholar
  21. Liu Y, Wu Y (2011) Simultaneous multiple non-crossing quantile regression estimation using kernel constraints. J Nonparametr Stat 23:415–437MathSciNetCrossRefMATHGoogle Scholar
  22. Miller R, Halpern J (1982) Regression with censored data. Biometrika 69(3):521–531MathSciNetCrossRefMATHGoogle Scholar
  23. Park JY, Lee J-L, Baek S, Eo S-H, Ro JY, Cho YM (2014) Sarcomatoid features, necrosis, and grade are prognostic factors in metastatic clear cell renal cell carcinoma with vascular endothelial growth factor-targeted therapy. Hum Pathol 45(7):1437–1444CrossRefGoogle Scholar
  24. Portnoy S (2003) Censored regression quantiles. J Am Stat Assoc 98:1001–1012MathSciNetCrossRefMATHGoogle Scholar
  25. R Core Team (2013) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org/
  26. Reich BJ, Fuentes M, Dunson DB (2011) Bayesian spatial quantile regression. J Am Stat Assoc 106:6–20MathSciNetCrossRefMATHGoogle Scholar
  27. Scholkopf B, Smola A (2002) Learning with kernels support vector machines, regularization, optimization and beyond. MIT Press, Cambridge, MAGoogle Scholar
  28. Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6:461–464Google Scholar
  29. Shim J, Hwang C (2009) Support vector censored quantile regression under random censoring. Comput Stat Data Anal 53:912–919MathSciNetCrossRefMATHGoogle Scholar
  30. Sousa SK, Pires JCM, Martins FG, Pereira MC, Alvim-Ferraz MCM (2008) Potentialities of quantile regression to predict ozone concentrations. Environmetrics 20:147–158MathSciNetCrossRefGoogle Scholar
  31. Stute W (1993) Consistent estimation under random censorship when covariables are present. J Multivar Anal 45:89–103MathSciNetCrossRefMATHGoogle Scholar
  32. Takeuchi I, Le QV, Sears TD, Smola AJ (2006) Nonparametric quantile estimation. J Mach Learn Res 7:1231–1264MathSciNetMATHGoogle Scholar
  33. Therneau TM, Grambsch PM (2000) Modeling survival data: extending the Cox model. Springer, New YorkCrossRefMATHGoogle Scholar
  34. Turlach B, Weingessel A (2013) quadprog: Functions to solve quadratic programming problems. R package version 1.5-5. http://CRAN.R-project.org/package=quadprog
  35. Wang H, He X (2007) Detecting differential expressions in genechip microarray studies: a quantile approach. J Am Stat Assoc 102:104–112CrossRefMathSciNetMATHGoogle Scholar
  36. Wang H, Wang L (2009) Locally weighted censored quantile regression. J Am Stat Assoc 104:1117–1128CrossRefMathSciNetMATHGoogle Scholar
  37. Wu Y, Liu Y (2009) Stepwise multiple quantile regression estimation using non-crossing constraints. Stat Interface 2:299–310MathSciNetCrossRefMATHGoogle Scholar
  38. Yang S (1999) Censored median regression using weighted empirical survival and hazard functions. J Am Stat Assoc 94:137–145CrossRefMathSciNetMATHGoogle Scholar
  39. Ying Z, Jung SH, Wei LJ (1995) Survival analysis with median regression models. J Am Stat Assoc 90:178–184MathSciNetCrossRefMATHGoogle Scholar
  40. Yuan M (2006) GACV for quantile smoothing splines. Comput Stat Data An 50:813–829Google Scholar
  41. Zhou L (2006) A simple censored median regression estimator. Stat Sin 16:1043–1058MathSciNetMATHGoogle Scholar
  42. Zhou M (1992) M-estimation in censored linear models. Biometrika 79:837–841MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Sungwan Bang
    • 1
  • Soo-Heang Eo
    • 2
  • Yong Mee Cho
    • 3
  • Myoungshic Jhun
    • 2
  • HyungJun Cho
    • 2
  1. 1.Department of MathematicsKorea Military AcademySeoulRepublic of Korea
  2. 2.Department of StatisticsKorea UniversitySeoulRepublic of Korea
  3. 3.Department of PathologyAsan Medical CenterSeoulRepublic of Korea

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