Skip to main content
SpringerLink
Log in

Hierarchically penalized additive hazards model with diverging number of parameters

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

In many applications, covariates can be naturally grouped. For example, for gene expression data analysis, genes belonging to the same pathway might be viewed as a group. This paper studies variable selection problem for censored survival data in the additive hazards model when covariates are grouped. A hierarchical regularization method is proposed to simultaneously estimate parameters and select important variables at both the group level and the within-group level. For the situations in which the number of parameters tends to ∞ as the sample size increases, we establish an oracle property and asymptotic normality property of the proposed estimators. Numerical results indicate that the hierarchically penalized method performs better than some existing methods such as lasso, smoothly clipped absolute deviation (SCAD) and adaptive lasso.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aalen O. A model for non-parametric regression analysis of counting processes. In: Klonecki W, Kozek A, Rosinski J, eds. Mathematical Statistics and Probability Theory. Lecture Notes in Statist, vol. 2. New York: Springer-Verlag, 1980, 1–25

    Google Scholar 

  2. Antoniadis A, Fan J. Regularization of wavelet approximations. J Amer Statist Assoc, 2001, 96: 939–967

    Article  MATH  MathSciNet  Google Scholar 

  3. Cai J, Fan J, Li R, et al. Variable selection for multivariate failure time data. Biometrika, 2005, 92: 303–316

    Article  MATH  MathSciNet  Google Scholar 

  4. Cox D R. Regression models and life-table (with discussion). J Royal Statist Soc Ser B, 1972, 4: 187–220

    Google Scholar 

  5. Dave S S, Wright G, Tan B, et al. Prediction of survival in follicular lymphoma based on molecular features of tumor-infiltrating immune cells. N Engl J Med, 2004, 351: 2159–2169

    Article  Google Scholar 

  6. Fan J, Li R. Variable selection via nonconcave penalized likelihood and its oracle properties. J Amer Statist Assoc, 2001, 96: 1348–1360

    Article  MATH  MathSciNet  Google Scholar 

  7. Fan J, Lv J. Non-concave penalized likelihood with NP-dimensionality. IEEE Trans Inform Theory, 2011, 57: 5467–5484

    Article  MathSciNet  Google Scholar 

  8. Fan J, Peng H. Nonconcave penalized likelihood with a diverging number of parameters. Ann Statist, 2004, 32: 928–961

    Article  MATH  MathSciNet  Google Scholar 

  9. Frank I E, Friedman J H. A statistical view of some chemometrics regression tools (with discussion). Technometrics, 1993, 35: 109–148

    Article  MATH  Google Scholar 

  10. Friedman J, Hastie T, Höfling H, et al. Pathwise coordinate optimization. Ann Appl Statist, 2007, 1: 302–332

    Article  MATH  Google Scholar 

  11. Huang J, Ma S, Xie H, et al. A group bridge approach for variable selection. Biometrika, 2009, 96: 339–355

    Article  MATH  MathSciNet  Google Scholar 

  12. Leng C, Ma S. Path consistent model selection in additive risk model via lasso. Stat Med, 2007, 26: 3753–3770

    Article  MathSciNet  Google Scholar 

  13. Lin D Y, Ying Z L. Semiparametric analysis of the additive risk model. Biometrika, 1994, 81: 61–71

    Article  MATH  MathSciNet  Google Scholar 

  14. Ma S, Huang J. Clustering threshold gradient descent regularization: with applications to microarray studies. Bioinformatics, 2007, 23: 466–472

    Article  Google Scholar 

  15. Martinussen T, Scheike T. Covariate selection for the semiparametric additive risk model. Scand J Statist, 2009, 36: 602–619

    Article  MATH  MathSciNet  Google Scholar 

  16. Wang S, Nan B, Zhou N, et al. Hierarchically penalized Cox regression for censored data with grouped variables and its oracle property. Biometrika, 2009, 96: 307–322

    Article  MATH  MathSciNet  Google Scholar 

  17. Yuan M, Lin Y. Model selection and estimation in regression with grouped variables. J Royal Statist Soc Ser B, 2006, 68: 49–67

    Article  MATH  MathSciNet  Google Scholar 

  18. Zhao P, Rocha G, Yu B. Grouped and hierarchical model selection through composite absolute penalties. Ann Statist, 2009, 37: 3468–3497

    Article  MATH  MathSciNet  Google Scholar 

  19. Zhou N, Zhu J. Group variable selection via hierarchical lasso and its oracle property. Stat Interface, 2010, 3: 557–574

    Article  MATH  MathSciNet  Google Scholar 

  20. Zou H. The adaptive lasso and its oracle properties. J Amer Statist Assoc, 2006, 101: 1418–1429

    Article  MATH  MathSciNet  Google Scholar 

  21. Zou H, Zhang H. On the adaptive elastic-net with a diverging number of parameters. Ann Statist, 2009, 37: 1733–1751

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Finance and Statistics, East China Normal University, Shanghai, 200241, China

    JiCai Liu, RiQuan Zhang & WeiHua Zhao

  2. Department of Mathematics, Shanxi Datong University, Datong, 037009, China

    RiQuan Zhang

  3. School of Science, Nantong University, Nantong, 226007, China

    WeiHua Zhao

Authors
  1. JiCai Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. RiQuan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. WeiHua Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to RiQuan Zhang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, J., Zhang, R. & Zhao, W. Hierarchically penalized additive hazards model with diverging number of parameters. Sci. China Math. 57, 873–886 (2014). https://doi.org/10.1007/s11425-013-4679-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-013-4679-9

Keywords

MSC(2010)

Search

Navigation