Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Variable selection for survival data with a class of adaptive elastic net techniques

Abstract

The accelerated failure time (AFT) models have proved useful in many contexts, though heavy censoring (as for example in cancer survival) and high dimensionality (as for example in microarray data) cause difficulties for model fitting and model selection. We propose new approaches to variable selection for censored data, based on AFT models optimized using regularized weighted least squares. The regularized technique uses a mixture of \(\ell _1\) and \(\ell _2\) norm penalties under two proposed elastic net type approaches. One is the adaptive elastic net and the other is weighted elastic net. The approaches extend the original approaches proposed by Ghosh (Adaptive elastic net: an improvement of elastic net to achieve oracle properties, Technical Reports 2007) and Hong and Zhang (Math Model Nat Phenom 5(3):115–133 2010), respectively. We also extend the two proposed approaches by adding censoring observations as constraints into their model optimization frameworks. The approaches are evaluated on microarray and by simulation. We compare the performance of these approaches with six other variable selection techniques-three are generally used for censored data and the other three are correlation-based greedy methods used for high-dimensional data.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. Akaike, H.: Information theory as an extension of the maximum likelihood principle. In: Petrov, B.N., Csaki, F. (eds.) Second International Symposium on Information Theory, pp. 267–281. Akademiai Kiado, Budapest (1973)

  2. Antoniadis, A., Fryzlewicz, P., Letue, F.: The Dantzig selector in Cox’s proportional hazards model. Scand. J. Stat. 37(4), 531–552 (2010)

  3. Buckley, J., James, I.: Linear regression with censored data. Biometrika 66, 429–436 (1979)

  4. Bühlmann, P., Kalisch, M., Maathuis, M.H.: Variable selection in high-dimensional linear models: partially faithful distributions and the PC-simple algorithm. Biometrika 97(2), 261–278 (2010)

  5. Cai, T., Huang, J., Tian, L.: Regularized estimation for the accelerated failure time model. Biometrics 65, 394–404 (2009)

  6. Candes, E., Tao, T.: The Dantzig selector: statistical estimation when \(p\) is much larger than \(n\). Ann. Stat. 35(6), 2313–2351 (2007)

  7. Cho, H., Fryzlewicz, P.: High dimensional variable selection via tilting. J. R. Stat. Soc. Ser. B 74(3), 593–622 (2012)

  8. Cox, D.R.: Regression models and life-tables. J. R. Stat. Soc. Ser. B 34, 187–220 (1972)

  9. Datta, S., Le-Rademacher, J., Datta, S.: Predicting patient survival from microarray data by accelerated failure time modeling using partial least squares and LASSO. Biometrics 63, 259–271 (2007)

  10. Efron, B.: The two sample problem with censored data. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 4, pp. 831–853. Prentice Hall, New York (1967)

  11. Efron, B., Tibshirani, R.: An Introduction to the Bootstrap. Chapman and Hall, New York (1993)

  12. Efron, B., Hastie, T., Johnstone, I., Tibshirani, R.: Least angle regression. Ann. Stat. 32, 407–499 (2004)

  13. Engler, D., Li, Y.: Survival analysis with high-dimensional covariates: an application in microarray studies. Stat. Appl. Genet. Mol. Biol. 8(1), 1–22 (2009). (Article 14)

  14. Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96, 1348–1360 (2001)

  15. Fan, J., Li, R.: Variable selection for Cox’s proportional hazards model and frailty model. Ann. Stat. 30, 74–99 (2002)

  16. Fan, J., Lv, J.: Sure independence screening for ultrahigh dimensional feature space. J. R. Stat. Soc. Ser. B 70(5), 849–911 (2008)

  17. Faraggi, D., Simon, R.: Bayesian variable selection method for censored survival data. Biometrics 54, 1475–1485 (1998)

  18. Frank, I.E., Friedman, J.H.: A statistical view of some chemometrics regression tools. Technometrics 35(2), 109–135 (1993)

  19. Gehan, E.A.: A generalized Wilcoxon test for comparing arbitrarily singlecensored samples. Biometrika 52, 203–223 (1965)

  20. Ghosh, S.: On the grouped selection and model complexity of the adaptive elastic net. Stat. Comput. 21(3), 451–462 (2011)

  21. Ghosh, S.: Adaptive elastic net: an improvement of elastic net to achieve oracle properties. Technical Reports, Indiana University-Purdue University, Indianapolis, (PR no. 07–01) (2007)

  22. Gui, J., Li, H.: Penalized Cox regression analysis in the high-dimensional and low-sample size settings, with applications to microarray gene expression data. Bioinformatics 21, 3001–3008 (2005)

  23. Hong, D., Zhang, F.: Weighted elastic net model for mass spectrometry imaging processing. Math. Model. Nat. Phenom. 5(3), 115–133 (2010)

  24. Hu, S., Rao, J.S.: Sparse penalization with censoring constraints for estimating high dimensional AFT models with applications to microarray data analysis. Technical Reports, University of Miami (2010)

  25. Huang, J., Harrington, D.: Iterative partial least squares with rightcensored data analysis: a comparison to other dimension reduction techniques. Biometrics 61(1), 17–24 (2005)

  26. Huang, J., Ma, S.: Variable selection in the accelerated failure time model via the bridge method. Lifetime Data Anal. 16, 176–195 (2010)

  27. Huang, J., Ma, S., Xie, H.: Regularized estimation in the accelerated failure time model with high-dimensional covariates. Biometrics 62, 813–820 (2006)

  28. Hunter, D.R., Li, R.: Variable selection using MM algorithms. Ann. Stat. 33(4), 1617–1642 (2005)

  29. Jin, Z., Lin, D., Wei, L.J., Ying, Z.L.: Rank-based inference for the accelerated failure time model. Biometrika 90, 341–353 (2003)

  30. Jin, Z., Lin, D.Y., Ying, Z.: On least-squares regression with censored data. Biometrika 93(1), 147–161 (2006)

  31. Khan, M.H.R., Shaw, J.E.H.: AdapEnetClass: a class of adaptive elastic net methods for censored data. R package version 1.1 (2014)

  32. Khan, M.H.R.: Variable selection and estimation procedures for high-dimensional survival data. Ph.D. Thesis, Department of Statistics, University of Warwick (2013)

  33. Khan, M.H.R., Shaw, J.E.H.: On dealing with censored largest observations under weighted least squares. CRiSM Working Paper, No 13–07 Department of Statistics, University of Warwick (2013b)

  34. Khan, M.H.R., Shaw, J.E.H.: Variable selection with the modified Buckley- James method and the dantzig selector for high-dimensional survival data. In: 59th ISI World Statistics Congress Proceedings, Hong Kong, pp. 4239–4244, 25–30 Aug 2013c

  35. Kriegeskorte, N., Simmons, W.K., Bellgowan, P.S.F., Baker, C.I.: Circular analysis in systems neuroscience: the dangers of double dipping. Nat. Neurosci. 12(5), 535–540 (2009)

  36. Li, H., Luan, Y.: Kernel Cox regression models for linking gene expression profiles to censored survival data. Pac. Symp. Biocomput. 8, 65–76 (2003)

  37. Meinshausen, N., Bühlmann, P.: Stability selection. J. R. Stat. Soc. Ser. B 72(4), 417–473 (2010)

  38. Peduzzi, P.N., Hardy, R.J., Holford, T.R.: A stepwise variable selection procedure for nonlinear regression models. Biometrics 36, 511–516 (1980)

  39. Radchenko, P., James, G.M.: Improved variable selection with Forward-Lasso adaptive shrinkage. Ann. Appl. Stat. 5(1), 427–448 (2011)

  40. Rosenwald, A., Wright, G., Wiestner, A., Chan, W., Connors, J., Campo, E., Gascoyne, R., Grogan, T., Muller Hermelink, H., Smeland, E., Chiorazzi, M., Giltnane, J., Hurt, E., Zhao, H., Averett, L., Henrickson, S., Yang, L., Powell, J., Wilson, W., Jaffe, E., Simon, R., Klausner, R., Montserrat, E., Bosch, F., Greiner, T., Weisenburger, D., Sanger, W., Dave, B., Lynch, J., Vose, J., Armitage, J., Fisher, R., Miller, T., LeBlanc, M., Ott, G., Kvaloy, S., Holte, H., Delabie, J., Staudt, L.: The proliferation gene expression signature is a quantitative integrator of oncogenic events that predicts survival in mantle cell lymphoma. Cancer Cell 3, 185–197 (2003)

  41. Sha, N., Tadesse, M.G., Vannucci, M.: Bayesian variable selection for the analysis of microarray data with censored outcome. Bioinformatics 22(18), 2262–2268 (2006)

  42. Stute, W.: Consistent estimation under random censorship when covariables are available. J. Multivar. Anal. 45, 89–103 (1993)

  43. Stute, W.: Distributional convergence under random censorship when covariables are present. Scand. J. Stat. 23, 461–471 (1996)

  44. Swerdlow, S., Williams, M.: From centrocytic to mantle cell lymphoma: a clinicopathologic and molecular review of 3 decades. Hum. Pathol. 33, 7–20 (2002)

  45. Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B 58, 267–288 (1996)

  46. Tibshirani, R.: The lasso method for variable selection in the Cox model. Stat. Med. 16, 385–395 (1997)

  47. Wang, S., Nan, B., Zhu, J., Beer, D.G.: Doubly penalized Buckley-James method for survival data with high-dimensional covariates. Biometrics 64, 132–140 (2008)

  48. Wu, Y.: Elastic net for Cox’s proportional hazards model with a solution path algorithm. Stat. Sin. 22, 271–294 (2012)

  49. Ying, Z.: A large sample study of rank estimation for censored regression data. Ann. Stat. 21(1), 76–99 (1993)

  50. Yuan, M., Lin, Y.: Model selection and estimation in regression with grouped variables. J. R. Stat. Soc. Ser. B 68, 49–67 (2006)

  51. Zhang, C.H.: Nearly unbiased variable selection under minimax concave penalty. Ann. Stat. 38(2), 894–942 (2010)

  52. Zou, H.: The adaptive lasso and its oracle properties. J. Am. Stat. Assoc. 101, 1418–1429 (2006)

  53. Zou, H., Hastie, T.: Regularization and variable selection via the elastic net. J. R. Stat. Soc. Ser. B 67, 301–320 (2005)

  54. Zou, H., Zhang, H.H.: On the adaptive elastic-net with a diverging number of parameters. Ann. Stat. 37(4), 1733–1751 (2009)

Download references

Acknowledgments

The first author is grateful to the centre for research in Statistical Methodology (CRiSM), Department of Statistics, University of Warwick, UK for offering research funding for his PhD study.

Author information

Correspondence to Md Hasinur Rahaman Khan.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Khan, M.H.R., Shaw, J.E.H. Variable selection for survival data with a class of adaptive elastic net techniques. Stat Comput 26, 725–741 (2016). https://doi.org/10.1007/s11222-015-9555-8

Download citation

Keywords

  • Adaptive elastic net
  • AFT
  • Variable selection
  • Stute’s weighted least squares
  • Weighted elastic net