Regularization for Cox’s proportional hazards model with NP-dimensionality. (English) Zbl 1246.62202

Summary: High throughput genetic sequencing arrays with thousands of measurements per sample and a great amount of related censored clinical data have increased demanding the need for better measurement specific model selection. We establish strong oracle properties of non-concave penalized methods for nonpolynomial (NP) dimensional data with censoring in the framework of Cox’s proportional hazards model. A class of folded-concave penalties are employed and both LASSO and SCAD are discussed specifically. We unveil the question under which dimensionality and correlation restrictions can an oracle estimator be constructed and grasped. It is demonstrated that non-concave penalties lead to significant reduction of the “irrepresentable condition” needed for LASSO model selection consistency. The large deviations result for martingales, bearing interests of its own, is developed for characterizing the strong oracle property. Moreover, the non-concave regularized estimator is shown to achieve asymptotically the information bound of the oracle estimator. A coordinate-wise algorithm is developed for finding the grid of solution paths for penalized hazard regression problems, and its performance is evaluated on simulated and gene association study examples.


62P10 Applications of statistics to biology and medical sciences; meta analysis
62N02 Estimation in survival analysis and censored data
62N01 Censored data models
60G44 Martingales with continuous parameter
60F10 Large deviations
92C50 Medical applications (general)


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[1] Andersen, P. K. and Gill, R. D. (1982). Cox’s regression model for counting processes: A large sample study. Ann. Statist. 10 1100-1120. · Zbl 0526.62026
[2] Bertsekas, D. P. (2003). Nonlinear programming. Athena Scientific, Nashua, NH. · Zbl 0935.90037
[3] Bhatia, R. (1997). Matrix Analysis. Graduate Texts in Mathematics 169 . Springer, New York. · Zbl 0863.15001
[4] Bickel, P. J., Ritov, Y. and Tsybakov, A. B. (2009). Simultaneous analysis of lasso and Dantzig selector. Ann. Statist. 37 1705-1732. · Zbl 1173.62022
[5] Bradic, J., Fan, J. and Wang, W. (2011). Penalized composite quasi-likelihood for ultrahigh dimensional variable selection. J. R. Stat. Soc. Ser. B Stat. Methodol. 73 325-349.
[6] Bradic, J., Fan, J. and Jiang, J. (2011). Supplement to “Regularization for Cox’s proportional hazards model with NP-dimensionality.” . · Zbl 1246.62202
[7] Bunea, F., Tsybakov, A. and Wegkamp, M. (2007). Sparsity oracle inequalities for the Lasso. Electron. J. Stat. 1 169-194 (electronic). · Zbl 1146.62028
[8] Cai, J., Fan, J., Li, R. and Zhou, H. (2005). Variable selection for multivariate failure time data. Biometrika 92 303-316. · Zbl 1094.62123
[9] Candes, E. and Tao, T. (2007). The Dantzig selector: Statistical estimation when p is much larger than n. Ann. Statist. 35 2313-2351. · Zbl 1139.62019
[10] Daubechies, I., Defrise, M. and De Mol, C. (2004). An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57 1413-1457. · Zbl 1077.65055
[11] Dave, S. S. et al. (2004). Prediction of survival in follicular lymphoma based on molecular features of tumor-infiltrating immune cells. N. Engl. J. Med. 351 2159-2169.
[12] de la Peña, V. H. (1999). A general class of exponential inequalities for martingales and ratios. Ann. Probab. 27 537-564. · Zbl 0942.60004
[13] Du, P., Ma, S. and Liang, H. (2010). Penalized variable selection procedure for Cox models with semiparametric relative risk. Ann. Statist. 38 2092-2117. · Zbl 1202.62132
[14] Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc. 96 1348-1360. · Zbl 1073.62547
[15] Fan, J. and Li, R. (2002). Variable selection for Cox’s proportional hazards model and frailty model. Ann. Statist. 30 74-99. · Zbl 1012.62106
[16] Fan, J. and Lv, J. (2011). Non-concave penalized likelihood with NP-dimensionality. IEEE Trans. Inform. Theory 57 5467-5484. · Zbl 1365.62277
[17] Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis . Wiley, New York. · Zbl 0727.62096
[18] Friedman, J., Hastie, T. and Tibshirani, R. (2010). Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software 33 1-22.
[19] Friedman, J., Hastie, T., Höfling, H. and Tibshirani, R. (2007). Pathwise coordinate optimization. Ann. Appl. Stat. 1 302-332. · Zbl 1378.90064
[20] Johnson, B. A. (2009). On lasso for censored data. Electron. J. Stat. 3 485-506. · Zbl 1326.62201
[21] Juditsky, A. B. and Nemirovski, A. S. (2011). Large deviations of vector-valued martingales in 2-smooth normed spaces. Ann. Appl. Probab. To appear. Available at .
[22] Kim, Y., Choi, H. and Oh, H. (2008). Smoothly clipped absolute deviation on high dimensions. J. Amer. Statist. Assoc. 103 1656-1673. · Zbl 1286.62062
[23] Koltchinskii, V. (2009). The Dantzig selector and sparsity oracle inequalities. Bernoulli 15 799-828. · Zbl 1452.62486
[24] Lv, J. and Fan, Y. (2009). A unified approach to model selection and sparse recovery using regularized least squares. Ann. Statist. 37 3498-3528. · Zbl 1369.62156
[25] Massart, P. and Meynet, C. (2010). An l 1 oracle inequality for the LASSO. Available at . · Zbl 1274.62468
[26] Meinshausen, N. and Bühlmann, P. (2006). High-dimensional graphs and variable selection with the lasso. Ann. Statist. 34 1436-1462. · Zbl 1113.62082
[27] Meinshausen, N. and Yu, B. (2009). Lasso-type recovery of sparse representations for high-dimensional data. Ann. Statist. 37 246-270. · Zbl 1155.62050
[28] Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58 267-288. · Zbl 0850.62538
[29] Tibshirani, R. (1997). The LASSO method for variable selection in the Cox model. Stat. Med. 16 385-395.
[30] van de Geer, S. (1995). Exponential inequalities for martingales, with application to maximum likelihood estimation for counting processes. Ann. Statist. 23 1779-1801. · Zbl 0852.60019
[31] van de Geer, S. and Bühlmann, P. (2009). On conditions used to prove oracle results for the LASSO. Electron. J. Stat. 3 1360-1392. · Zbl 1327.62425
[32] Wang, S., Nan, B., Zhou, N. and Zhu, J. (2009). Hierarchically penalized Cox regression with grouped variables. Biometrika 96 307-322. · Zbl 1163.62089
[33] Wu, T. T. and Lange, K. (2008). Coordinate descent algorithms for lasso penalized regression. Ann. Appl. Stat. 2 224-244. · Zbl 1137.62045
[34] Yuan, M. and Lin, Y. (2007). On the non-negative garrote estimator. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 143-161. · Zbl 1120.62052
[35] Zhang, C.-H. (2010). Nearly unbiased variable selection under minimax concave penalty. Ann. Statist. 38 894-942. · Zbl 1183.62120
[36] Zhang, C.-H. and Huang, J. (2008). The sparsity and bias of the LASSO selection in high-dimensional linear regression. Ann. Statist. 36 1567-1594. · Zbl 1142.62044
[37] Zhao, P. and Yu, B. (2006). On model selection consistency of Lasso. J. Mach. Learn. Res. 7 2541-2563. · Zbl 1222.62008
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