An efficient algorithm for joint feature screening in ultrahigh-dimensional Cox’s model. (English) Zbl 1505.62099
Summary: The Cox model is an exceedingly popular semiparametric hazard regression model for the analysis of time-to-event accompanied by explanatory variables. Within the ultrahigh-dimensional data setting, not like the marginal screening strategy, there is a joint feature screening method based on the partial likelihood of the Cox model but it leaves computational feasibility unsolved. In this paper, we develop an enhanced iterative hard-thresholding algorithm by adapting the non-monotone proximal gradient method under the Cox model. The proposed algorithm is efficient because it is computationally both effective and fast. Meanwhile, our proposed algorithm begins with a LASSO initial estimator rather than the naive zero initial and still enjoys sure screening in theory and further enhances the computational efficiency in practice. We also give a rigorous theory proof. The advantage of our proposed work is demonstrated by numerical studies and illustrated by the diffuse large B-cell lymphoma data example.
MSC:
| 62-08 | Computational methods for problems pertaining to statistics |
| 62G08 | Nonparametric regression and quantile regression |
| 62J07 | Ridge regression; shrinkage estimators (Lasso) |
| 62N01 | Censored data models |
| 62P10 | Applications of statistics to biology and medical sciences; meta analysis |
Keywords:
Cox’s model; Lasso initial; locally Lipschitz optimization; non-monotone proximal gradient; joint feature screeningReferences:
| [1] | Andersen, PK; Gill, RD, Cox’s regression model for counting processes: a large sample study, Ann Stat, 10, 4, 1100-1120 (1982) · Zbl 0526.62026 · doi:10.1214/aos/1176345976 |
| [2] | Barzilai, J.; Borwein, JM, Two-point step size gradient methods, IMA J Numer Anal, 8, 1, 141-148 (1988) · Zbl 0638.65055 · doi:10.1093/imanum/8.1.141 |
| [3] | Bickel, PJ; Ritov, Y.; Tsybakov, AB, Simultaneous analysis of Lasso and Dantzig selector, Ann Stat, 37, 4, 1705-1732 (2009) · Zbl 1173.62022 · doi:10.1214/08-AOS620 |
| [4] | Bradic, J.; Fan, J.; Jiang, J., Regularization for Cox’s proportional hazards model with NP-dimensionality, Ann Stat, 39, 6, 3092-3120 (2011) · Zbl 1246.62202 · doi:10.1214/11-AOS911 |
| [5] | Breslow, N., Covariance analysis of censored survival data, Biometrics, 30, 1, 89-99 (1974) · doi:10.2307/2529620 |
| [6] | Chen, X.; Lu, Z.; Pong, TK, Penalty methods for a class of non-Lipschitz optimization problems, SIAM J Optim, 26, 3, 1465-1492 (2016) · Zbl 1342.90181 · doi:10.1137/15M1028054 |
| [7] | Chen, X.; Chen, X.; Wang, H., Robust feature screening for ultra-high dimensional right censored data via distance correlation, Comput Stat Data Anal, 119, 118-138 (2018) · Zbl 1469.62043 · doi:10.1016/j.csda.2017.10.004 |
| [8] | Cox, DR, Partial likelihood, Biometrika, 62, 2, 269-276 (1975) · Zbl 0312.62002 · doi:10.1093/biomet/62.2.269 |
| [9] | Fan, J.; Lv, J., Sure independence screening for ultrahigh dimensional feature space, J R Stat Soc Ser B, 70, 5, 849-911 (2008) · Zbl 1411.62187 · doi:10.1111/j.1467-9868.2008.00674.x |
| [10] | Fan, J.; Song, R., Sure independence screening in generalized linear models with NP-dimensionality, Ann Stat, 38, 6, 3567-3604 (2010) · Zbl 1206.68157 · doi:10.1214/10-AOS798 |
| [11] | Fan, J.; Samworth, R.; Wu, Y., Ultrahigh dimensional variable selection: beyond the linear model, J Mach Learn Res, 70, 2013-2038 (2009) · Zbl 1235.62089 |
| [12] | Fan J, Feng Y, Wu Y (2010) High-dimensional variable selection for Cox’s proportional hazards model. In: Borrowing strength: theory powering applications—a festschrift for Lawrence D. Brown, Institute of Mathematical Statistics, pp 70-86 |
| [13] | Fleming, TR; Harrington, DP, Counting processes and survival analysis (2011), New York: Wiley, New York · Zbl 1079.62093 |
| [14] | Gong P, Zhang C, Lu Z, Huang J, Ye J (2013) A general iterative shrinkage and thresholding algorithm for non-convex regularized optimization problems. In: International conference on machine learning, pp 37-45 |
| [15] | Gorst-Rasmussen, A.; Scheike, T., Independent screening for single-index hazard rate models with ultrahigh dimensional features, J R Stat Soc Ser B, 75, 2, 217-245 (2013) · Zbl 07555446 · doi:10.1111/j.1467-9868.2012.01039.x |
| [16] | He, X.; Wang, L.; Hong, HG, Quantile-adaptive model-free variable screening for high-dimensional heterogeneous data, Ann Stat, 41, 1, 342-369 (2013) · Zbl 1295.62053 · doi:10.1214/13-AOS1087 |
| [17] | Hong, HG; Chen, X.; Christiani, DC; Li, Y., Integrated powered density: screening ultrahigh dimensional covariates with survival outcomes, Biometrics, 74, 2, 421-429 (2018) · Zbl 1415.62101 · doi:10.1111/biom.12820 |
| [18] | Huang, J.; Sun, T.; Ying, Z.; Yu, Y.; Zhang, CH, Oracle inequalities for the Lasso in the Cox model, Ann Stat, 41, 3, 1142-1165 (2013) · Zbl 1292.62135 · doi:10.1214/13-AOS1098 |
| [19] | Lai, P.; Liu, Y.; Liu, Z.; Wan, Y., Model free feature screening for ultrahigh dimensional data with responses missing at random, Comput Stat Data Anal, 105, 201-216 (2017) · Zbl 1466.62125 · doi:10.1016/j.csda.2016.08.008 |
| [20] | Liu, Y.; Chen, X., Quantile screening for ultra-high-dimensional heterogeneous data conditional on some variables, J Stat Comput Simul, 88, 2, 329-342 (2018) · Zbl 07192555 · doi:10.1080/00949655.2017.1389944 |
| [21] | Liu, Y.; Zhang, J.; Zhao, X., A new nonparametric screening method for ultrahigh-dimensional survival data, Comput Stat Data Anal, 119, 74-85 (2018) · Zbl 1469.62107 · doi:10.1016/j.csda.2017.10.003 |
| [22] | Ma, S.; Li, R.; Tsai, CL, Variable screening via quantile partial correlation, J Am Stat Assoc, 112, 518, 650-663 (2017) · doi:10.1080/01621459.2016.1156545 |
| [23] | Meinshausen, N.; Bühlmann, P., High-dimensional graphs and variable selection with the Lasso, Ann Stat, 34, 1436-1462 (2006) · Zbl 1113.62082 |
| [24] | Metzeler, KH; Hummel, M.; Bloomfield, CD; Spiekermann, K.; Braess, J.; Sauerland, MC; Heinecke, A.; Radmacher, M.; Marcucci, G.; Whitman, SP, An 86-probe-set gene-expression signature predicts survival in cytogenetically normal acute myeloid leukemia, Blood, 112, 10, 4193-4201 (2008) · doi:10.1182/blood-2008-02-134411 |
| [25] | Oberthuer, A.; Berthold, F.; Warnat, P.; Hero, B.; Kahlert, Y.; Spitz, R.; Ernestus, K.; Konig, R.; Haas, S.; Eils, R., Customized oligonucleotide microarray gene expression-based classification of neuroblastoma patients outperforms current clinical risk stratification, J Clin Oncol, 24, 31, 5070-5078 (2006) · doi:10.1200/JCO.2006.06.1879 |
| [26] | Rosenwald, A.; Wright, G.; Chan, WC; Connors, JM; Campo, E.; Fisher, RI; Gascoyne, RD; Muller-Hermelink, HK; Smeland, EB; Giltnane, JM, The use of molecular profiling to predict survival after chemotherapy for diffuse large-B-cell lymphoma, N Engl J Med, 346, 25, 1937-1947 (2002) · doi:10.1056/NEJMoa012914 |
| [27] | Rosenwald, A.; Wright, G.; Wiestner, A.; Chan, WC; Connors, JM; Campo, E.; Gascoyne, RD; Grogan, TM; Muller-Hermelink, HK; Smeland, EB, The proliferation gene expression signature is a quantitative integrator of oncogenic events that predicts survival in mantle cell lymphoma, Cancer Cell, 3, 2, 185-197 (2003) · doi:10.1016/S1535-6108(03)00028-X |
| [28] | She, Y., Thresholding-based iterative selection procedures for model selection and shrinkage, Electron J Stat, 3, 384-415 (2009) · Zbl 1326.62158 · doi:10.1214/08-EJS348 |
| [29] | Song, R.; Lu, W.; Ma, S.; Jessie Jeng, X., Censored rank independence screening for high-dimensional survival data, Biometrika, 101, 4, 799-814 (2014) · Zbl 1306.62207 · doi:10.1093/biomet/asu047 |
| [30] | Tibshirani, R., The lasso method for variable selection in the Cox model, Stat Med, 16, 4, 385-395 (1997) · doi:10.1002/(SICI)1097-0258(19970228)16:4<385::AID-SIM380>3.0.CO;2-3 |
| [31] | van Wieringen, WN; Kun, D.; Hampel, R.; Boulesteix, AL, Survival prediction using gene expression data: a review and comparison, Comput Stat Data Anal, 53, 5, 1590-1603 (2009) · Zbl 1453.62225 · doi:10.1016/j.csda.2008.05.021 |
| [32] | Wright, SJ; Nowak, RD; Figueiredo, MA, Sparse reconstruction by separable approximation, IEEE Trans Signal Process, 57, 7, 2479-2493 (2009) · Zbl 1391.94442 · doi:10.1109/TSP.2009.2016892 |
| [33] | Wu, Y.; Yin, G., Conditional quantile screening in ultrahigh-dimensional heterogeneous data, Biometrika, 102, 1, 65-76 (2015) · Zbl 1345.62097 · doi:10.1093/biomet/asu068 |
| [34] | Xu, C.; Chen, J., The sparse mle for ultrahigh-dimensional feature screening, J Am Stat Assoc, 109, 507, 1257-1269 (2014) · Zbl 1368.62295 · doi:10.1080/01621459.2013.879531 |
| [35] | Yang L (2017) Proximal gradient method with extrapolation and line search for a class of nonconvex and nonsmooth problems. ArXiv preprint arXiv:1711.06831 |
| [36] | Yang, G.; Yu, Y.; Li, R.; Buu, A., Feature screening in ultrahigh dimensional Cox’s model, Stat Sin, 26, 881 (2016) · Zbl 1356.62175 |
| [37] | Zhang T (2009) Multi-stage convex relaxation for learning with sparse regularization. In: Advances in neural information processing systems, pp 1929-1936 |
| [38] | Zhang, J.; Liu, Y.; Wu, Y., Correlation rank screening for ultrahigh-dimensional survival data, Comput Stat Data Anal, 108, 121-132 (2017) · Zbl 1466.62226 · doi:10.1016/j.csda.2016.11.005 |
| [39] | Zhao, SD; Li, Y., Principled sure independence screening for Cox models with ultra-high-dimensional covariates, J Multivar Anal, 105, 1, 397-411 (2012) · Zbl 1233.62173 · doi:10.1016/j.jmva.2011.08.002 |
| [40] | Zhao, P.; Yu, B., On model selection consistency of Lasso, J Mach Learn Res, 7, 2541-2563 (2006) · Zbl 1222.62008 |
| [41] | Zhou, T.; Zhu, L., Model-free feature screening for ultrahigh dimensional censored regression, Stat Comput, 27, 4, 947-961 (2017) · Zbl 1384.62144 · doi:10.1007/s11222-016-9664-z |
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