Abstract
This paper discusses variable or covariate selection for high-dimensional quadratic Cox model. Although many variable selection methods have been developed for standard Cox model or high-dimensional standard Cox model, most of them cannot be directly applied since they cannot take into account the important and existing hierarchical model structure. For the problem, we present a penalized log partial likelihood-based approach and in particular, generalize the regularization algorithm under marginality principle (RAMP) proposed in Hao et al. (J Am Stat Assoc 2018;113:615–25) under the context of linear models. An extensive simulation study is conducted and suggests that the presented method works well in practical situations. It is then applied to an Alzheimer’s Disease study that motivated this investigation.
Acknowledgments
The authors wish to thank the two reviewers for their many helpful and useful comments and suggestions that greatly improved the paper. The data used in preparation of this article were obtained from the Alzheimer’s Disease NeuroimagingInitiative (ADNI) database (adni.loni.ucla.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in the analysis or writing of this report. A complete listing of ADNI investigators can be found at https://adni.loni.usc.edu/wp-content/uploads/how–to–apply/ADNI–Acknowledgement–List.pdf.
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Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
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Research funding: This work was partly supported by the China Scholarship Council, No. 201806175024.
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Employment or leadership: None declared.
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Honorarium: None declared.
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
1. Cox, D., 1972. Regression models and life-tables. J Roy Stat Soc Ser B (Methodol) 34, 187–202. https://doi.org/10.1111/j.2517-6161.1972.tb00899.x.Search in Google Scholar
2. Cox, D., 1975. Partial likelihood. Biometrika 62, 269–76. https://doi.org/10.1093/biomet/62.2.269.Search in Google Scholar
3. Kooperberg, C., LeBlanc, M., 2008. Increasing the power of identifying gene×gene interactions in genome-wide association studies. Genet Epidemiol 32, 255–63. https://doi.org/10.1002/gepi.20300.Search in Google Scholar
4. Cordell, H., 2009. Detecting gene-gene interactions that underlie human diseases. Nat RevGenet 10, 392–404. https://doi.org/10.1038/nrg2579.Search in Google Scholar
5. Bien, J., Taylor, J., Tibshirani, R., 2013. A lasso for hierarchical interactions. Ann Stat 41, 1111–41. https://doi.org/10.1214/13-AOS1096.Search in Google Scholar
6. Hao N, Zhang HH. Interaction screening for ultrahigh-dimensional data. J Am Stat Assoc 2014;109:1285–301. https://doi.org/10.1080/01621459.2014.881741.Search in Google Scholar
7. Hao N, Feng Y, Zhang HH. Model selection for high-dimensional quadratic regression via regularization. J Am Stat Assoc 2018;113:615–25. https://doi.org/10.1080/01621459.2016.1264956.Search in Google Scholar
8. Tibshirani, R., 1997. The lasso method for variable selection in the Cox model. Stat Med 16, 385–95. https://doi.org/10.1002/(SICI)1097-0258(19970228)16:4<385::AID-SIM380>3.0.CO;2-3.10.1002/(SICI)1097-0258(19970228)16:4<385::AID-SIM380>3.0.CO;2-3Search in Google Scholar
9. Zhang, H., Lu, W., 2007. Adaptive Lasso for Cox’s proportional hazards model. Biometrika 94, 691–703. https://doi.org/10.1093/biomet/asm037.Search in Google Scholar
10. Fan, J., Li, R., 2002. Variable selection for Cox’s proportional hazards model and frailty model. Ann Stat 30, 74–99. https://www.jstor.org/stable/2700003.10.1214/aos/1015362185Search in Google Scholar
11. Shi, Y., Cao, Y., Jiao, Y., Liu, Y., 2014. SICA for Coxs proportional hazards model with a diverging number of parameters. Acta Math Appl Sin Eng Ser 30, 887–902. https://doi.org/10.1007/s10255-014-0402-z.Search in Google Scholar
12. Shi, Y., Xu, D., Cao, Y., Jiao, Y., 2019. Variable selection via generalized SELO-penalized Cox regression models. J Sys Sci Comp 32, 709–36. https://doi.org/10.1007/s11424-018-7276-8.Search in Google Scholar
13. Wang L, Shen J, Thall PF. A modified adaptive Lasso for identifying interactions in the Cox model with the heredity constraint. Stat Prob Lett 2014;93:126–33. https://doi.org/10.1016/j.spl.2014.06.024.Search in Google Scholar
14. Fan, Y., Tang, C., 2013. Tuning parameter selection in high dimensional penalized likelihood. J Roy Stat Soc Ser B (Stat Method) 75, 531–52. https://doi.org/10.1111/rssb.12001.Search in Google Scholar
15. Tibshirani, R., 1996. Regression shrinkage and selection via the lasso. J Roy Stat Soc Ser B (Methodol) 58, 267–88. https://doi.org/10.1111/j.2517-6161.1996.tb02080.x.Search in Google Scholar
16. Fan J, Li R. Variable selection via nonconcave penalized likelihood and its oracle properties. J Am Stat Assoc 2001;96:1348–60. https://doi.org/10.1198/016214501753382273.Search in Google Scholar
17. Zhang, C., 2010. Nearly unbiased variable selection under minimax concave penalty. Ann Stat 38, 894–942. https://doi.org/10.1214/09-AOS729.Search in Google Scholar
18. Lv, J., Fan, Y., 2009. A unified approach to model selection and sparse recovery using regularized least squares. Ann Stat 37, 3498–528. https://doi.org/10.1214/09-AOS683.Search in Google Scholar
19. Dicker, L., Huang, B., Lin, X., 2013. Variable selection and estimation with the seamless-L0 penalty. Stat Sin 23, 929–62. https://doi.org/10.1002/cjs.11165.Search in Google Scholar
20. Zou, H., Li, R., 2008. One-step sparse estimates in nonconcave penalized likelihood models. Ann Stat 36, 1509–33. https://doi.org/10.1214/009053607000000802.Search in Google Scholar
21. Weiner, M., Veitch, D., Aisen, P., Beckett, L., Cairns, N., Green, R., et al., 2013. The Alzheimer’s disease neuroimaging initiative: a review of papers published since its inception. Alzheimer’s Dementia 9, e111-94. https://doi.org/10.1016/j.jalz.2013.05.1769.Search in Google Scholar
22. Fan, J., Lv, J., 2008. Sure independence screening for ultrahigh dimensional feature space. J Roy Stat Soc Ser B (Stat Method) 70, 849–83. https://doi.org/10.1111/j.1467-9868.2008.00674.x.Search in Google Scholar
23. Sun J. The statistical analysis of interval-censored failure time data. Springer Science+Business Media Inc.; 2006.Search in Google Scholar
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