無

Assuming Cox’s regression model, we consider penalized likelihood approaches to conduct variable selection under nested case-control sampling or case-cohort sampling. Penalized non-parametric maximum likelihood estimate (PNPMLE) are characterized by self-consistency equations derived from score functions, which form the basis of the algorithm to compute PNPMLE. Consistency, asymptotic normality and oracle properties of the PNPMLE, the sparsity property of the penalty, and a consistent estimate of the asymptotic variance, based on observed profile likelihood, are established. A cross-validation method is used to choose the tuning parameter within a family of penalty function. Simulation studies indicate that the numerical performance of PNPMLE is satisfactory and that LASSO performs best when cohort size is small and SCAD performs best when cohort size is large and may eventually perform as well as the oracle estimator, resembling the findings when i.i.d. sampling is considered. This method is also illustrated in a real dataset.

References
Cai, T., and Zheng, Y. (2012), “Evaluating Prognostic Accuracy of Biomarkers in
Nested Case-Control Studies,” Biostatistics, 13, 89-100.
Chang, I. S., Hsiung, C. A., Wang, M. C., and Wen, C. C. (2005), “An Asymptotic Theory for the Nonparametric Maximum Likelihood Estimation in the Cox-Gene Model,” Bernoulli, 11, 863-892.
Chang, I. S., Hsiung, C. A., Wen, C. C., Wu, Y. J., and Yang, C. C. (2007), “Non-Parametric Maximum-Likelihood Estimation in a Semiparametric Mixture Model for Competing-Risks Data,” Scandinavian Journal of Statistics, 34, 870-895.
Chang, I. S., Wen, C. C., Wu, Y. J., and Yang, C. C. (2007), “A Fast Algorithm for the Nonparametric Maximum Likelihood Estimate in the Cox-Gene Model,” Statistica Sinca, 17, 841-856.
Chen, H. Y. (2002), “Double-Semiparametric Method for Missing Covariates in Cox Regression Models,” Journal of the American Statistical Association, 97, 565-576.
Donoho, D. L., and Johnstone, I. M. (1994), “Ideal Spatial Adaptation by Wavelet Shrinkage,” Biometrika, 81, 425-455.
Fan, J., and Li, R. (2001), “Variable Selection via Nonconcave Penalized Likelihood and its Oracle Peoperties,” Journal of the American Statistical Association, 96, 1348-1360.
Fan, J., and Li, R. (2002), “Variable Selection for Cox’s Proportional Hazards Model and Frailty Model,” The Annals of Statistics, 30, 74-99.
Hunter, D. R., and Li, R. (2005), “Variable Selection Using MM Algorithms,” The Annals of Statistics, 33, 1617-1642.
Läärä, E. (2011), “Study Designs for Biobank-Based Epidemiologic Research on Chronic Diseases,” Methods in Molecular Biology, 675, 165-178.
Murphy, S. A., and van der Vaart, A. W. (1999), “Observed Information in Semi-Parametric Models,” Bernoulli, 5, 381-412.
Murphy, S. A., and van der Vaart, A. W. (2000), “On Profile Likelihood,” Journal of the American Statistical Association, 95, 449-465.
Prentice, R. L. (1986), “A Case-Cohort Design for Epidemiologic Cohort Studies and Disease Prevention Trials,” Biometrika, 73, 1-11.
Rudin, W. (1973), Functional Analysis. New York: McGraw-Hill.
Rundle, A. G., Vineis, P., and Ahsa, H. (2005), “Design Options for Molecular Epidemiologic Research within Cohort Studies,” Cancer Epidemiology Biomarkers Prev., 14, 1899-1907.
Scheike, T. H., and Juul, A. (2004), “Maximum Likelihood Estimation for Cox’s Regression Model under Nested Case-Control Sampling,” Biostatistics, 5, 193-206.
Scheike, T. H., and Martinussen, T. (2004), “Maximum Likelihood Estimation for Cox’s Regression Model under Case-Cohorts Sampling,” Scandinavian Journal of Statistics, 31, 283-293.
Suzuki, M., and Yoshino, I. (2010), “Aberrant Methylation in Non-Small Cell Lung Cancer,” Surgery Today, 40, 602-607.
Thomas, D. C. (1977), Addendum to “Methods of Cohort Analysis: Appraisal by Application to Asbestos Mining," by F. D. K. Liddell, J. C. McDonald and D. C. Thomas. Journal of the Royal Statistical Society, Ser. A, 140, 483-485.
Tibshirani, R. (1996), “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, Ser. B, 58, 267-288.
Tibshirani, R. (1997), “The Lasso Method for Variable Selection in the Cox Model,” Statistics in Medicine, 16, 385-395.
van der Vaart, A. W. (1998), Asymptotic Statistics, Cambridge University Press.
van der Vaart, A. W., and Wellner, J. A. (1996), Weak Convergence and Empirical Processes with Application to Statistics, New York: Springer.
Verweij, P. J. M., and Van Houwelingen, H. C. (1993), “Cross-Validation in Survival Analysis,” Statistics in Medicine, 12, 2305-2314.
Wellner, J. A., and Zhan, Y. (1997), “A Hybrid Algorithm for Computing of the Nonparametric Maximum Likelihood Estimator from Censored Data,” Journal of the American Statistical Association, 92, 945-959.