## The Annals of Statistics

### Variable Selection for Cox's proportional Hazards Model and Frailty Model

#### Abstract

A class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed in Fan and Li (2001a). It has been shown there that the resulting procedures perform as well as if the subset of significant variables were known in advance. Such a property is called an oracle property. The proposed procedures were illustrated in the context of linear regression, robust linear regression and generalized linear models. In this paper, the nonconcave penalized likelihood approach is extended further to the Cox proportional hazards model and the Cox proportional hazards frailty model, two commonly used semi-parametric models in survival analysis. As a result, new variable selection procedures for these two commonly-used models are proposed. It is demonstrated how the rates of convergence depend on the regularization parameter in the penalty function. Further, with a proper choice of the regularization parameter and the penalty function, the proposed estimators possess an oracle property. Standard error formulae are derived and their accuracies are empirically tested. Simulation studies show that the proposed procedures are more stable in prediction and more effective in computation than the best subset variable selection, and they reduce model complexity as effectively as the best subset variable selection. Compared with the LASSO, which is the penalized likelihood method with the $L_1$ -penalty, proposed by Tibshirani, the newly proposed approaches have better theoretic properties and finite sample performance.

#### Article information

Source
Ann. Statist., Volume 30, Number 1 (2002), 74-99.

Dates
First available in Project Euclid: 5 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1015362185

Digital Object Identifier
doi:10.1214/aos/1015362185

Mathematical Reviews number (MathSciNet)
MR1892656

Zentralblatt MATH identifier
1012.62106

Subjects
Primary: 62F12: Asymptotic properties of estimators 62N02: Estimation

#### Citation

Fan, Jianqing; Li, Runze. Variable Selection for Cox's proportional Hazards Model and Frailty Model. Ann. Statist. 30 (2002), no. 1, 74--99. doi:10.1214/aos/1015362185. https://projecteuclid.org/euclid.aos/1015362185

#### References

• ANDERSEN, P. K, BORGAN, Ø., GILL, R. D. and KEIDING, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.
• ANDERSEN, P. K. and GILL, R. D. (1982). Cox's regression model for counting processes: a large sample study. Ann. Statist. 10 1100-1120.
• ANTONIADIS, A. (1997). Wavelets in Statistics: A review (with discussion). J. Italian Statist. Assoc. 6 97-144.
• ANTONIADIS, A. and FAN, J. (2001). Regularization of wavelet approximations (with discussion). J. Amer. Statist. Assoc. 96 939-967.
• BICKEL, P. J. (1975). One-step Huber estimates in the linear model. J. Amer. Statist. Assoc. 70 428- 434.
• BREIMAN, L. (1996). Heuristics of instability and stabilization in model selection. Ann. Statist. 24 2350-2383.
• COX, D. R. (1975). Partial likelihood. Biometrika 62 269-276.
• CRAVEN, P. and WAHBA, G. (1979). Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation. Numer. Math. 31 377-403.
• DONOHO, D. L. and JOHNSTONE, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425-455.
• FAN, J. (1997). Comment on "Wavelets in statistics: a review" by A. Antoniadis. J. Italian Statist. Assoc. 6 131-138.
• FAN, J. and LI, R. (2001a). Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc. 96 1348-1360.
• FAN, J. and LI, R. (2001b). Variable selection for Cox's proportional hazards model and frailty model. Institute of Statistic Mimeo Series #2372, Dept. Statistics, Univ. North Carolina, Chapel Hill.
• FARAGGI, D. and SIMON, R. (1998). Bayesian variable selection method for censored survival data. Biometrics 54 1475-1485.
• KNIGHT, K. and FU, W. (2000). Asymptotics for lasso-type estimators. Ann. Statist. 28 1356-1378.
• LEHMANN, E. L. (1983). Theory of Point Estimation. Wiley, New York.
• LI, K. C. (1987). Asymptotic optimality for Cp, Cl, cross-validation and generalized cross validation: discrete index set. Ann. Statist. 15 958-975.
• LINDLEY, D. V. (1968). The choice of variables in multiple regression (with discussion). J. Roy. Statist. Soc. Ser. B 30 31-66.
• MORRIS, C. N., NORTON, E. C. and ZHOU, X. H. (1994). Parametric duration analysis of nursing home usage. In Case Studies in Biometry (N. Lange, L. Ryan, L. Billard, D. Brillinger, L. Conquest and J. Greenhouse, eds.) 231-248. Wiley, New York.
• MURPHY, S. A. and VAN DER VAART, A. W. (1999). Observed information in semiparametric models. Bernoulli 5 381-412.
• MURPHY, S. A. and VAN DER VAART, A. W. (2000). On profile likelihood. J. Amer. Statist. Assoc. 95 449-465.
• NIELSEN, G. G., GILL, R. D., ANDERSEN, P. K. and SØ RENSEN, T. I. A. (1992). A counting process approach to maximum likelihood estimator in frailty models. Scand. J. Statist. 19 25-43.
• PARNER, E. (1998). Asymptotic theory for the correlated gamma-frailty model. Ann. Statist. 26 183- 214.
• ROBINSON, P. M. (1988). The stochastic difference between econometrics and statistics. Econometrica 56 531-548.
• SINHA, D. (1998). Posterior likelihood methods for multivariate survival data. Biometrics 54 1463- 1474.
• TIBSHIRANI, R. J. (1996). Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58 267-288.
• TIBSHIRANI, R. J. (1997). The lasso method for variable selection in the Cox model. Statistics in Medicine 16 385-395.
• WAHBA, G. (1985). A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem. Ann. Statist. 13 1378-1402.
• SHATIN, HONG KONG E-MAIL: jfan@sta.cuhk.edu.hk DEPARTMENT OF STATISTICS PENNSYLVANIA STATE UNIVERSITY
• UNIVERSITY PARK, PENNSYLVANIA 16802-2111 E-MAIL: rli@stat.psu.edu