Joint Variable Selection of Mean-Covariance Model for Longitudinal Data
Affiliation(s)
College of Applied Sciences, Beijing University of Technology, Beijing, China. Faculty of Science, Kunming University of Science and Technology, Kunming, China.
College of Applied Sciences, Beijing University of Technology, Beijing, China. Faculty of Science, Kunming University of Science and Technology, Kunming, China.
ABSTRACT
In this paper we reparameterize covariance structures in
longitudinal data analysis through the modified Cholesky decomposition of
itself. Based on this modified Cholesky decomposition, the within-subject
covariance matrix is decomposed into a unit lower triangular matrix involving
moving average coefficients and a diagonal matrix involving innovation variances,
which are modeled as linear functions of covariates. Then, we propose a
penalized maximum likelihood method for variable selection in joint
mean and covariance models based on this decomposition. Under certain regularity
conditions, we establish the consistency and asymptotic normality of the
penalized maximum likelihood estimators of parameters in the models. Simulation
studies are undertaken to assess the finite sample performance of the proposed
variable selection procedure.
Cite this paper
D. Xu, Z. Zhang and L. Wu, "Joint Variable Selection of Mean-Covariance Model for Longitudinal Data," Open Journal of Statistics, Vol. 3 No. 1, 2013, pp. 27-35. doi: 10.4236/ojs.2013.31004.
D. Xu, Z. Zhang and L. Wu, "Joint Variable Selection of Mean-Covariance Model for Longitudinal Data," Open Journal of Statistics, Vol. 3 No. 1, 2013, pp. 27-35. doi: 10.4236/ojs.2013.31004.
References
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[1] M. Pourahmadi, “Joint Mean-Covariance Models with Applications to Lontidinal Data: Unconstrained Parameterisation,” Biometrika, Vol. 86, No. 3, 1999, pp. 677- 690. doi:10.1093/biomet/86.3.677 [2] M. Pourahmadi, “Maximum Likelihood Estimation for Generalised Linear Models for Multivariate Normal Covariance Matrix,” Biometrika, Vol. 87, No. 2, 2000, pp. 425-435. doi:10.1093/biomet/87.2.425 [3] P.T. Diggle and A. Verbyla, “Nonparametric Estimation of Covariance Structure in Longitudinal Data,” Biometrics, Vol. 54, No. 2, 1998, pp. 401-415. doi:10.2307/3109751 [4] J. Q. Fan, T. Huang and R. Li, “Analysis of Longitudinal Data with Semiparametric Estimation of Covariance Function,” Journal of the American Statistical Association, Vol. 102, No. 478, 2007, pp. 632-641. doi:10.1198/016214507000000095 [5] J. Q. Fan and Y. Wu, “Semiparametric Estimation of Covariance Matrices for Longitudinal Data,” Journal of the American Statistical Association, Vol. 103, No. 484, 2008, pp. 1520-1533. doi:10.1198/016214508000000742 [6] A. J. Rothman, E. Levina and J. Zhu, “A New Approach to Cholesky-Based Covariance Regularization in High Dimensions,” Biometrika, Vol. 97, No. 3, 2010, pp. 539- 550. doi:10.1093/biomet/asq022 [7] W. P. Zhang and C. L. Leng, “A Moving Average Cholesky Factor Model in Covariance Modeling for Longitudinal Data,” Biometrika, Vol. 99, No. 1, 2012, pp. 141- 150. doi:10.1093/biomet/asr068 [8] L. Breiman, “Better Subset Selection Using Nonnegative Garrote,” Techonometrics, Vol. 37, No. 4, 1995, pp. 373- 384. doi:10.1080/00401706.1995.10484371 [9] R. Tibshirani, “Regression Shrinkage and Selection via the LASSO,” Journal of Royal Statistical Society, Series B, Vol. 58, No. 1, 1996, pp. 267-288. [10] W. J. Fu, “Penalized Regression: The Bridge versus the LASSO,” Journal of Computational and Graphical Statistics, Vol. 7, No. 3, 1998, pp.397-416. [11] J. Q. Fan and R. Li, “Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties,” Journal of American Statistical Association, Vol. 96, No. 456, 2001, pp. 1348-1360. doi:10.1198/016214501753382273 [12] H. Zou and R. Li, “One-Step Sparse Estimates in Nonconcave Penalized Likelihood Models,” The Annals of Statistics, Vol. 36, No. 4, 2008, pp. 1509-1533. doi:10.1214/009053607000000802 [13] Z. Z. Zhang and D. R. Wang, “Simultaneous Variable Selection for Heteroscedastic Regression Models,” Science China Mathematic, Vol. 54, No. 3, 2011, pp. 515- 530. doi:10.1007/s11425-010-4147-8 [14] P. X. Zhao and L. G. Xue, “Variable Selection in Semiparametric Regression Analysis for Longitudinal Data,” Annals of the Institute of Statistical Mathematics, Vol. 64, No. 1, 2012, pp. 213-231. doi:10.1007/s10463-010-0312-7 [15] H. J. Ye and J. X. Pan, “Modelling of Covariance Structures in Generalized Estimating Equations for Longitudinal Data,” Biometrika, Vol. 93, No. 4, 2006, pp. 927-941. doi:10.1093/biomet/93.4.927 [16] H. Wang, G. Li and C. L. Tsai, “Tuning Parameter Selectors for the Smoothly Clipped Absolute Deviation Method,” Biometrika, Vol. 94, No. 3, 2007, pp. 553-568. doi:10.1093/biomet/asm053 [17] H. Zou, “The Adaptive Lasso and Its Oracle Properties,” Journal of American Statistical Association, Vol. 101, No. 476, 2006, pp. 1418-1429. doi:10.1198/016214506000000735