Model Detection for Additive Models with Longitudinal Data
Affiliation(s)
1 College of Applied Sciences, Beijing University of Technology, Beijing, China. 2 College of Science, Northeastern University, Shenyang, China.
1 College of Applied Sciences, Beijing University of Technology, Beijing, China. 2 College of Science, Northeastern University, Shenyang, China.
ABSTRACT
In this paper, we consider the problem of variable selection and model detection in additive models with longitudinal data. Our approach is based on spline approximation for the components aided by two Smoothly Clipped Absolute Deviation (SCAD) penalty terms. It can perform model selection (finding both zero and linear components) and estimation simultaneously. With appropriate selection of the tuning parameters, we show that the proposed procedure is consistent in both variable selection and linear components selection. Besides, being theoretically justified, the proposed method is easy to understand and straightforward to implement. Extensive simulation studies as well as a real dataset are used to illustrate the performances.
In this paper, we consider the problem of variable selection and model detection in additive models with longitudinal data. Our approach is based on spline approximation for the components aided by two Smoothly Clipped Absolute Deviation (SCAD) penalty terms. It can perform model selection (finding both zero and linear components) and estimation simultaneously. With appropriate selection of the tuning parameters, we show that the proposed procedure is consistent in both variable selection and linear components selection. Besides, being theoretically justified, the proposed method is easy to understand and straightforward to implement. Extensive simulation studies as well as a real dataset are used to illustrate the performances.
Cite this paper
Wu, J. and Xue, L. (2014) Model Detection for Additive Models with Longitudinal Data. Open Journal of Statistics, 4, 868-878. doi: 10.4236/ojs.2014.410082.
Wu, J. and Xue, L. (2014) Model Detection for Additive Models with Longitudinal Data. Open Journal of Statistics, 4, 868-878. doi: 10.4236/ojs.2014.410082.
References
[1] Hastie, T.J. and Tibshirani, R.J. (1990) Generalized Additive Models. Chapman and Hall, London. [2] Berhane, K. and Tibshirani, R.J. (1998) Generalized Additive Models for Longitudinal Data. The Canadian Journal of Statistics, 26, 517-535.
http://dx.doi.org/10.2307/3315715 [3] Martinussen, T. and Scheike, T.H. (1999) A Semiparametric Additive Regression Model for Longitudinal Data. Biometrika, 86, 691-702.
http://dx.doi.org/10.1093/biomet/86.3.691 [4] Xue, L. (2010) Consistent Model Selection for Marginal Generalized Additive Model for Correlated Data. Journal of the American Statistical Association, 105, 1518-1530.
http://dx.doi.org/10.1198/jasa.2010.tm10128 [5] Hu, T. and Xia, Y.C. (2012) Adaptive Semi-Varying Coefficient Model Selection. Statistica Sinica, 22, 575-599.
http://dx.doi.org/10.5705/ss.2010.105 [6] Tang, Y.L., Wang, H.X., Zhu, Z.Y. and Song, X.Y. (2012) A Unified Variable Selection Approach for Varying Coefficient Models. Statistica Sinica, 22, 601-628.
http://dx.doi.org/10.5705/ss.2010.121 [7] Lian, H. (2012) Shrinkage Estimation for Identification of Linear Components in Additive Models. Statistics and Probability Letters, 82, 225-231.
http://dx.doi.org/10.1016/j.spl.2011.10.009 [8] Qu, A., Lindsay, B.G. and Li, B. (2000) Improving Generalised Estimating Equations Using Quadratic Inference Functions. Biometrika, 87, 823-836.
http://dx.doi.org/10.1093/biomet/87.4.823 [9] Huang, J.Z. (1998) Projection Estimation in Multiple Regression with Application to Functional ANOVA Models. The Annals of Statistics, 26, 242-272.
http://dx.doi.org/10.1214/aos/1030563984 [10] Xue, L. (2009) A Root-N Consistent Backfitting Estimator for Semiparametric Additive Modeling. Statistica Sinica, 19, 1281-1296. [11] Wang, H., Li, R. and Tsai, C.L. (2007) Tuning Parameter Selectors for the Smoothly Clipped Absolute Deviation Method. Biometrika, 94, 553-568.
http://dx.doi.org/10.1093/biomet/asm053 [12] Xue, L.G. and Zhu, L.X. (2007) Empirical Likehood for a Varying Coefficient Model with Longitudinal Data. Journal of the American Statistical Association, 102, 642-654. [13] Wu, C.O., Chiang, C.T. and Hoover, D.R. (2010) Asymptotic Confidence Regions for Kernel Smoothing of a Varying-Coefficient Model with Longitudinal Data. Journal of the American Statistical Association, 93, 1388-1402. [14] De Boor, C. (2001) A Practical Guide to Splines. Springer, New York.
[1] Hastie, T.J. and Tibshirani, R.J. (1990) Generalized Additive Models. Chapman and Hall, London. [2] Berhane, K. and Tibshirani, R.J. (1998) Generalized Additive Models for Longitudinal Data. The Canadian Journal of Statistics, 26, 517-535.
http://dx.doi.org/10.2307/3315715 [3] Martinussen, T. and Scheike, T.H. (1999) A Semiparametric Additive Regression Model for Longitudinal Data. Biometrika, 86, 691-702.
http://dx.doi.org/10.1093/biomet/86.3.691 [4] Xue, L. (2010) Consistent Model Selection for Marginal Generalized Additive Model for Correlated Data. Journal of the American Statistical Association, 105, 1518-1530.
http://dx.doi.org/10.1198/jasa.2010.tm10128 [5] Hu, T. and Xia, Y.C. (2012) Adaptive Semi-Varying Coefficient Model Selection. Statistica Sinica, 22, 575-599.
http://dx.doi.org/10.5705/ss.2010.105 [6] Tang, Y.L., Wang, H.X., Zhu, Z.Y. and Song, X.Y. (2012) A Unified Variable Selection Approach for Varying Coefficient Models. Statistica Sinica, 22, 601-628.
http://dx.doi.org/10.5705/ss.2010.121 [7] Lian, H. (2012) Shrinkage Estimation for Identification of Linear Components in Additive Models. Statistics and Probability Letters, 82, 225-231.
http://dx.doi.org/10.1016/j.spl.2011.10.009 [8] Qu, A., Lindsay, B.G. and Li, B. (2000) Improving Generalised Estimating Equations Using Quadratic Inference Functions. Biometrika, 87, 823-836.
http://dx.doi.org/10.1093/biomet/87.4.823 [9] Huang, J.Z. (1998) Projection Estimation in Multiple Regression with Application to Functional ANOVA Models. The Annals of Statistics, 26, 242-272.
http://dx.doi.org/10.1214/aos/1030563984 [10] Xue, L. (2009) A Root-N Consistent Backfitting Estimator for Semiparametric Additive Modeling. Statistica Sinica, 19, 1281-1296. [11] Wang, H., Li, R. and Tsai, C.L. (2007) Tuning Parameter Selectors for the Smoothly Clipped Absolute Deviation Method. Biometrika, 94, 553-568.
http://dx.doi.org/10.1093/biomet/asm053 [12] Xue, L.G. and Zhu, L.X. (2007) Empirical Likehood for a Varying Coefficient Model with Longitudinal Data. Journal of the American Statistical Association, 102, 642-654. [13] Wu, C.O., Chiang, C.T. and Hoover, D.R. (2010) Asymptotic Confidence Regions for Kernel Smoothing of a Varying-Coefficient Model with Longitudinal Data. Journal of the American Statistical Association, 93, 1388-1402. [14] De Boor, C. (2001) A Practical Guide to Splines. Springer, New York.