Skip to main content

Advertisement

SpringerLink
Log in

Model detection and variable selection for varying coefficient models with longitudinal data

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we consider the problem of variable selection and model detection in varying coefficient models with longitudinal data. We propose a combined penalization procedure to select the significant variables, detect the true structure of the model and estimate the unknown regression coefficients simultaneously. With appropriate selection of the tuning parameters, we show that the proposed procedure is consistent in both variable selection and the separation of varying and constant coefficients, and the penalized estimators have the oracle property. Finite sample performances of the proposed method are illustrated by some simulation studies and the real data analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. de Boor, C.: A Practical Guide to Spilnes, Revised Edition, Springer, New York, 2001

    Google Scholar 

  2. Chiang, C. T., Rice, J. A., Wu, C. O.: Smoothing spline estimation for varying-coefficient models with repeatedly measured dependent variables. J. Amer. Statist. Assoc., 96, 605–619 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Diggle, P. J., Liang, K. Y., Zeger, S. L.: Analysis of Longitudinal Data, Oxford University Press, England, 1994

    Google Scholar 

  4. Fan, J. Q., Li, R. Z.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc., 96, 1348–1360 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fan, J. Q., Huang, T.: Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Bernoulli, 11, 1031–1057 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fan, J. Q., Zhang, J.: Two-step estimation of functional linear models with applications to longitudinal data. J. R. Stat. Soc. Ser. B Stat. Methodol., 62, 303–322 (2000)

    Article  MathSciNet  Google Scholar 

  7. Fan, J. Q., Zhang, W. Y.: Simultaneous confidence bands and hypothesis testing in varying-coefficient models. Scand. J. Statist., 27, 715–731 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fan, J. Q., Zhang, C., Zhang, J.: Generalized likelihood ratio statistics and Wilks phenomenon. Ann. Statist., 29, 153–193 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hoover, D. R., Rice, J. A., Wu, C. O., et al.: Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika, 85, 809–822 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hu, T., Xia, Y.: Adaptive semi-varying coefficient model selection. Statist. Sinica, 22, 575–599 (2012)

    MathSciNet  MATH  Google Scholar 

  11. Huang, J. Z., Wu, C. O., Zhou, L.: Varying-coefficient models and basis function approximation for the analysis of repeated measurements. Biometrika, 89, 111–128 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Huang, J. Z., Wu, C. O., Zhou, L.: Polynomial spline estimation and inference for varying coefficient models with longitudinal data. Statist. Sinica, 14, 763–788 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Johnson, B. A., Lin, D. Y., Zeng, D. L.: Penalized estimating functions and variable selection in semiparametric regression models. J. Amer. Statist. Assoc., 103, 672–680 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kaslow, R. A., Ostrow, D. G., Detels, R., et al.: The multicenter AIDS cohort study: rationale, organization and selected characteristics of the participants. Amer. J. Epidemiology, 126, 310–318 (1987)

    Article  Google Scholar 

  15. Leng, C. L.: A simple approach for varying-coefficient model selection. J. Statist. Plann. Inference, 139, 2138–2146 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lian, H.: Semiparametric estimation of additive quantile regression models by two-fold penalty. J. Bus. Econom. Statist., 30, 337–350 (2012)

    Article  MathSciNet  Google Scholar 

  17. Lin, D. Y., Ying, Z.: Semiparametric and nonparametric regression analysis of longitudinal data (with Discussion). J. Amer. Statist. Assoc., 96, 103–26 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Schumaker, L. L.: Spline Functions, Wiley Press, New York, 1981

    MATH  Google Scholar 

  19. Tang, Y. L., Wang, H. X., Zhu, Z. Y., et al.: A unified variable selection approach for varying coefficient models. Statist. Sinica, 22, 601–628 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Verbeke, G., Molenberghs, G.: Linear Mixed Models for Longitudinal Data, Springer-Verlag, New York, 2000

    MATH  Google Scholar 

  21. Wang, D., Kulasekera, K. B.: Parametric component detection and variable selection in varying-coefficient partially linear models. J. Multivariate Anal., 112, 117–129 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang, H., Li, R., Tsai, C.: Tuning parameter selectors for the smoothly clipped absolute deviation method. Biometrika, 94, 553–568 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wang, H., Xia, Y.: Shrinkage estimation of the varying coefficient model. J. Amer. Statist. Assoc., 104, 747–757 (2009)

    Article  MathSciNet  Google Scholar 

  24. Wang, L., Li, H., Huang, J. Z.: Variable selection in nonparametric varying coefficient models for analysis of repeated measurements. J. Amer. Statist. Assoc., 103, 1556–1569 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wei, F. R., Huang, J., Li, H. Z.: Variable selection and estimation in high-dimensional varying coefficient models. Statist. Sinica, 21, 1515–1540 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wu, C. O., Chiang, C. T., Hoover, D. R.: Asymptotic confidence regions for kernel smoothing of a varying coefficient model with longitudinal data. J. Amer. Statist. Assoc., 93, 1388–402 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Xia, Y., Zhang, W., Tong, H.: Efficient estimation for semivarying-coefficient models. Biometrika, 91, 661–681 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Xue, L. G., Zhu, L. X.: Empirical likelihood for a varying coefficient model with longitudinal data. J. Amer. Statist. Assoc., 102, 642–652 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhang, H. H., Cheng, G., Liu, Y. F.: Linear or nonlinear? Automatic structure discovery for partially linear models. J. Amer. Statist. Assoc., 106, 1099–1112 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhao, P. X., Xue, L. G.: Variable selection for semiparametric varying coefficient partially linear errors-invariables models. J. Multivariate Anal., 101, 1872–1883 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zou, H.: The adaptive Lasso and its oracle properties. J. Amer. Statist. Assoc., 101, 1418–1429 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, P. R. China

    San Ying Feng & Yu Ping Hu

  2. College of Applied Sciences, Beijing University of Technology, Beijing, 100124, P. R. China

    San Ying Feng & Yu Ping Hu

  3. College of Applied Sciences, Beijing University of Technology, Beijing, 100124, P. R. China

    Liu Gen Xue

Authors
  1. San Ying Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu Ping Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liu Gen Xue
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to San Ying Feng.

Additional information

Supported by National Natural Science Foundation of China (Grant Nos. 11501522, 11101014, 11001118 and 11171012), National Statistical Research Projects (Grant No. 2014LZ45), the Doctoral Fund of Innovation of Beijing University of Technology, the Science and Technology Project of the Faculty Adviser of Excellent PhD Degree Thesis of Beijing (Grant No. 20111000503) and the Beijing Municipal Education Commission Foundation (Grant No. KM201110005029)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feng, S.Y., Hu, Y.P. & Xue, L.G. Model detection and variable selection for varying coefficient models with longitudinal data. Acta. Math. Sin.-English Ser. 32, 331–350 (2016). https://doi.org/10.1007/s10114-016-4639-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-016-4639-8

Keywords

MR(2010) Subject Classification

Search

Navigation