Advertisement

Tree-structured modelling of varying coefficients

  • Moritz Berger
  • Gerhard Tutz
  • Matthias Schmid
Article
  • 69 Downloads

Abstract

The varying-coefficient model is a strong tool for the modelling of interactions in generalized regression. It is easy to apply if both the variables that are modified as well as the effect modifiers are known. However, in general one has a set of explanatory variables, and it is unknown which covariates are modified by which variables. A recursive partitioning strategy is proposed that is able to deal with this complex selection problem. The tree-structured modelling yields for each covariate, which is modified by other variables, a tree that visualizes the modified effects. The performance of the method is investigated in simulations, and two applications illustrate its usefulness.

Keywords

Varying-coefficient models Interactions Recursive partitioning Tree-based models 

Notes

Acknowledgements

The authors thank the reviewers for their thorough reading and helpful suggestions for improving the article.

Supplementary material

11222_2018_9804_MOESM1_ESM.gz (17 kb)
Supplementary material 1 (gz 16 KB)

References

  1. Berger, M.: TSVC: Tree-Structured Modelling of Varying Coefficients. R package version, vol. 1 (2018)Google Scholar
  2. Breiman, L., Friedman, J.H., Olshen, R.A., Stone, J.C.: Classification and Regression Trees. Wadsworth, Monterey (1984)zbMATHGoogle Scholar
  3. Bürgin, R., Ritschard, G.: Tree-based varying coefficient regression for longitudinal ordinal responses. Comput. Stat. Data Anal. 86(C), 65–80 (2015)MathSciNetCrossRefGoogle Scholar
  4. Bürgin, R., Ritschard, G.: Coefficient-wise tree-based varying coefficient regression with vcrpart. J. Stat. Softw. 80(6), 1–33 (2017)CrossRefGoogle Scholar
  5. Cameron, A.C., Trivedi, P.K.: Econometric models based on count data: comparisons and applications of some estimators and tests. J. Appl. Econom. 1(1), 29–53 (1986)CrossRefGoogle Scholar
  6. Cameron, A.C., Trivedi, P.K.: Regression Analysis of Count Data. Econometric Society Monographs No. 30. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  7. Fan, J., Zhang, W.: Statistical estimation in varying coefficient models. Ann. Stat. 27(5), 1491–1518 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Fan, J., Zhang, W.: Statistical methods with varying coefficient models. Stat. Interface 1(1), 179–195 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Gerfin, M.: Parametric and semi-parametric estimation of the binary response model of labour market participation. J. Appl. Econom. 11(3), 321–339 (1996)CrossRefGoogle Scholar
  10. Gertheiss, J., Tutz, G.: Regularization and model selection with categorial effect modifiers. Stat. Sin. 22(3), 957–982 (2012)MathSciNetzbMATHGoogle Scholar
  11. Hastie, T., Tibshirani, R.: Varying-coefficient models. J. R. Stat. Soc. B 55, 757–796 (1993)MathSciNetzbMATHGoogle Scholar
  12. Hofner, B., Hothorn, T., Kneib, T.: Variable selection and model choice in structured survival models. Comput. Stat. 28(3), 1079–1101 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Hoover, D., Rice, J.A., Wu, C., Yang, L.: Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika 85(4), 809–822 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Hothorn, T., Lausen, B.: On the exact distribution of maximally selected rank statistics. Comput. Stat. Data Anal. 43(2), 121–137 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Hothorn, T., Hornik, K., Zeileis, A.: Unbiased recursive partitioning: a conditional inference framework. J. Comput. Graph. Stat. 15(3), 651–674 (2006)MathSciNetCrossRefGoogle Scholar
  16. Kauermann, G., Tutz, G.: Local likelihood estimation in varying coefficient models including additive bias correction. J. Nonparametric Stat. 12(3), 343–371 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Kleiber, C., Zeileis, A.: Applied Econometrics with R. New York. ISBN: 978-0-387-77316-2. http://CRAN.R-project.org/package=AER (2008)
  18. Leng, C.: A simple approach for varying-coefficient model selection. J. Stat. Plan. Inference 139(7), 2138–2146 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Lu, Y., Zhang, R., Zhu, L.: Penalized spline estimation for varying-coefficient models. Commun. Stat. Theory Methods 37(14), 2249–2261 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Oelker, M.R., Gertheiss, J., Tutz, G.: Regularization and model selection with categorical predictors and effect modifiers in generalized linear models. Stat. Model. 14(2), 157–177 (2014)MathSciNetCrossRefGoogle Scholar
  21. Ripley, B.D.: Pattern Recognition and Neural Networks. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  22. Shih, Y.S.: A note on split selection bias in classification trees. Comput. Stat. Data Anal. 45(3), 457–466 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Shih, Y.S., Tsai, H.: Variable selection bias in regression trees with constant fits. Comput. Stat. Data Anal. 45(3), 595–607 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Su, X., Meneses, K., McNees, P., Johnson, W.O.: Interaction trees: exploring the differential effects of an intervention programme for breast cancer survivors. J. R. Stat. Soc. C 60(3), 457–474 (2011)MathSciNetCrossRefGoogle Scholar
  25. Wang, H., Xia, Y.: Shrinkage estimation of the varying coefficient model. J. Am. Stat. Assoc. 104(486), 747–757 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Wang, J.C., Hastie, T.: Boosted varying-coefficient regression models for product demand prediction. J. Comput. Graph. Stat. 23(2), 361–382 (2014)MathSciNetCrossRefGoogle Scholar
  27. Wang, L., Li, H., Haung, J.: Variable selection in nonparametric varying-coefficient models for analysis of repeated measurements. J. Am. Stat. Assoc. 103(484), 1556–1569 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Wedderburn, R.W.M.: Quasilikelihood functions, generalized linear models and the Gauss–Newton method. Biometrika 61(3), 439–447 (1974)MathSciNetzbMATHGoogle Scholar
  29. Wong, H., Guo, S., Chen, M., Wai-Cheung, I.P.: On locally weighted estimation and hypothesis testing of varying-coefficient models with missing covariates. J. Stat. Plan. Inference 139(9), 2933–2951 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Wu, C., Chiang, C., Hoover, D.: Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data. J. Am. Stat. Assoc. 93(444), 1388–1402 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Zhao, P., Xue, L.: Variable selection for semiparametric varying coefficient partially linear models. Stat. Probab. Lett. 79(20), 2148–2157 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für Medizinische Biometrie, Informatik und EpidemiologieUniversitätsklinikum BonnBonnGermany
  2. 2.Institut für StatistikLudwig-Maximilians-Universität MünchenMunichGermany

Personalised recommendations