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Tree-structured modelling of varying coefficients

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.

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References

  1. Berger, M.: TSVC: Tree-Structured Modelling of Varying Coefficients. R package version, vol. 1 (2018)

  2. Breiman, L., Friedman, J.H., Olshen, R.A., Stone, J.C.: Classification and Regression Trees. Wadsworth, Monterey (1984)

  3. Bürgin, R., Ritschard, G.: Tree-based varying coefficient regression for longitudinal ordinal responses. Comput. Stat. Data Anal. 86(C), 65–80 (2015)

  4. Bürgin, R., Ritschard, G.: Coefficient-wise tree-based varying coefficient regression with vcrpart. J. Stat. Softw. 80(6), 1–33 (2017)

  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)

  6. Cameron, A.C., Trivedi, P.K.: Regression Analysis of Count Data. Econometric Society Monographs No. 30. Cambridge University Press, Cambridge (1998)

  7. Fan, J., Zhang, W.: Statistical estimation in varying coefficient models. Ann. Stat. 27(5), 1491–1518 (1999)

  8. Fan, J., Zhang, W.: Statistical methods with varying coefficient models. Stat. Interface 1(1), 179–195 (2008)

  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)

  10. Gertheiss, J., Tutz, G.: Regularization and model selection with categorial effect modifiers. Stat. Sin. 22(3), 957–982 (2012)

  11. Hastie, T., Tibshirani, R.: Varying-coefficient models. J. R. Stat. Soc. B 55, 757–796 (1993)

  12. Hofner, B., Hothorn, T., Kneib, T.: Variable selection and model choice in structured survival models. Comput. Stat. 28(3), 1079–1101 (2013)

  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)

  14. Hothorn, T., Lausen, B.: On the exact distribution of maximally selected rank statistics. Comput. Stat. Data Anal. 43(2), 121–137 (2003)

  15. Hothorn, T., Hornik, K., Zeileis, A.: Unbiased recursive partitioning: a conditional inference framework. J. Comput. Graph. Stat. 15(3), 651–674 (2006)

  16. Kauermann, G., Tutz, G.: Local likelihood estimation in varying coefficient models including additive bias correction. J. Nonparametric Stat. 12(3), 343–371 (2000)

  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)

  19. Lu, Y., Zhang, R., Zhu, L.: Penalized spline estimation for varying-coefficient models. Commun. Stat. Theory Methods 37(14), 2249–2261 (2008)

  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)

  21. Ripley, B.D.: Pattern Recognition and Neural Networks. Cambridge University Press, Cambridge (1996)

  22. Shih, Y.S.: A note on split selection bias in classification trees. Comput. Stat. Data Anal. 45(3), 457–466 (2004)

  23. Shih, Y.S., Tsai, H.: Variable selection bias in regression trees with constant fits. Comput. Stat. Data Anal. 45(3), 595–607 (2004)

  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)

  25. Wang, H., Xia, Y.: Shrinkage estimation of the varying coefficient model. J. Am. Stat. Assoc. 104(486), 747–757 (2009)

  26. Wang, J.C., Hastie, T.: Boosted varying-coefficient regression models for product demand prediction. J. Comput. Graph. Stat. 23(2), 361–382 (2014)

  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)

  28. Wedderburn, R.W.M.: Quasilikelihood functions, generalized linear models and the Gauss–Newton method. Biometrika 61(3), 439–447 (1974)

  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)

  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)

  31. Zhao, P., Xue, L.: Variable selection for semiparametric varying coefficient partially linear models. Stat. Probab. Lett. 79(20), 2148–2157 (2009)

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Acknowledgements

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

Author information

Correspondence to Moritz Berger.

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Berger, M., Tutz, G. & Schmid, M. Tree-structured modelling of varying coefficients. Stat Comput 29, 217–229 (2019). https://doi.org/10.1007/s11222-018-9804-8

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Keywords

  • Varying-coefficient models
  • Interactions
  • Recursive partitioning
  • Tree-based models