Statistics and Computing

, Volume 28, Issue 3, pp 579–594 | Cite as

Objective Bayesian transformation and variable selection using default Bayes factors

Article
  • 112 Downloads

Abstract

In this work, the problem of transformation and simultaneous variable selection is thoroughly treated via objective Bayesian approaches by the use of default Bayes factor variants. Four uniparametric families of transformations (Box–Cox, Modulus, Yeo-Johnson and Dual), denoted by T, are evaluated and compared. The subjective prior elicitation for the transformation parameter \(\lambda _T\), for each T, is not a straightforward task. Additionally, little prior information for \(\lambda _T\) is expected to be available, and therefore, an objective method is required. The intrinsic Bayes factors and the fractional Bayes factors allow us to incorporate default improper priors for \(\lambda _T\). We study the behaviour of each approach using a simulated reference example as well as two real-life examples.

Keywords

Bayesian model selection Fractional Bayes factor Intrinsic Bayes factor Posterior model probabilities Transformation family selection Variable selection 

References

  1. Bartlett, M.S.: Comment on D.V. Lindley’s statistical paradox. Biometrika 44, 533–534 (1957)CrossRefMATHGoogle Scholar
  2. Berger, J.O., Pericchi, L.R.: The intrinsic Bayes factor for linear models. In: Bernardo, J., Berger, J., Dawid, A., Smith, A. (eds.) Bayesian Statistics, vol. 5, pp. 25–44. Oxford University Press, Oxford (1996a)Google Scholar
  3. Berger, J.O., Pericchi, L.R.: The intrinsic Bayes factor for model selection and prediction. J. Am. Stat. Assoc. 91, 109–122 (1996b)MathSciNetCrossRefMATHGoogle Scholar
  4. Berger, J.O., Pericchi, L.R.: Objective Bayesian methods for model selection: Introduction and comparison. In: Lahiri, P. (ed.) ‘Model selection’, Lecture Notes-Monograph Series, vol. 38, Institute of Mathematical Statistics, pp. 135–207 (2001)Google Scholar
  5. Berger, J.O., Pericchi, L.R.: Training samples in objective model selection. Ann. Stat. 32, 841–869 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. Box, G.E.P., Cox, D.R.: An analysis of transformations (with discussion). J. R. Stat. Soc. Ser. B 26, 211–252 (1964)MathSciNetMATHGoogle Scholar
  7. Casella, G., Moreno, E.: Objective Bayesian variable selection. J. Am. Stat. Assoc. 101, 157–167 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. Charitidou, E., Fouskakis, D., Ntzoufras, I.: Bayesian transformation family selection: moving towards a transformed Gaussian universe. Can. J. Stat. 43, 600–623 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. Consonni, G., Veronese, P.: Compatibility of prior specifications accross linear models. Stat. Sci. 23, 332–363 (2008)CrossRefMATHGoogle Scholar
  10. Gottardo, R., Raftery, A.E.: Bayesian robust variable and transformation selection: a unified approach. Can. J. Stat. 37, 1–20 (2009)CrossRefMATHGoogle Scholar
  11. Hald, A.: Statistical Theory with Engineering Applications. Wiley, New York (1952)MATHGoogle Scholar
  12. Hoeting, J.A., Ibrahim, J.G.: Bayesian predictive simultaneous variable and transformation selection in the linear model. J. Comput. Stat. Data Anal. 28, 87–103 (1998)CrossRefMATHGoogle Scholar
  13. Hoeting, J.A., Raftery, A.E., Madigan, D.: A method for simultaneous variable and transformation selection in linear regression. J. Comput. Graph. Stat. 11, 485–507 (2002)CrossRefMATHGoogle Scholar
  14. Ibrahim, J.G., Chen, M.H.: Power prior distributions for regression models. Stat. Sci. 15, 46–60 (2000)MathSciNetCrossRefGoogle Scholar
  15. John, J.A., Draper, N.R.: An alternative family of transformations. Appl. Stat. 29, 190–197 (1980)CrossRefMATHGoogle Scholar
  16. Kim, S., Chen, M.H., Ibrahim, J.G., Shah, A.K., Lin, J.: Bayesian inference for multivariate meta-analysis Box–Cox transformation models for individual patient data with applications to evaluation of cholesterol-lowering drugs. Stat. Med. 32, 3972–3990 (2013)Google Scholar
  17. Kuo, L., Mallick, B.: Variable selection for regression models. Sankhyā B 60, 65–81 (1998)MathSciNetMATHGoogle Scholar
  18. Laud, P.W., Ibrahim, J.G.: Predictive model selection. J. R. Stat. Soc. B 57, 247–262 (1995)MathSciNetMATHGoogle Scholar
  19. Ley, E., Steel, M.F.J.: On the effect of prior assumptions in Bayesian model averaging with applications to growth regression. J. Appl. Econom. 24, 651–674 (2009)MathSciNetCrossRefGoogle Scholar
  20. Madigan, D., York, J.: Bayesian graphical models for discrete data. Int. Stat. Rev. 63, 215–232 (1995)CrossRefMATHGoogle Scholar
  21. Miranda, M.F., Zhu, H., Ibrahim, J.G.: Bayesian spatial transformation models with applications in neuroimaging data. Biometrics 69, 1074–1083 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. O’Hagan, A.: Fractional Bayes factors for model comparison. J. R. Stat. Soc. B 57, 99–138 (1995)MathSciNetMATHGoogle Scholar
  23. Pérez, J.M., Berger, J.O.: Expected-posterior prior distributions for model selection. Biometrika 89, 491–511 (2002)MathSciNetCrossRefMATHGoogle Scholar
  24. Pericchi, L.R.: A Bayesian approach to transformations to normality. Biometrika 68, 35–43 (1981)MathSciNetCrossRefMATHGoogle Scholar
  25. Scott, J.G., Berger, J.O.: Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem. Ann. Stat. 38, 2587–2619 (2010)MathSciNetCrossRefMATHGoogle Scholar
  26. Sweeting, T.J.: On the choice of the prior distribution for the Box–Cox transformed linear model. Biometrika 71, 127–134 (1984)MathSciNetCrossRefGoogle Scholar
  27. Thall, P.F., Russell, K.E., Simon, R.M.: Variable selection in regression via repeated data splitting. J. Am. Stat. Assoc. 6, 416–434 (1997)Google Scholar
  28. Villa, C., Lee, J.E.: Model prior distribution for variable selection in linear regression models (2016). arXiv:1512.08077. http://arxiv.org/abs/1104.0861
  29. Weisberg, S.: Applied Linear Regression, 3rd edn. Wiley-Interscience, New Jersey (2005)CrossRefMATHGoogle Scholar
  30. Westerberg, I., Guerrero, J.L., Seibert, J., Beven, K.J., Halldin, S.: Stage-discharge uncertaintly derived with a non-stationary rating curve in the Choluteca river, honduras. Hydrol. Process. 25, 603–613 (2011)CrossRefGoogle Scholar
  31. Yang, Y., Christensen, O.F., Sorensen, D.: Analysis of a genetically structured variance heterogeneity model using the Box–Cox transformation. Genet. Res. 93, 33–46 (2011)CrossRefGoogle Scholar
  32. Yang, Z.: A modified family of power transformations. Econ. Lett. 92, 14–19 (2006)MathSciNetCrossRefMATHGoogle Scholar
  33. Yeo, I.K., Johnson, R.A.: A new family of power transformations to improve normality or symmetry. Biometrika 87, 954–959 (2000)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsNational Technical University of AthensAthensGreece
  2. 2.Department of StatisticsAthens University of Economics and Business10434Greece

Personalised recommendations