Bayesian fractional polynomials

Statistics and Computing volume 21, pages309–324(2011)Cite this article

Abstract

This paper sets out to implement the Bayesian paradigm for fractional polynomial models under the assumption of normally distributed error terms. Fractional polynomials widen the class of ordinary polynomials and offer an additive and transportable modelling approach. The methodology is based on a Bayesian linear model with a quasi-default hyper-g prior and combines variable selection with parametric modelling of additive effects. A Markov chain Monte Carlo algorithm for the exploration of the model space is presented. This theoretically well-founded stochastic search constitutes a substantial improvement to ad hoc stepwise procedures for the fitting of fractional polynomial models. The method is applied to a data set on the relationship between ozone levels and meteorological parameters, previously analysed in the literature.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1964) (ninth Dover printing, tenth GPO printing edn.)

    Google Scholar 

  2. Albert, J.H., Chib, S.: Bayesian analysis of binary and polychotomous response data. J. Am. Stat. Assoc. 88(422), 669–679 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  3. Ambler, G., Royston, P.: Fractional polynomial model selection procedures: Investigation of type I error rate. J. Stat. Comput. Simul. 69(1), 89–108 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  4. Anderson, I.J.: A distillation algorithm for floating-point summation. SIAM J. Sci. Comput. 20(5), 1797–1806 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  5. Barbieri, M.M., Berger, J.O.: Optimal predictive model selection. Ann. Stat. 32(3), 870–897 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  6. Besag, J., Green, P., Higdon, D., Mengersen, K.: Bayesian computation and stochastic systems (with discussion). Stat. Sci. 10(1), 3–66 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  7. Box, G.E.P., Tidwell, P.W.: Transformation of the independent variables. Technometrics 4(4), 531–550 (1962)

    MathSciNet  MATH  Article  Google Scholar 

  8. Breiman, L., Friedman, J.H.: Estimating optimal transformations for multiple regression and correlation. J. Am. Stat. Assoc. 80(391), 580–598 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  9. Brooks, S.P., Friel, N., King, R.: Classical model selection via simulated annealing. J. R. Stat. Soc., Ser. B Stat. Methodol. 65(2), 503–520 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  10. Denison, D.G.T., Holmes, C.C., Mallick, B.K., Smith, A.F.M.: Bayesian Methods for Nonlinear Classification and Regression. Wiley Series in Probability and Statistics. Wiley, Chichester (2002)

    Google Scholar 

  11. Fouskakis, D., Ntzoufras, I., Draper, D.: Bayesian variable selection using cost-adjusted BIC, with application to cost-effective measurement of quality of health care. Ann. Appl. Stat. 3(2), 663–690 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  12. Frühwirth-Schnatter, S., Wagner, H.: Auxiliary mixture sampling for parameter-driven models of time series of counts with applications to state space modelling. Biometrika 93(4), 827–841 (2006)

    MathSciNet  Article  Google Scholar 

  13. Frühwirth-Schnatter, S., Frühwirth, R., Held, L., Rue, H.: Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data. Stat. Comput. 19(4), 479–492 (2009)

    MathSciNet  Article  Google Scholar 

  14. George, E.I., McCulloch, R.E.: Approaches for Bayesian variable selection. Stat. Sin. 7(2), 339–373 (1997)

    MATH  Google Scholar 

  15. Gottardo, R., Raftery, A.: Bayesian robust transformation and variable selection: a unified approach. Can. J. Stat. 37(3), 361–380 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  16. Govindarajulu, U.S., Malloy, E.J., Ganguli, B., Spiegelman, D., Eisen, E.A.: The comparison of alternative smoothing methods for fitting non-linear exposure-response relationships with Cox models in a simulation study. Int. J. Biostat. 5(1), 1–19 (2009)

    MathSciNet  Google Scholar 

  17. Hans, C., Dobra, A., West, M.: Shotgun stochastic search for “large p” regression. J. Am. Stat. Assoc. 102(478), 507–516 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  18. Hastie, T.J., Tibshirani, R.J.: Generalized Additive Models. Chapman & Hall, London (1990)

    Google Scholar 

  19. Hoeting, J.A., Ibrahim, J.G.: Bayesian predictive simultaneous variable and transformation selection in the linear model. J. Comput. Stat. Data Anal. 28(1), 87–103 (1998)

    MATH  Article  Google Scholar 

  20. Hoeting, J.A., Raftery, A.E., Madigan, D.: Bayesian variable and transformation selection in linear regression. J. Comput. Graph. Stat. 11(3), 485–507 (2002)

    MathSciNet  Article  Google Scholar 

  21. Holmes, C.C., Held, L.: Bayesian auxiliary variable models for binary and multinomial regression. Bayesian Anal. 1(1), 145–168 (2006)

    MathSciNet  Google Scholar 

  22. Jasra, A., Stephens, D.A., Holmes, C.C.: Population-based reversible jump Markov chain Monte Carlo. Biometrika 94(4), 787–807 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  23. Liang, F., Paulo, R., Molina, G., Clyde, M., Berger, J.: Mixtures of g priors for Bayesian variable selection. J. Am. Stat. Assoc. 103(481), 410–423 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  24. Madigan, D., York, J.: Bayesian graphical models for discrete data. Int. Stat. Rev. 63(2), 215–232 (1995)

    MATH  Article  Google Scholar 

  25. Raftery, A.E., Madigan, D., Hoeting, J.A.: Bayesian model averaging for linear regression models. J. Am. Stat. Assoc. 92(437), 179–191 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  26. Royston, P., Altman, D.G.: Regression using fractional polynomials of continuous covariates: Parsimonious parametric modelling. J. R. Stat. Soc., Ser. C, Appl. Stat. 46(3), 429–467 (1994)

    Google Scholar 

  27. Royston, P., Altman, D.: Approximating statistical functions by using fractional polynomial regression. J. R. Stat. Soc., Ser. D Stat. 46(3), 411–422 (1997)

    Article  Google Scholar 

  28. Royston, P., Sauerbrei, W.: Multivariable Model-building: A Pragmatic Approach to Regression Analysis based on Fractional Polynomials for Modelling Continuous Variables. Wiley Series in Probability and Statistics. Wiley, Chichester (2008)

    Google Scholar 

  29. Ruppert, D., Wand, M.P., Carroll, R.J.: Semiparametric Regression. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  30. Sauerbrei, W., Royston, P.: Building multivariable prognostic and diagnostic models: transformation of the predictors by using fractional polynomials. J. R. Stat. Soc., Ser. A, Stat. Soc. 162(1), 71–94 (1999)

    Article  Google Scholar 

  31. Sauerbrei, W., Meier-Hirmer, C., Benner, A., Royston, P.: Multivariable regression model building by using fractional polynomials: Description of SAS, STATA and R programs. J. Comput. Stat. Data Anal. 50(12), 3464–3485 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  32. Shkedy, Z., Aerts, M., Molenberghs, G., Beutels, P., van Damme, P.: Modelling force of infection from prevalence data using fractional polynomials. Stat. Med. 25(9), 1577–1591 (2006)

    MathSciNet  Article  Google Scholar 

  33. Sutradhar, B.C.: On the characteristic function of multivariate Student t-distribution. Can. J. Stat. 14(4), 329–337 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  34. Zellner, A.: On assessing prior distributions and Bayesian regression analysis with g-prior distributions. In: Goel, P.K., Zellner, A. (eds.) Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti. Studies in Bayesian Econometrics and Statistics, vol. 6, pp. 233–243. North-Holland, Amsterdam (1986)

    Google Scholar 

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Affiliations

  1. Institute of Social and Preventive Medicine, Biostatistics Unit, University of Zurich, Zurich, Switzerland

    Daniel Sabanés Bové & Leonhard Held

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  1. Daniel Sabanés Bové
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  2. Leonhard Held
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Correspondence to Leonhard Held.

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Sabanés Bové, D., Held, L. Bayesian fractional polynomials. Stat Comput 21, 309–324 (2011). https://doi.org/10.1007/s11222-010-9170-7

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Keywords

  • Bayesian linear model
  • Fractional polynomials
  • Hyper-g prior
  • Stochastic search algorithm