Skip to main content
SpringerLink
Log in

Bayesian fractional polynomials

  • Published:
Statistics and Computing Aims and scope Submit manuscript

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.

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

Access this article

Log in via an institution

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

References

  • 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.)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • 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)

    Article  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • 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 

  • 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)

    MATH  Google Scholar 

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

    Google Scholar 

  • 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 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

Download references

Author information

Authors and Affiliations

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

    Daniel Sabanés Bové & Leonhard Held

Authors
  1. Daniel Sabanés Bové
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leonhard Held
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leonhard Held.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11222-010-9170-7

Keywords

Search

Navigation