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Bayesian fractional polynomials

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.)

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

  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)

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

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

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

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

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

  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)

  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)

  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)

  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)

  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)

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

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

  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)

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

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

  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)

  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)

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

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

  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)

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

  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)

  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)

  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)

  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)

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

  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)

  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)

  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)

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

  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)

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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