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Objective Testing Procedures in Linear Models: Calibration of the p‐values

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  • F. JAVIER GIRÓN
  • M. LINA MARTÍNEZ
  • ELÍAS MORENO
  • FRANCISCO TORRES

Abstract

. An optimal Bayesian decision procedure for testing hypothesis in normal linear models based on intrinsic model posterior probabilities is considered. It is proven that these posterior probabilities are simple functions of the classical F‐statistic, thus the evaluation of the procedure can be carried out analytically through the frequentist analysis of the posterior probability of the null. An asymptotic analysis proves that, under mild conditions on the design matrix, the procedure is consistent. For any testing hypothesis it is also seen that there is a one‐to‐one mapping – which we call calibration curve– between the posterior probability of the null hypothesis and the classical bip‐value. This curve adds substantial knowledge about the possible discrepancies between the Bayesian and the p‐value measures of evidence for testing hypothesis. It permits a better understanding of the serious difficulties that are encountered in linear models for interpreting the p‐values. A specific illustration of the variable selection problem is given.

Suggested Citation

  • F. Javier Girón & M. Lina Martínez & Elías Moreno & Francisco Torres, 2006. "Objective Testing Procedures in Linear Models: Calibration of the p‐values," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(4), pages 765-784, December.
  • Handle: RePEc:bla:scjsta:v:33:y:2006:i:4:p:765-784
    DOI: 10.1111/j.1467-9469.2006.00514.x
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    Cited by:

    1. Antonio Barrera & Patricia Román-Román & Juan José Serrano-Pérez & Francisco Torres-Ruiz, 2021. "Two Multi-Sigmoidal Diffusion Models for the Study of the Evolution of the COVID-19 Pandemic," Mathematics, MDPI, vol. 9(19), pages 1-29, September.
    2. Cano, J.A. & Carazo, C. & Salmerón, D., 2016. "Linear contrasts for the one way analysis of variance: A Bayesian approach," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 54-62.
    3. Diego Salmeron & Juan Antonio Cano & Christian Robert, 2013. "Objective bayesian Hypothesis Testing in Binomial Regression Models with Integral Prior Distributions," Working Papers 2013-44, Center for Research in Economics and Statistics.
    4. Guido Consonni & Roberta Paroli, 2017. "Objective Bayesian Comparison of Constrained Analysis of Variance Models," Psychometrika, Springer;The Psychometric Society, vol. 82(3), pages 589-609, September.
    5. J. Cano & C. Carazo & D. Salmerón, 2013. "Bayesian model selection approach to the one way analysis of variance under homoscedasticity," Computational Statistics, Springer, vol. 28(3), pages 919-931, June.
    6. Sang Gil Kang & Woo Dong Lee & Yongku Kim, 2022. "Objective Bayesian group variable selection for linear model," Computational Statistics, Springer, vol. 37(3), pages 1287-1310, July.
    7. Pérez, María-Eglée & Pericchi, Luis Raúl, 2014. "Changing statistical significance with the amount of information: The adaptive α significance level," Statistics & Probability Letters, Elsevier, vol. 85(C), pages 20-24.
    8. Elías Moreno & Carmen Martínez, 2022. "Bayesian and frequentist evidence in one-sided hypothesis testing," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(1), pages 278-297, March.
    9. Rui Wang & Xingzhong Xu, 2021. "A Bayesian-motivated test for high-dimensional linear regression models with fixed design matrix," Statistical Papers, Springer, vol. 62(4), pages 1821-1852, August.
    10. T S Shively & S G Walker, 2018. "On Bayes factors for the linear model," Biometrika, Biometrika Trust, vol. 105(3), pages 739-744.
    11. Miguel A. Negrín & Francisco J. Vázquez-Polo & María Martel & Elías Moreno & Francisco J. Girón, 2010. "Bayesian Variable Selection in Cost-Effectiveness Analysis," IJERPH, MDPI, vol. 7(4), pages 1-20, April.

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