, Volume 71, Issue 2, pp 125–138 | Cite as

Optimal hypothesis testing: from semi to fully Bayes factors

  • Albert Vexler
  • Chengqing Wu
  • Kai Fun YuEmail author


We propose and examine statistical test-strategies that are somewhat between the maximum likelihood ratio and Bayes factor methods that are well addressed in the literature. The paper shows an optimality of the proposed tests of hypothesis. We demonstrate that our approach can be easily applied to practical studies, because execution of the tests does not require deriving of asymptotical analytical solutions regarding the type I error. However, when the proposed method is utilized, the classical significance level of tests can be controlled.


Likelihood ratio Maximum likelihood Bayes factor Most powerful Hypotheses testing Significance level Type I error 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aitkin M (1991) Posterior Bayes factors. J R Stat Soc B 53: 111–142zbMATHGoogle Scholar
  2. Berger JO (1993) Statistical decision theory and bayesian analysis, 2nd edn. Springer Series in Statistics, New YorkGoogle Scholar
  3. Brown RL, Durbin J, Evans JM (1975) Techniques for testing the constancy of regression relationships over time (with discussion). J R Stat Soc B 37: 149–192zbMATHMathSciNetGoogle Scholar
  4. Fisher RA (1925) Statistical methods for research workers, 1st edn. Oliver and Boyd, EdinburghGoogle Scholar
  5. Gelfand AE, Dey DK (1994) Bayesian model choice: asymptotics and exact calculations. J R Stat Soc B 56: 501–514zbMATHMathSciNetGoogle Scholar
  6. Green PJ (1990) On use of the EM for penalized likelihood estimation.. J R Stat Soc B 52: 443–452zbMATHGoogle Scholar
  7. Kass RE, Wasserman L (1995) A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion. J Am Stat Assoc 90: 928–934zbMATHCrossRefMathSciNetGoogle Scholar
  8. Krieger AM, Pollak M, Yakir B (2003) Surveillance of a simple linear regression. J Am Stat Assoc 98: 456–469zbMATHCrossRefMathSciNetGoogle Scholar
  9. Lehmann EL, Romano JP (2005) Testing statistical hypotheses. Springer, New YorkzbMATHGoogle Scholar
  10. Marden JI (2000) Hypothesis testing: from p values to Bayes factors. J Am Stat Assoc 95: 1316–1320zbMATHCrossRefMathSciNetGoogle Scholar
  11. Neyman J, Pearson E (1928) On the use and interpretation of certain test criteria for purposes of statistical inference: Part I. Biometrika 20A: 175–240Google Scholar
  12. Neyman J, Pearson E (1928) On the use and interpretation of certain test criteria for purposes of statistical inference: Part II. Biometrika 20A: 263–294Google Scholar
  13. Neyman J, Pearson E (1933) On the testing of statistical hypotheses in relation to probability a priori. Proc Camb Philos Soc 29: 492–510CrossRefGoogle Scholar
  14. Neyman J, Pearson E (1933) On the problem of the most efficient tests of statistical hypotheses. Philos Trans R Soc A 231: 289–337zbMATHCrossRefGoogle Scholar
  15. Neyman J, Pearson E (1936) Contributions to the theory of testing statistical hypotheses. I. Unbiased critical region of type A and type A. Stat Res Mem 1: 1–37Google Scholar
  16. Neyman J, Pearson E (1936) Sufficient statistics and uniformly most powerful tests of statistical hypotheses. Stat Res Mem 1: 113–137Google Scholar
  17. Neyman J, Pearson E (1938) Contributions to the theory of testing statistical hypotheses. II. Stat Res Mem 2: 25–57Google Scholar
  18. O’Hagan A (1995) Fractional Bayes factors for model comparison. J R Stat Soc B 57: 99–138zbMATHMathSciNetGoogle Scholar
  19. Pearson K (1900) On the criterion that a given system of deviation from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philos Mag Ser 5 50: 157–172CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of BiostatisticsThe New York State University at BuffaloBuffaloUSA
  2. 2.Department of Epidemiology and Public HealthYale UniversityNew HavenUSA
  3. 3.Department of Health and Human ServicesBiostatistics and Bioinformatics Branch, Eunice Kennedy Shriver National Institute of Child Health and Human DevelopmentBethesdaUSA

Personalised recommendations