The Bayesian t-Test and Beyond

  • Mithat Gönen
Part of the Methods in Molecular Biology book series (MIMB, volume 620)


In this chapter we will explore Bayesian alternatives to the t-test. We saw in Chapter 1 how t-test can be used to test whether the expected outcomes of the two groups are equal or not. In Chapter 3 we saw how to make inferences from a Bayesian perspective in principle. In this chapter we will put these together to develop a Bayesian procedure for a t-test. This procedure depends on the data only through the t-statistic. It requires prior inputs and we will discuss how to assign them. We will use an example from a microarray study as to demonstrate the practical issues. The microarray study is an important application for the Bayesian t-test as it naturally brings up the question of simultaneous t-tests. It turns out that the Bayesian procedure can easily be extended to carry several t-tests on the same data set, provided some attention is paid to the concept of the correlation between tests.

Key words

Two-sample comparisons prior correlations multivariate testing simultaneous inferences multiple testing 


  1. 1.
    Antonescu, C.R., Viale, A., Sarran, L., Tschernyavsky, S.J., Gönen, M., Segal, N.H., Maki, R.G., Socci, N.D., DeMatteo, R.P., and Besmer, P. (2004) Gene expression in gastrointestinal stromal tumors is distinguished by KIT genotype and anatomic site. Clinical Cancer Reserch 10, 3282–3290.CrossRefGoogle Scholar
  2. 2.
    Berger, J.O. (1985) Statistical Decision Theory and Bayesian Analysis. New York: Springer-Verlag.CrossRefGoogle Scholar
  3. 3.
    Bernardo, J.M. and Smith, A.F.M. (1994) Bayesian Theory. New York: John Wiley and Sons.CrossRefGoogle Scholar
  4. 4.
    Berry, D.A. (1996) Statistics: A Bayesian Perspective. Belmont, CA: Wadsworth.Google Scholar
  5. 5.
    Berry, D.A. (1997) Teaching elementary Bayesian statistics with real applications in science. The American Statistician 51, 241–246.Google Scholar
  6. 6.
    Carlin, B. and Louis, T. (2001) Bayes and Empirical Bayes Methods for Data Analysis. London: Chapman and Hall.Google Scholar
  7. 7.
    Jeffreys, H. (1961) Theory of Probability. Oxford: Oxford University Press.Google Scholar
  8. 8.
    Cohen, J. (1988) Statistical Power Analysis for the Behavioral Sciences New York: Academic Press.Google Scholar
  9. 9.
    Westfall, P.H., Johnson, W.O., and Utts, J.M. (1997) A Bayesian perspective on the Bonferroni adjustment. Biometrika 84, 419–427.CrossRefGoogle Scholar
  10. 10.
    Efron, B., Tibshirani, R., Storey, J.D., and Tusher, V. (2001) Empirical Bayes analysis of a microarray experiment. Journal of the American Statistical Association 96, 1151–1160.CrossRefGoogle Scholar
  11. 11.
    Gönen M., Johnson, W.O., Lu, T., and Westfall, P.H. (2005) The Bayesian two-sample t-test. The American Statistician 59, 252–257.CrossRefGoogle Scholar
  12. 12.
    Berger, J.O. and Sellke, T. (1987) Testing a point null hypothesis: The irreconcilability of p values and evidence. Journal of the American Statistical Association 82, 112–122.Google Scholar
  13. 13.
    Miller, R. (1981) Simultaneous Statistical Inference. New York: Springer.CrossRefGoogle Scholar
  14. 14.
    Hochberg, Y. and Tamhane, A. (1987) Multiple Comparison Procedures. New York: Wiley.CrossRefGoogle Scholar
  15. 15.
    Westfall, P.H. and Young, S.S. (1993). Resampling-Based Multiple Testing. New York: Wiley.Google Scholar
  16. 16.
    Westfall, P.H., Tobias, R., Rom, D., Wolfinger, R., and Hochberg, Y. (1999) Multiple Comparisons and Multiple Tests Using the SAS(R) System. Cary: SAS Institute.Google Scholar
  17. 17.
    Gönen, M., Westfall, P.H., and Johnson, W.O. (2003) Bayesian multiple testing for two-sample multivariate endpoints. Biometrics 59, 76–82.PubMedCrossRefGoogle Scholar
  18. 18.
    Kass, R.E. and Raftery, A.E. (1995) Bayes factors. Journal of the American Statistical Association 90, 773–795.CrossRefGoogle Scholar
  19. 19.
    Genz, A. (1992) Numerical computation of the multivariate normal probabilities. Journal of Computational and Graphical Statistics 1, 141–150.Google Scholar
  20. 20.
    Gönen, M. and Westfall, P.H. (1998) Bayesian multiple testing of multiple endpoints in clinical trials. Proceedings of the American Statistical Association, Biopharmaceutical Subsection, 108–113.Google Scholar

Copyright information

© Humana Press, a part of Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Mithat Gönen
    • 1
  1. 1.Department of Epidemiology and BiostatisticsMemorial Sloan-Kettering Cancer CenterNew YorkUSA

Personalised recommendations