Statistical Methods & Applications

, Volume 22, Issue 3, pp 319–340 | Cite as

Predictive control of posterior robustness for sample size choice in a Bernoulli model

  • Fulvio De Santis
  • Maria Clara Fasciolo
  • Stefania GubbiottiEmail author


In this article we consider the sample size determination problem in the context of robust Bayesian parameter estimation of the Bernoulli model. Following a robust approach, we consider classes of conjugate Beta prior distributions for the unknown parameter. We assume that inference is robust if posterior quantities of interest (such as point estimates and limits of credible intervals) do not change too much as the prior varies in the selected classes of priors. For the sample size problem, we consider criteria based on predictive distributions of lower bound, upper bound and range of the posterior quantity of interest. The sample size is selected so that, before observing the data, one is confident to observe a small value for the posterior range and, depending on design goals, a large (small) value of the lower (upper) bound of the quantity of interest. We also discuss relationships with and comparison to non robust and non informative Bayesian methods.


Bayesian robustness Clinical trials Conjugate analysis  Sample size determination 



The authors would like to thank the editor and two anonymous reviewers for their interesting comments and helpful suggestions.


  1. Adcock CJ (1997) Sample size determination: a review. Statistician 46:261–283Google Scholar
  2. Berger JO (1984) The robust Bayesian viewpoint (with discussion). In: Kadane J (ed) Robustness of Bayesian analysis. North-Holland, AmsterdamGoogle Scholar
  3. Berger JO (1990) Robust Bayesian analysis: sensitivity to the prior. J Stat Plan Inference 25(3):303–328zbMATHCrossRefGoogle Scholar
  4. Brutti P, De Santis F (2008) Avoiding the range of equivalence in clinical trials: Robust Bayesian sample size determination for credible intervals. J Stat Plan Inference 138:1577–1591zbMATHCrossRefGoogle Scholar
  5. Brutti P, De Santis F, Gubbiotti S (2008) Robust Bayesian sample size determination in clinical trials. Stat Med 27(13):2290–2306MathSciNetCrossRefGoogle Scholar
  6. Brutti P, De Santis F, Gubbiotti S (2009) Mixtures of prior distributions for predictive Bayesian sample size calculations in clinical trials. Stat Med 28(17):2185–2201MathSciNetCrossRefGoogle Scholar
  7. Brutti P, Gubbiotti S, Sambucini V (2011) An extension of the single threshold design for monitoring efficacy and safety in phase II clinical trials. Stat Med 30(14):1648–1664MathSciNetCrossRefGoogle Scholar
  8. Chaloner K, Verdinelli I (1995) Bayesian experimental design: a review. Stat Sci 10:237–308MathSciNetCrossRefGoogle Scholar
  9. DasGupta A, Mukhopadhyay S (1994) Uniform and subuniform posterior robustness: the sample size problem. J Stat Plan Inference 40:189–200MathSciNetzbMATHCrossRefGoogle Scholar
  10. De Santis F (2006) Sample size determination for robust Bayesian analysis. J Am Stat Assoc 101(473):278–291zbMATHCrossRefGoogle Scholar
  11. De Santis F (2007) Using historical data for Bayesian sample size determination. J R Stat Soc Ser A 170(1):95–113CrossRefGoogle Scholar
  12. Etzioni R, Kadane JB (1993) Optimal experimental design for another’s analysis. J Am Stat Assoc 88(424):1404–1411MathSciNetzbMATHCrossRefGoogle Scholar
  13. Gubbiotti S, De Santis F (2011) A Bayesian method for the choice of the sample size in equivalence trials. Aust N Z J Stat 53(4):443–460MathSciNetCrossRefGoogle Scholar
  14. Ibrahim JG, Chen M (2000) Power prior distributions for regression models. Stat Sci 15(1):46–60MathSciNetCrossRefGoogle Scholar
  15. Joseph L, Belisle P (1997) Bayesian sample size determination for normal means and difference between normal means. Statistician 46(2):209–226Google Scholar
  16. Kerman J (2011) Neutral noninformative and informative conjugate beta and gamma prior distributions. Electron J Stat 5:1450–1470MathSciNetzbMATHCrossRefGoogle Scholar
  17. Neuenschwander B, Branson M, Spiegelhalter D (2009) A note on the power prior. Stat Med 28: 3562–3566MathSciNetCrossRefGoogle Scholar
  18. Sambucini V (2008) A Bayesian predictive two-stage design for phase ii clinical trials. Stat Med 27: 1199–1224MathSciNetCrossRefGoogle Scholar
  19. Simon R (1989) Optimal two-stage designs for phase II clinical trials. Control Clin Trials 10:1–10CrossRefGoogle Scholar
  20. Sivaganesan S, Berger JO (1989) Ranges of posterior measures for priors with unimodal contaminations. Ann Stat 17(2):868–889MathSciNetzbMATHCrossRefGoogle Scholar
  21. Spiegelhalter DJ, Abrams K, Myles JP (2004) Bayesian approaches to clinical trials and health-care evaluation. Wiley, New YorkGoogle Scholar
  22. Tsutakawa RK (1972) Design of experiment for bioassay. J Am Stat Assoc 67(339):585–590CrossRefGoogle Scholar
  23. Wang F, Gelfand AE (2002) A simulation-based approach to Bayesian sample size determination for performance under a given model and for separating models. Stat Sci 17(2):193–208MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Fulvio De Santis
    • 1
  • Maria Clara Fasciolo
    • 1
  • Stefania Gubbiotti
    • 1
    Email author
  1. 1.Dipartimento di Scienze StatisticheSapienza Università di RomaRomaItaly

Personalised recommendations