AStA Advances in Statistical Analysis

, Volume 102, Issue 1, pp 1–20 | Cite as

Assessment of vague and noninformative priors for Bayesian estimation of the realized random effects in random-effects meta-analysis

  • Olha Bodnar
  • Clemens Elster
Original Paper


Random-effects meta-analysis has become a well-established tool applied in many areas, for example, when combining the results of several clinical studies on a treatment effect. Typically, the inference aims at the common mean and the amount of heterogeneity. In some applications, the laboratory effects are of interest, for example, when assessing uncertainties quoted by laboratories participating in an interlaboratory comparison in metrology. We consider the Bayesian estimation of the realized random effects in random-effects meta-analysis. Several vague and noninformative priors are examined as well as a proposed novel one. Conditions are established that ensure propriety of the posteriors for the realized random effects. We present extensive simulation results that assess the inference in dependence on the choice of prior as well as mis-specifications in the statistical model. Overall good performance is observed for all priors with the novel prior showing the most promising results. Finally, the uncertainties reported by eleven national metrology institutes and universities for their measurements on the Newtonian constant of gravitation are assessed.


Random-effects model Bayesian estimation Reference prior Newtonian constant of gravitation 

Mathematics Subject Classification

62F15 62P35 



The authors are thankful to Professor Göran Kauermann and two anonymous Reviewers for careful reading of the paper and for their suggestions which have improved an earlier version of this paper. They also thank Alfred Link for the validation of the numerical results and for helpful discussions.


  1. Berger, J., Bernardo, J.M.: On the development of reference priors. In: Bernardo, J.M., Berger, J., Dawid, A.P., Smith, A.F.M. (eds.) Bayesian statistics, vol. 4, pp. 35–60. University Press, Oxford (1992)Google Scholar
  2. Berger, J., Bernardo, J.M.: Reference priors in a variance components problem. In: Goel, P. (ed.) Proceedings of the Indo-USA Workshop on Bayesian Analysis in Statistics and Econometrics, pp. 323–340. Springer, New-York (1992)Google Scholar
  3. Bodnar, O., Elster, C.: Analytical derivation of the reference prior by sequential maximization of shannon’s mutual information in the multi-group parameter case. J. Stat. Plan. Inference 147, 106–116 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bodnar, O., Elster, C., Fischer, J., Possolo, A., Toman, B.: Evaluation of uncertainty in the adjustment of fundamental constants. Metrologia 53, S46–S54 (2016)CrossRefGoogle Scholar
  5. Bodnar, O., Link, A., Arendacká, B., Possolo, A., Elster, C.: Improved estimation in random effects meta-analysis. Stat. Med. (2016). doi: 10.1002/sim.7156 Google Scholar
  6. Bodnar, O., Link, A., Elster, C.: Objective bayesian inference for a generalized marginal random effects model. Bayesian Anal. 11, 25–45 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Cochran, W.G.: Problems arising in the analysis of a series of similar experiments. J. R. Stat. Soc. Suppl. 4, 102–118 (1937)CrossRefzbMATHGoogle Scholar
  8. Cochran, W.G.: The combination of estimates from different experiments. Biometrics 10, 109–129 (1954)Google Scholar
  9. DerSimonian, R., Laird, N.: Meta-analysis in clinical trials. Control. Clin. Trials 7, 177–188 (1986)CrossRefGoogle Scholar
  10. Gelman, A.: Prior distributions for variance parameters in hierarchical models. Bayesian Anal. 1, 515–533 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B.: Bayesian data analysis. Taylor & Francis, UK (2013)zbMATHGoogle Scholar
  12. Guolo, A., Varin, C.: Random-effects meta-analysis: the number of studies matters. Stat. Methods Med. Res. (2015). doi: 10.1177/0962280215583568 Google Scholar
  13. Higgins, J., Thompson, S.G., Spiegelhalter, D.J.: A re-evaluation of random-effects meta-analysis. J. R. Stat. Soc. Ser. A (Stat. Soc.) 172, 137–159 (2009)Google Scholar
  14. Higgins, J., Whitehead, A.: Borrowing strength from external trials in a meta-analysis. Stat. Med. 15, 2733–2749 (1996)CrossRefGoogle Scholar
  15. Hill, B.M.: Inference about variance components in the one-way model. J. Am. Stat. Assoc. 60, 806–825 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Hurtado Rúa, S.M., Mazumdar, M., Strawderman, R.L.: The choice of prior distribution for a covariance matrix in multivariate meta-analysis: a simulation study. Stat. Med. 34, 4083–4104 (2015)Google Scholar
  17. Kacker, R.N.: Combining information from interlaboratory evaluations using a random effects model. Metrologia 41, 132–136 (2004)CrossRefGoogle Scholar
  18. Klein, N., Kneib, T.: Scale-dependent priors for variance parameters in structured additive distributional regression. Bayesian Anal. 11(4), 1071–1106 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Knapp, G., Hartung, J.: Improved tests for a random effects meta-regression with a single covariate. Stat. Med. 22, 2693–2710 (2003)CrossRefGoogle Scholar
  20. Lambert, P.C., Sutton, A.J., Burton, P.R., Abrams, K.R., Jones, D.R.: How vague is vague? a simulation study of the impact of the use of vague prior distributions in mcmc using winbugs. Stat. Med. 24, 2401–2428 (2005)MathSciNetCrossRefGoogle Scholar
  21. McCulloch, C.E., Neuhaus, J.M.: Generalized linear mixed models. Wiley Online Library, New York (2001)Google Scholar
  22. Mohr, P.J., Taylor, B.N., Newell, D.B.: Codata recommended values of the fundamental physical constants: 2010. J. Phys. Chem. Ref. Data 41, 043109 (2012)CrossRefGoogle Scholar
  23. Mohr, P.J., Taylor, B.N., Newell, D.B.: Codata recommended values of the fundamental physical constants: 2010. Rev. Mod. Phys. 84, 1527–1605 (2012)CrossRefGoogle Scholar
  24. Müller, I., Brade, V., Hagedorn, H.-J., Straube, E., Schörner, C., Frosch, M., Hlobil, H., Stanek, G., Hunfeld, K.-P.: Is serological testing a reliable tool in laboratory diagnosis of syphilis? Meta-analysis of eight external quality control surveys performed by the german infection serology proficiency testing program. J. Clin. Microbiol. 44, 1335–1341 (2006)CrossRefGoogle Scholar
  25. Ohlssen, D.I., Sharples, L.D., Spiegelhalter, D.J.: Flexible random-effects models using Bayesian semi-parametric models: applications to institutional comparisons. Stat. Med. 26, 2088–2112 (2007)MathSciNetCrossRefGoogle Scholar
  26. Paule, R.C., Mandel, J.: Consensus values and weighting factors. J. Res. Natl. Bureau Stand. 87, 377–385 (1982)CrossRefzbMATHGoogle Scholar
  27. Pullenayegum, E.M.: An informed reference prior for between-study heterogeneity in meta-analyses of binary outcomes. Stat. Med. 30, 3082–3094 (2011)MathSciNetCrossRefGoogle Scholar
  28. Rao, P.S.R.S.: Variance components estimation: mixed models, methodologies, and applications. Chapman and Hall, London (1997)zbMATHGoogle Scholar
  29. Sahai, H., Ojeda, M.: Analysis of variance for random models. Unbalanced data, vol. 2. Birkhauser, Boston, Basel, Berlin (2004)Google Scholar
  30. Searle, S.R., Casella, G., McCulloch, C.E.: Variance components, vol. 391. Wiley, New York (2009)zbMATHGoogle Scholar
  31. Simpson, D.P., Rue, H., Martins, T.G., Riebler, A., Sørbye, S.H.: Penalising model component complexity: a principled, practical approach to constructing priors Stat. Sci. (2016, to appear)Google Scholar
  32. Smith, T.C., Spiegelhalter, D.J., Thomas, A.: Bayesian approaches to random-effects meta-analysis: a comparative study. Stat. Med. 14, 2685–2699 (1995)CrossRefGoogle Scholar
  33. Stone, M., Springer, B.G.F.: A paradox involving quasi-prior distributions. Biometrika 52, 623–627 (1965)Google Scholar
  34. Sun, D., Berger, J.: Reference priors with partial information. Biometrika 85, 55–71 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  35. Sutton, A.J., Higgins, J.: Recent developments in meta-analysis. Stat. Med. 27, 625–650 (2008)MathSciNetCrossRefGoogle Scholar
  36. Tiao, G.C., Tan, W.Y.: Bayesian analysis of random-effect models in the analysis of variance. i: Posterior distribution of variance components. Biometrika 52, 37–53 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  37. Toman, B.: Bayesian approaches to calculating a reference value in key comparison experiments. Technometrics 49, 81–87 (2007)MathSciNetCrossRefGoogle Scholar
  38. Toman, B., Fischer, J., Elster, C.: Alternative analyses of measurements of the planck constant. Metrologia 49, 567–571 (2012)CrossRefGoogle Scholar
  39. Toman, B., Possolo, A.: Laboratory effects models for interlaboratory comparisons. Accredit. Qual. Assur. 14, 553–563 (2009)CrossRefGoogle Scholar
  40. Turner, R.M., Davey, J., Clarke, M.J., Thompson, S.G., Higgins, J.: Predicting the extent of heterogeneity in meta-analysis, using empirical data from the cochrane database of systematic reviews. Int. J. Epidemiol. 41, 818–827 (2012)CrossRefGoogle Scholar
  41. Turner, R.M., Jackson, D., Wei, Y., Thompson, S.G., Higgins, J.: Predictive distributions for between-study heterogeneity and simple methods for their application in bayesian meta-analysis. Stat. Med. 34, 984–998 (2015)MathSciNetCrossRefGoogle Scholar
  42. Warn, D.E., Thompson, S.G., Spiegelhalter, D.J.: Bayesian random effects meta-analysis of trials with binary outcomes: methods for the absolute risk difference and relative risk scales. Stat. Med. 21, 1601–1623 (2002)CrossRefGoogle Scholar
  43. Yates, F., Cochran, W.G.: The analysis of groups of experiments. J. Agric. Sci. 28, 556–580 (1938)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Physikalisch-Technische BundesanstaltBerlinGermany

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