On modeling the correlation as an additional parameter in random effects model

Muhumuza, Rebecca; Bodnar, Olha

doi:10.1090/tpms/1137

Authors: Rebecca Nalule Muhumuza and Olha Bodnar
Journal: Theor. Probability and Math. Statist. 103 (2020), 121-136
MSC (2020): Primary 62F15, 62H10, 62H12, 62F10
DOI: https://doi.org/10.1090/tpms/1137
Published electronically: June 16, 2021
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we develop objective Bayesian inference, i. e., Bayesian procedures based on non-informative priors, for the parameters in a generalized marginal random effects model where the correlation between random effects is introduced as an additional model parameter. This approach provides a generalization of the classical random effects model based on the assumption of normality and on the assumption that the random effects are independent. Assuming a generalized marginal random effects model, we derive the closed-form expression of the Fisher information matrix and the analytical expressions of both the Jeffreys prior and the Berger & Bernardo reference prior. Furthermore, the posterior distributions are obtained and it is shown that the conditional posteriors for the location parameter of the model under the Jeffreys prior and under the Berger & Bernardo reference prior are elliptically contoured distributed.

References

James O. Berger and José M. Bernardo, On the development of reference priors, Bayesian statistics, 4 (Peñíscola, 1991) Oxford Univ. Press, New York, 1992, pp. 35–60. MR 1380269
O. Bodnar, Non-informative Bayesian inference for heterogeneity in a generalized marginal random effects meta-analysis, Teor. Ĭmovīr. Mat. Stat. 100 (2019), 7–23 (English, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 100 (2020), 3–19. MR 3992990, DOI 10.1090/tpms/1095
O. Bodnar and C. Elster, Analysis of key comparisons with two reference standards: Extended random effects meta-analysis, Advanced Mathematical and Computational Tools in Metrology and Testing XI, 2018, pp. 1–8.
O. Bodnar, C. Elster, J. Fischer, A. Possolo, and B. Toman, Evaluation of uncertainty in the adjustment of fundamental constants, Metrologia 53 (2016), S46–S54.
Olha Bodnar, Alfred Link, Barbora Arendacká, Antonio Possolo, and Clemens Elster, Bayesian estimation in random effects meta-analysis using a non-informative prior, Stat. Med. 36 (2017), no. 2, 378–399. MR 3582981, DOI 10.1002/sim.7156
O. Bodnar, A. Link, and C. Elster, Objective Bayesian inference for a generalized marginal random effects model, Bayesian Anal. 11 (2016), no. 1, 25–45. MR 3447090, DOI 10.1214/14-BA933
William J. Browne and David Draper, A comparison of Bayesian and likelihood-based methods for fitting multilevel models, Bayesian Anal. 1 (2006), no. 3, 473–513. MR 2221283, DOI 10.1214/06-BA117
F. Yates, Obituary: William Gemmell Cochran, 1909–1980, J. Roy. Statist. Soc. Ser. A 145 (1982), no. 4, 521–523. MR 696709
William G. Cochran, Some methods for strengthening the common $\chi ^2$ tests, Biometrics 10 (1954), 417–451. MR 67428, DOI 10.2307/3001616
Andrew Gelman, Analysis of variance—why it is more important than ever, Ann. Statist. 33 (2005), no. 1, 1–53. With discussions and a rejoinder by the author. MR 2157795, DOI 10.1214/009053604000001048
Andrew Gelman, Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper), Bayesian Anal. 1 (2006), no. 3, 515–533. MR 2221284, DOI 10.1214/06-BA117A
Andrew Gelman, John B. Carlin, Hal S. Stern, and Donald B. Rubin, Bayesian data analysis, Texts in Statistical Science Series, Chapman & Hall, London, 1995. MR 1385925
Arjun K. Gupta, Tamas Varga, and Taras Bodnar, Elliptically contoured models in statistics and portfolio theory, 2nd ed., Springer, New York, 2013. MR 3112145
Leonhard Held and Daniel Sabanés Bové, Applied statistical inference, Springer, Heidelberg, 2014. Likelihood and Bayes. MR 3155114
Bruce M. Hill, Inference about variance components in the one-way model, J. Amer. Statist. Assoc. 60 (1965), 806–825. MR 187319
Stephen Portnoy, Formal Bayes estimation with application to a random effects model, Ann. Math. Statist. 42 (1971), 1379–1402. MR 323016, DOI 10.1214/aoms/1177693250
Poduri S. R. S. Rao, Variance components estimation, Monographs on Statistics and Applied Probability, vol. 78, Chapman & Hall, London, 1997. Mixed models, methodologies and applications. MR 1466964
Andrew L. Rukhin, Estimating heterogeneity variance in meta-analysis, J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 (2013), no. 3, 451–469. MR 3065475, DOI 10.1111/j.1467-9868.2012.01047.x
Andrew L. Rukhin, Estimation of the common mean from heterogeneous normal observations with unknown variances, J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 (2017), no. 5, 1601–1618. MR 3731678, DOI 10.1111/rssb.12227
Andrew L. Rukhin and Antonio Possolo, Laplace random effects models for interlaboratory studies, Comput. Statist. Data Anal. 55 (2011), no. 4, 1815–1827. MR 2748682, DOI 10.1016/j.csda.2010.11.016
Hardeo Sahai and Mario Miguel Ojeda, Analysis of variance for random models. Vol. II. Unbalanced data, Birkhäuser Boston, Inc., Boston, MA, 2005. Theory, methods, applications, and data analysis. MR 2152283
M. Stone and B. G. F. Springer, A paradox involving quasi prior distributions, Biometrika 52 (1965), 623–627. MR 210233, DOI 10.1093/biomet/52.3-4.623
George E. P. Box and George C. Tiao, Bayesian inference in statistical analysis, Addison-Wesley Series in Behavioral Science: Quantitative Methods, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1973. MR 0418321
George C. Tiao and W. Y. Tan, Bayesian analysis of random-effect models in the analysis of variance. I. Posterior distribution of variance-components, Biometrika 52 (1965), 37–53. MR 208765, DOI 10.1093/biomet/52.1-2.37
Blaza Toman, Bayesian approaches to calculating a reference value in key comparison experiments, Technometrics 49 (2007), no. 1, 81–87. MR 2345454, DOI 10.1198/004017006000000273
Rebecca M. Turner, Dan Jackson, Yinghui Wei, Simon G. Thompson, and Julian P. T. Higgins, Predictive distributions for between-study heterogeneity and simple methods for their application in Bayesian meta-analysis, Stat. Med. 34 (2015), no. 6, 984–998. MR 3310676, DOI 10.1002/sim.6381
David A. van Dyk and Xiao-Li Meng, The art of data augmentation, J. Comput. Graph. Statist. 10 (2001), no. 1, 1–111. With discussions, and a rejoinder by the authors. MR 1936358, DOI 10.1198/10618600152418584
Wolfgang Viechtbauer, Choosing between the fixed-, random-, and mixed-effects model in meta-analysis: An analysis of existing and new model selection methods, ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Ph.D.)–University of Illinois at Urbana-Champaign. MR 2717217
Wolfgang Viechtbauer, Confidence intervals for the amount of heterogeneity in meta-analysis, Stat. Med. 26 (2007), no. 1, 37–52. MR 2312698, DOI 10.1002/sim.2514
F. Yates, Obituary: William Gemmell Cochran, 1909–1980, J. Roy. Statist. Soc. Ser. A 145 (1982), no. 4, 521–523. MR 696709