Using Copulas for Bayesian Meta-analysis


Specific bivariate classes of distributions with given marginals can be used for contribution of the linking distribution between conditional and unconditional effectiveness using copulas. In this paper, a Bayesian model is proposed for meta-analysis of treatment effectiveness data which are generally discrete Binomial and sparse. A bivariate class of priors is imposed to accommodate a wide range of heterogeneity between the multicenter clinical trials involved in the study. Applications to real data are provided.

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  1. 1.

    Sweeting MJ, Sutton JA, Lambert PC (2004) What to add to nothing? Use and avoidance of continuity corrections in meta-analysis of sparse data. Stat Med 23(9):1351–1375

    Article  Google Scholar 

  2. 2.

    Lambert PC, Sutton AJ, Burton PR, Abrams KR, Jones DR (2005) How vague is vague? A simulation study of the impact of the use of vague prior distributions in mcmc using winbugs. Stat Med 24(15):2401–2428

    MathSciNet  Article  Google Scholar 

  3. 3.

    Joe H (1997) Multivariate models and multivariate dependence concepts. CRC Press, Boca Raton

    Book  Google Scholar 

  4. 4.

    Casella G, Moreno E (2005) Intrinsic meta-analysis of contingency tables. Stat Med 24(4):583–604

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chu H, Nie L, Chen Y, Huang Y, Sun W (2012) Bivariate random effects models for meta-analysis of comparative studies with binary outcomes: methods for the absolute risk difference and relative risk. Stat Methods Med Res 21(6):621–633

    MathSciNet  Article  Google Scholar 

  6. 6.

    Chen Y, Chu H, Luo S, Nie L, Chen S (2015) Bayesian analysis on meta-analysis of case-control studies accounting for within-study correlation. Stat Methods Med Res 24(6):836–855

    MathSciNet  Article  Google Scholar 

  7. 7.

    Fienberg S (1992) Comment to hierarchical models for combining information and for meta-analyses. Bayesian Stat 4(70):321–344

    Google Scholar 

  8. 8.

    Fisher LD, Van Belle G (1993) Biostatistics. A Methodology for the Health Sciences. Wiley, New York

  9. 9.

    Collins R, Peto R, Flather M, Parish S, Sleight P, Conway M, Pipilis A, Baigent C, Barnett D, Boissel J et al (1995). Isis-4-a randomised factorial assessing early oral captopril, oral mononitrate, and intravenous magnesium sulphate in 58.050 patient with suspected acute myocardial-infarction. Lancet 345(8951):669–685

  10. 10.

    Egger M, Davey Smith G (1995) Misleading meta-analysis. lessons from “an effective, safe, simple” intervention that wasn’t. BMJ 310:752–754

    Article  Google Scholar 

  11. 11.

    Egger M, Smith GD, Schneider M, Minder C (1997) Bias in meta-analysis detected by a simple, graphical test. BMJ 315(7109):629–634

    Article  Google Scholar 

  12. 12.

    Fisher N (1997) Copulas. In: Kotz S, Read CB, Banks DL (eds) Encyclopedia of statistical sciences, vol 1, pp 159–163

  13. 13.

    Nelsen R, Quesada-Molina J, Rodríguez-Lallena J (1997) Bivariate copulas with cubic sections. J Nonparametric Stat 7(3):205–220

    MathSciNet  Article  Google Scholar 

  14. 14.

    Quesada-Molina J, Rodríguez-Lallena J (1995) Bivariate Copulas with quadratic sections. J Nonparametric Stat 5(4):323–337

    MathSciNet  Article  Google Scholar 

  15. 15.

    Nelsen RB (2007) An introduction to copulas. Springer, New York

    MATH  Google Scholar 

  16. 16.

    Sklar M (1959) Fonctions de repartition an dimensions et leurs marges. Publ Inst Stat Univ Paris 8:229–231

    MATH  Google Scholar 

  17. 17.

    Moreno E, Vázquez-Polo F, Negrín M (2014) Objective Bayesian meta-analysis for sparse discrete data. Stat Med 33(21):3676–3692

    MathSciNet  Article  Google Scholar 

  18. 18.

    Lapuyade-Lahorgue J, Xue JH, Ruan S (2017) Segmenting multi-source images using hidden Markov fields with Copula-based multivariate statistical distributions. IEEE Trans Image Process 26(7):3187–3195

    MathSciNet  Article  Google Scholar 

  19. 19.

    Zhang A, Fang J, Calhoun VD, Wang Yp (2018) High dimensional latent gaussian copula model for mixed data in imaging genetics. In: 2018 IEEE 15th International Symposium on Biomedical Imaging

  20. 20.

    Bahrami M, Hossein-Zadeh GA (2015) Assortativity changes in alzheimer’s diesease: A resting-state fmri study. In: 2015 23rd Iranian Conference on Electrical Engineering. IEEE

  21. 21.

    Eban E, Rothschild G, Mizrahi A, Nelken I, Elidan G (2013) Dynamic copula networks for modeling real-valued time series. In: Artificial Intelligence and Statistics, PMLR

  22. 22.

    Onken A, Grünewälder S, Munk MH, Obermayer K (2009) Analyzing short-term noise dependencies of spike-counts in macaque prefrontal cortex using copulas and the flashlight transformation. PLoS Comput Biol 5(11):e1000577

    MathSciNet  Article  Google Scholar 

  23. 23.

    Bao L, Zhu Z, Ye J (2009) Modeling oncology gene pathways network with multiple genotypes and phenotypes via a copula method. In: 2009 IEEE Symposium on Computational Intelligence in Bioinformatics and Computational Biology. IEEE

  24. 24.

    Kwitt R, Uhl A, Häfner M, Gangl A, Wrba F, Vécsei A (2010) Predicting the histology of colorectal lesions in a probabilistic framework. In: 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition-Workshops. IEEE

  25. 25.

    Kon MA, Nikolaev N (2011) Empirical normalization for quadratic discriminant analysis and classifying cancer subtypes. In: 2011 10th International Conference on Machine Learning and Applications and Workshops. IEEE

  26. 26.

    Iyengar SG, Varshney PK, Damarla T, Bhanu B, Govindaraju V (2011) Biometric authentication: a copula-based approach. Cambridge University Press, Cambridge

    Google Scholar 

  27. 27.

    Susyanto N, Veldhuis R, Spreeuwers L, Klaassen C (2019) Semiparametric likelihood-ratio-based biometric score-level fusion via parametric copula. IET Biometrics 8(4):277–283

    Article  Google Scholar 

  28. 28.

    Tuyl F, Gerlach R, Mengersen K (2008) A comparison of Bayes-Laplace, Jeffreys, and other priors: the case of zero events. Am Stat 62(1):40–44

    MathSciNet  Article  Google Scholar 

  29. 29.

    Bairamov I, Kotz S (2002) Dependence structure and symmetry of Huang-Kotz FGM distributions and their extensions. Metrika 56(1):55–72

    MathSciNet  Article  Google Scholar 

  30. 30.

    Lin G, Huang J (2011) Maximum correlation for the generalized Sarmanov bivariate distributions. J Stat Plan Inference 141(8):2738–2749

    MathSciNet  Article  Google Scholar 

  31. 31.

    Bairamov I, Kotz S, Bekci M (2001) New generalized Farlie-Gumbel-Morgenstern distributions and concomitants of order statistics. J Appl Stat 28(5):521–536

    MathSciNet  Article  Google Scholar 

  32. 32.

    Bairamov I, Altinsoy B, Kerns GJ (2011) On generalized Sarmanov bivariate distributions. TWMS J Appl Eng Math 1(1):86

    MathSciNet  MATH  Google Scholar 

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The first author is grateful to Department of Science and Technology, Govt. of India for providing financial assistance for carrying out this work. All the authors also acknowledge the support provided by DST under PURSE grants.

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Correspondence to Savita Jain.

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Jain, S., Sharma, S.K. & Jain, K. Using Copulas for Bayesian Meta-analysis. Stat Biosci (2021).

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  • Meta-Analysis
  • Copula
  • Frechet Class
  • Uniform Prior
  • Jeffrey’s Prior
  • Linking Distribution