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Bivariate beta-binomial model using Gaussian copula for bivariate meta-analysis of two binary outcomes with low incidence

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Abstract

In meta-analysis of rare-event outcomes, an additional statistical consideration is necessary due to the occurrence of studies with no event. The traditional approaches of adding a correction factor or omitting these studies are known to result in misleading conclusions. Furthermore, studies involved in the meta-analysis often report results for more than one outcome. Bivariate meta-analysis is known as a promising approach for jointly combining two outcomes whilst incorporating correlations between outcomes. However, there has not been sufficient discussion on a bivariate extension in the context of meta-analysis for rare-event outcomes. We consider a joint synthesis of two binary outcomes with low incidence, and propose a novel bivariate meta-analysis method using copula. The method assumes marginal beta-binomial distributions for the two outcomes, and links these margins by a bivariate copula which identifies an overall dependence structure between outcomes. A simulation study suggested that the method could provide a robust estimation for the incidence of rare events and have potential benefits of bivariate meta-analysis such as an improvement of precision of pooled estimates. We illustrated the method through an application to a meta-analysis of 48 studies that evaluated a potential risk of rosiglitazone on myocardial infection and cardiovascular death.

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References

  • Barnett, V. (1980). Some bivariate uniform distributions. Communications in Statistics—Theory and Methods, 9, 453–461.

    MathSciNet  MATH  Google Scholar 

  • Bennetts, M., Whalen, E., Ahadiehc, S., & Cappelleri, J. C. (2017). An appraisal of meta-analysis guidelines: How do they relate to safety outcomes? Research Synthesis Methods, 8, 64–78.

    Google Scholar 

  • Berlin, J. A., Crowe, B. J., Whalen, E., Xia, H. A., Koro, C. E., & Kuebler, J. (2013). Meta-analysis of clinical trial safety data in a drug development program: Answer to frequently asked questions. Clinical Trials, 10, 20–31.

    Google Scholar 

  • Bohning, D., Mylona, K., & Kimber, A. (2015). Meta-analysis of clinical trials with rare events. Biometrical Journal, 4, 633–648.

    MathSciNet  MATH  Google Scholar 

  • Bradburn, M. J., Deeks, J. J., Berlin, J. A., & Localio, A. R. (2007). Much ado about nothing: A comparison of the performance of meta-analytical methods with rare events. Statistics in Medicine, 26, 53–77.

    MathSciNet  Google Scholar 

  • Cai, T., Parast, L., & Ryan, L. (2010). Meta-analysis for rare events. Statistics in Medicine, 29, 2078–2089.

    MathSciNet  Google Scholar 

  • Chen, Y., Hong, C., Ning, Y., & Su, X. (2016). Meta-analysis of studies with bivariate binary outcomes: A marginal beta-binomial model approach. Statistics in Medicine, 35, 21–40.

    MathSciNet  Google Scholar 

  • Cheng, J., Pullenayegum, E., Marshall, J. K., Iorio, A., & Thabane, L. (2015). Impact of including or excluding both-armed zero-event studies on using standard meta-analysis methods for rare event outcome: A simulation study. British Medical Journal Open, 6, e010983. https://doi.org/10.1136/bmjopen-2015-010983.

    Article  Google Scholar 

  • 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. Statistical Methods in Medical Research, 21, 621–633.

    MathSciNet  Google Scholar 

  • Clayton, D. (1978). A model for association in bivariate life tables and its application in epidemiological studies of family tendency in chronic disease incidence. Biometrika, 65, 141–151.

    MathSciNet  MATH  Google Scholar 

  • Cook, R. D., & Johnson, M. E. (1981). A family of distributions for modeling non-elliptically symmetric multivariate data. Journal of the Royal Statistical Society: Series B, 43, 210–218.

    MATH  Google Scholar 

  • Frank, M. J. (1979). On the simultaneous associativity of F(x, y) and x + y-F(x, y). Aequationes mathematicae, 21, 194–226.

    MathSciNet  MATH  Google Scholar 

  • Friedrich, J. O., Adhikari, N. K. J., & Beyene, J. (2007). Inclusion of zero total event trials in meta-analyses maintains analytic consistency and incorporates all available data. BMC Medical Research Methodology, 7, 5.

    Google Scholar 

  • Gumbel, E. J. (1960). Bivariate exponential distributions. Journal of the American Statistical Association, 55, 698–707.

    MathSciNet  MATH  Google Scholar 

  • Hammad, T. A., Laughren, T., & Racoosin, J. (2006). Suicidality in pediatric patients treated with antidepressant drugs. Archives of General Psychiatry, 63, 332–339.

    Google Scholar 

  • Higgins, J. P. T., & Green, S. (editors). (2011). Cochrane Handbook for Systematic Reviews of Interventions Version 5.1.0 [updated March 2011]. The Cochrane Collaboration.

  • Hoyer, A., & Kuss, O. (2015). Meta-analysis of diagnostic tests accounting for disease prevalence: A new model using trivariate copulas. Statistics in Medicine, 34, 1912–1924.

    MathSciNet  Google Scholar 

  • Jackson, D., Riley, R. D., & White, I. R. (2011). Multivariate meta-analysis: Potential and promise. Statistics in Medicine, 30, 2481–2498.

    MathSciNet  Google Scholar 

  • Joe, H. (2014). Dependence modeling with copulas. London: Chapman and Hall/CRC.

    MATH  Google Scholar 

  • Kirkham, J. J., Riley, R. D., & Williamson, P. R. (2012). A multivariate meta-analysis approach for reducing the impact of outcome reporting bias in systematic reviews. Statistics in Medicine, 31, 2179–2195.

    MathSciNet  Google Scholar 

  • Kuss, O. (2015). Statistical methods for meta-analyses including information from studies without any events-add nothing to nothing and succeed nevertheless. Statistics in Medicine, 34, 1097–1116.

    MathSciNet  Google Scholar 

  • Kuss, O., Hoyer, A., & Solms, A. (2014). Meta-analysis for diagnostic accuracy studies: A new statistical model using beta-binomial distributions and bivariate copulas. Statistics in Medicine, 33, 17–30.

    MathSciNet  Google Scholar 

  • Lane, P. W. (2011). Meta-analysis of incidence of rare events. Statistical Methods in Medical Research, 22, 117–132.

    MathSciNet  Google Scholar 

  • Loke, Y. K., Price, D., & Herxheimer, A. (2011). Chapter 14: adverse effects. In Higgins, J.P.T., Green, S. (eds.), Cochrane Handbook for Systematic Reviews of Interventions. Version 5.1.0. The Cochrane Collaboration. www.handbook.cochrane.org. Updated March, 2011.

  • Mavridis, D., & Salanti, G. (2013). A practical introduction to multivariate meta-analysis. Statistical Methods in Medical Research, 22, 133–158.

    MathSciNet  Google Scholar 

  • Nelsen, R. B. (2006). An introduction to copulas (2nd ed.). New York: Springer.

    MATH  Google Scholar 

  • Nikoloulopoulos, A. K. (2015a). A mixed effect model for bivariate meta-analysis of diagnostic test accuracy studies using a copula representation of the random effects distribution. Statistics in Medicine, 34, 3842–3865.

    MathSciNet  Google Scholar 

  • Nikoloulopoulos, A. K. (2015b). A vine copula mixed effect model for trivariate meta-analysis of diagnostic test accuracy studies accounting for disease prevalence. Statistical Methods in Medical Research. https://doi.org/10.1177/0962280215596769.

    Article  MathSciNet  Google Scholar 

  • Nissen, S. E., & Wolski, K. (2007). Effect of rosiglitazone on the risk of myocardial infarction and death from cardiovascular causes. New England Journal of Medicine, 356, 2457–2471.

    Google Scholar 

  • Nissen, S. E., & Wolski, K. (2010). Rosiglitazone revisited: An updated meta-analysis for myocardial infarction and cardiovascular mortality. Archives of Internal Medicine, 170, 1191–1201.

    Google Scholar 

  • Oakes, D. (1982). A model for association in bivariate survival data. Journal of the Royal Statistical Society: Series B, 44, 414–422.

    MathSciNet  MATH  Google Scholar 

  • Riley, R. D. (2009). Multivariate meta-analysis: The effect of ignoring within-study correlation. Journal of the Royal Statistical Society: Series A, 172, 789–811.

    MathSciNet  Google Scholar 

  • Riley, R. D., Abrams, K. R., Lambert, P. C., Sutton, A. J., & Thompson, J. R. (2007). An evaluation of bivariate random-effects meta-analysis for the joint synthesis of two correlated outcomes. Statistics in Medicine, 26, 78–97.

    MathSciNet  Google Scholar 

  • Riley, R. D., Thompson, J. R., & Abrams, K. R. (2008). An alternative model for bivariate random-effects meta-analysis when the within-study correlations are unknown. Biostatistics, 9, 172–186.

    MATH  Google Scholar 

  • Rucker, G., Schwarzer, G., Carpenter, J. R., & Olkin, I. (2009). Why add anything to nothing? The arcsine difference as a measure of treatment effect in meta-analysis with zero cells. Statistics in Medicine, 28, 721–738.

    MathSciNet  Google Scholar 

  • Shuster, J. J. (2010). Empirical vs natural weighting in random effects meta-analysis. Statistics in Medicine, 29, 1259–1265.

    MathSciNet  Google Scholar 

  • Shuster, J. J., Guo, J. D., & Skyler, J. S. (2012). Meta-analysis of safety for low event-rate binomial trials. Research Synthesis Methods, 3, 30–50.

    Google Scholar 

  • Shuster, J. J., Jones, L. S., & Salmon, D. A. (2007). Fixed vs random effects meta-analysis in rare event studies: The rosiglitazone link with myocardial infarction and cardiac death. Statistics in Medicine, 26, 4375–4385.

    MathSciNet  Google Scholar 

  • Shuster, J. J., & Walker, M. A. (2016). Low-event-rate meta-analyses of clinical trials: Implementing good practices. Statistics in Medicine, 35, 2467–2478.

    MathSciNet  Google Scholar 

  • Sklar, A. (1973). Random variables, joint distributions, and copulas. Kybernetica, 9, 449–460.

    MathSciNet  MATH  Google Scholar 

  • Song, P. X.-K. (2000). Multivariate dispersion models generated from Gaussian copula. Scandinavian Journal of Statistics, 27, 305–320.

    MathSciNet  MATH  Google Scholar 

  • Song, P. X.-K., Li, M., & Yuan, Y. (2009). Joint regression analysis of correlated data using Gaussian copulas. Biometrics, 65, 60–68.

    MathSciNet  MATH  Google Scholar 

  • Spittal, M. J., Pirkis, J., & Gurrin, L. C. (2015). Meta-analysis of incidence rate data in the presence of zero events. BMC Medical Research Methodology. https://doi.org/10.1186/s12874-015-0031-0.

    Article  Google Scholar 

  • Stijnen, T., Hamza, T. H., & Ozdemir, P. (2010). Random effects meta-analysis of event outcome in the framework of the generalized linear mixed model with applications in sparse data. Statistics in Medicine, 29, 3046–3067.

    MathSciNet  Google Scholar 

  • Sweeting, M., Sutton, A. J., & Lambert, P. C. (2004). What to add to nothing? Use and avoidance of continuity corrections in meta-analysis of sparse data. Statistics in Medicine, 23, 1351–1375.

    Google Scholar 

  • Tian, L., Cai, T., Pfeffer, M. A., Piankov, N., Cremieux, P. Y., & Wei, L. J. (2009). Exact and efficient inference procedure for meta-analysis and its application to the analysis of independent 2 × 2 tables with all available data but without artificial continuity correction. Biostatistics, 10, 275–281.

    MATH  Google Scholar 

  • Yusuf, S., Peto, R., Lewis, J., Collins, R., & Sleight, P. (1985). Beta blockade during and after myocardial infarction: An overview of the randomized trials. Progress in Cardiovascular Disease, 27, 335–371.

    Google Scholar 

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Authors and Affiliations

  1. Japan-Asia Data Science, Development, Astellas Pharma Inc., 2-5-1, Nihonbashi-Honcho, Chuo-ku, Tokyo, 103-8411, Japan

    Yusuke Yamaguchi

  2. Faculty of Medicine, University of Tsukuba, Tsukuba, Ibaraki, Japan

    Kazushi Maruo

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  1. Yusuke Yamaguchi
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  2. Kazushi Maruo
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Correspondence to Yusuke Yamaguchi.

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Yamaguchi, Y., Maruo, K. Bivariate beta-binomial model using Gaussian copula for bivariate meta-analysis of two binary outcomes with low incidence. Jpn J Stat Data Sci 2, 347–373 (2019). https://doi.org/10.1007/s42081-019-00037-z

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