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Range of correlation matrices for dependent Bernoulli random variables

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  • N. Rao Chaganty
  • Harry Joe

Abstract

We say that a pair (p, R) is compatible if there exists a multivariate binary distribution with mean vector p and correlation matrix R. In this paper we study necessary and sufficient conditions for compatibility for structured and unstructured correlation matrices. We give examples of correlation matrices that are incompatible with any p. Using our results we show that the parametric binary models of Emrich & Piedmonte (1991) and Qaqish (2003) allow a good range of correlations between the binary variables. We also obtain necessary and sufficient conditions for a matrix of odds ratios to be compatible with a given p. Our findings support the popular belief that the odds ratios are less constrained and more flexible than the correlations. Copyright 2006, Oxford University Press.

Suggested Citation

  • N. Rao Chaganty & Harry Joe, 2006. "Range of correlation matrices for dependent Bernoulli random variables," Biometrika, Biometrika Trust, vol. 93(1), pages 197-206, March.
    • Handle: RePEc:oup:biomet:v:93:y:2006:i:1:p:197-206
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    File URL: http://hdl.handle.net/10.1093/biomet/93.1.197
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    Cited by:

    1. Alessandro Barbiero & Asmerilda Hitaj, 2020. "Goodman and Kruskal’s Gamma Coefficient for Ordinalized Bivariate Normal Distributions," Psychometrika, Springer;The Psychometric Society, vol. 85(4), pages 905-925, December.
    2. Roy T. Sabo & N. Rao Chaganty, 2011. "Letter to the Editor of Biometrics on “Joint Regression Analysis for Discrete Longitudinal Data” by Madsen and Fang," Biometrics, The International Biometric Society, vol. 67(4), pages 1669-1670, December.
    3. Oman, Samuel D., 2009. "Easily simulated multivariate binary distributions with given positive and negative correlations," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 999-1005, February.
    4. Shults, Justine, 2017. "Simulating longer vectors of correlated binary random variables via multinomial sampling," Computational Statistics & Data Analysis, Elsevier, vol. 114(C), pages 1-11.
    5. Bruce J. Swihart & Brian S. Caffo & Ciprian M. Crainiceanu, 2014. "A Unifying Framework for Marginalised Random-Intercept Models of Correlated Binary Outcomes," International Statistical Review, International Statistical Institute, vol. 82(2), pages 275-295, August.
    6. Fangya Mao & Richard J. Cook, 2023. "Spatial dependence modeling of latent susceptibility and time to joint damage in psoriatic arthritis," Biometrics, The International Biometric Society, vol. 79(3), pages 2605-2618, September.
    7. Sergei Leonov & Bahjat Qaqish, 2020. "Correlated endpoints: simulation, modeling, and extreme correlations," Statistical Papers, Springer, vol. 61(2), pages 741-766, April.
    8. Huihui Lin & N. Rao Chaganty, 2021. "Multivariate distributions of correlated binary variables generated by pair-copulas," Journal of Statistical Distributions and Applications, Springer, vol. 8(1), pages 1-14, December.
    9. Wang, Bin & Wang, Ruodu & Wang, Yuming, 2019. "Compatible matrices of Spearman’s rank correlation," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 67-72.
    10. Deng, Yihao & Sabo, Roy T. & Chaganty, N. Rao, 2012. "Multivariate probit analysis of binary familial data using stochastic representations," Computational Statistics & Data Analysis, Elsevier, vol. 56(3), pages 656-663.
    11. Fontana, Roberto & Semeraro, Patrizia, 2018. "Representation of multivariate Bernoulli distributions with a given set of specified moments," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 290-303.
    12. Nooraee, Nazanin & Molenberghs, Geert & van den Heuvel, Edwin R., 2014. "GEE for longitudinal ordinal data: Comparing R-geepack, R-multgee, R-repolr, SAS-GENMOD, SPSS-GENLIN," Computational Statistics & Data Analysis, Elsevier, vol. 77(C), pages 70-83.
    13. Mark Huber & Nevena Marić, 2019. "Admissible Bernoulli correlations," Journal of Statistical Distributions and Applications, Springer, vol. 6(1), pages 1-8, December.
    14. Alessandro Barbiero, 2021. "Inducing a desired value of correlation between two point-scale variables: a two-step procedure using copulas," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 105(2), pages 307-334, June.
    15. Oliver Williams & Stephen Satchell, 2011. "Social welfare issues of financial literacy and their implications for regulation," Journal of Regulatory Economics, Springer, vol. 40(1), pages 1-40, August.
    16. Berman, Oded & Krass, Dmitry & Menezes, Mozart B.C., 2013. "Location and reliability problems on a line: Impact of objectives and correlated failures on optimal location patterns," Omega, Elsevier, vol. 41(4), pages 766-779.
    17. Modarres, Reza, 2011. "High-dimensional generation of Bernoulli random vectors," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1136-1142, August.
    18. Darolles, Serge & Fol, Gaëlle Le & Lu, Yang & Sun, Ran, 2019. "Bivariate integer-autoregressive process with an application to mutual fund flows," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 181-203.
    19. Anestis Touloumis & Alan Agresti & Maria Kateri, 2013. "GEE for Multinomial Responses Using a Local Odds Ratios Parameterization," Biometrics, The International Biometric Society, vol. 69(3), pages 633-640, September.
    20. Krause, Daniel & Scherer, Matthias & Schwinn, Jonas & Werner, Ralf, 2018. "Membership testing for Bernoulli and tail-dependence matrices," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 240-260.

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