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Biometrika 2006 93(1):197-206; doi:10.1093/biomet/93.1.197
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© 2006 Biometrika Trust

Range of correlation matrices for dependent Bernoulli random variables

N. Rao Chaganty1 and Harry Joe2

1 Department of Mathematics and Statistics, Old Dominion University, Norfolk, Virginia 23529-0077, U.S.A. rchagant{at}odu.edu, 2 Department of Statistics, University of British Columbia, 6356 Agricultural Road, Vancouver, British Columbia, Canada V6T1Z2 harry{at}stat.ubc.ca

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.

Key Words: Fréchet bound; Generalised estimating equation; Multivariate binary; Odds ratio

Received June 2005. Revised October 2005.


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