Range of correlation matrices for dependent Bernoulli random variables

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Range of correlation matrices for dependent Bernoulli random variables

TitleRange of correlation matrices for dependent Bernoulli random variables
Publication TypeJournal Article
Year of Publication2006
AuthorsChaganty, NR, Joe, H
JournalBiometrika
Volume93
Pagination197-206
Date PublishedMAR
ISSN0006-3444
AbstractWe 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.
DOI10.1093/biomet/93.1.197