Range of correlation matrices for dependent Bernoulli random variables

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.