An extension of Yule's Q to multivariate binary data

In this note we describe a summary measure of pairwise association for multivariate binary data based on the conditional odds ratio. The proposed measure is an extension of Yule's Q to more than two binary random variables. Unlike marginal measures of association, this measure is not constrained by the marginal probabilities of success. For example, when each binary variable has a different probability of success, the upper limit of the pairwise marginal correlation coefficient is constrained to be less than 1. If one prefers a measure of association that is unconstrained, then with only two binary variables, Bishop, Feinberg, and Holland (1975, Discrete Multivariate Analysis: Theory and Practice, Cambridge, Massachusetts: MIT Press) suggest the use of the odds ratio or, equivalently, Yule's Q. Yule's Q transforms the odds ratio between the two binary variables from [0, infinity) to [-1, 1]. We propose an extension of Yule's Q to more than two binary random variables. This measure of pairwise association is based on the conditional odds ratio from a log-linear model.