On a New Correlation Coefficient, its Orthogonal Decomoposition, and Associated Tests of Independence

A possible drawback of the ordinary correlation coefficient p for two real random variables X and Y is that zero correlation does not imply independence. In this paper we introduce a new correlation coefficient p* which assumes values between zero and one, equalling zero iff the two variables are independent and equalling one iff the two variables are linearly related. The coefficients p* and p2 are shown to be closely related algebraically, and they coincide for distributions on a 2x2 contingency table. We derive an orthogonal decomposition of p* as a positively weighted sum of squared ordinary correlations between certain marginal eigenfunctions. Estimation of p* and its component correlations and their asymptotic distributions are discussed, and we develop visual tools for assessing the nature of a possible association in a bivariate data set. The paper includes consideration of grade (rank) versions of p* as well as the use of p* for contingency table analysis. As a special case a new generalization of the Cramer-von Mises test to K ordered samples is obtained