On a New Correlation Coefficient, Its Orthogonal Decomposition and Associated Tests of Independence

2A possible drawback of the ordinary correlation coefficient ρ 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 ρ∗ 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 ρ∗ and ρ2 are shown to be closely related algebraically, and they coincide for distributions on a 2× 2 contingency table. We derive an orthogonal decomposition of ρ∗ as a positively weighted sum of squared ordinary correlations between certain marginal eigenfunctions. Estimation of ρ∗ 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 ρ∗ as well as the use of ρ∗ for contingency table analysis. As a special case a new generalization of the Cramér-von Mises test to K ordered samples is obtained.