Evaluating Correlation with Proper Bounds

Abstract

In the medical literature we often find the use of the Pearson correlation coefficient to describe the association between two variables. However, unless the marginal distributions of the two random variables can be different only in location or scale parameters, the range of the correlation is smaller than the usual reference interval [1, 1]. To assist proper interpretation of the strength of an empirical correlation, we devise a computationally convenient method to obtain the maximum and minimum correlation coefficients for random variables with prescribed marginals. The result is an application of method by Cambanis, Simons, and Stout (1976, Zeitschrift fur Wahrscheinlichkeitstheorie 36, 285-294) and is a generalization of that of Gradstein (1986, Journal qf Educational Statistics 11, 259-261), who has investigated the maximum correlation between normal and dichotomous random variables. We provide numerical results for various pairs of distributions and also illustrate the method by one example from Young et al. (1986, Diabetes 35, 192-197) for different syndromes of diabetic complications.

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