Some results on the maximal correlation in 2 x k contingency tables

For 2 x k contingency tables, we consider the statistic r*, the maximal correlation between the row and column variables, where the maximum is taken over all possible sets of scores (or "scales" or "weights") assigned to the k categories. For general m x k contingency tables, methods involving the maximization over sets of scores assigned to the categories (called dual-scaling methods) have been criticized for lack of statistical interpretation and for difficulty of computation. For the case m = 2, however, where nominal categorical data on two populations are compared, this article shows that r* has meaningful interpretations as a multiple correlation coefficient, as a numerical measure of association, and as an upper bound on correlation for reduced tables. These interpretations lead to a better understanding of the nature of the association between the two variables. These interpretations also yield insight into the role of the usual chi-square statistic for 2 x k tables. Furthermore, both r" and the set of scores at which this maximum is achieved are shown to have simple closed-form expressions. These scores are used to furnish a simple proof that the asymptotic distribution of nr*(2), based on a sample of size n, is a chi(2) distribution with k - 1 degrees of freedom.