The Relation Between Measures of Correlation in the Universe of Sample Permutations

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Abstract
Consider the 2 sets [chi]1, [chi]2, . . . [chi]n and y1, y2, . . . yn, both arranged in some given order relative to each other. They may be permuted to give n different ways of grouping the [chi]''s with the y''s. To each pair ([chi]i, [chi]j) assign a convenient score aij and to each pair (yi, yj) a score bij, where aij = aji and bij = bji. Let [image] where i and j run from 1 to n. The author shows that [GAMMA], a general correlation coeff., for special cases is equal to Kendall''s [tau] correlation coeff., the linear correlation coeff. ryx and the Spearman''s Rank correlation coefficient [rho]. He finds the relationship between [tau] and [rho] and the association between the variates before and after transformations. For large values of n, the number in the sample, the distribution of [GAMMA] approaches the normal curve. It is interesting that a general correlation coeff. has been defined.

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