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Correlation and regression in contingency tables. A measure of association or correlation in nominal data (contingency tables), using determinants

Colignatus, Thomas (2007): Correlation and regression in contingency tables. A measure of association or correlation in nominal data (contingency tables), using determinants.

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Nominal data in contingency tables currently lack a correlation coefficient, such as has already been defined for real data. A measure can be designed using the determinant, with the useful interpretation that the determinant gives the ratio between volumes. A contingency table by itself gives all connections between the variables. Required operations are only normalization and aggregation by means of that determinant, so that, in fact, a contingency table is its own correlation matrix. The idea for the normalization is that the conditional probabilities given the row and column sums can also be seen as regression coefficients that hence depend upon correlations. With M a m × n contingency table and n ≤ m the suggested measure is r = Sqrt[det[A'A]] with A = Normalized[M]. The sign can be recovered from a generalization of the determinant to non-square matrices. With M an n1 × n2 × ... × nk contingency matrix, we can construct a matrix of pairwise correlations R. A matrix of such pairwise correlations is called an association matrix. If that matrix is also positive semi-definite (PSD) then it is a proper correlation matrix. The overall correlation then is R = f[R] where f can be chosen to impose PSD-ness. An option is to use f[R] = Sqrt[1 - det[R]]. However, for both nominal and cardinal data the advisable choice is to take the maximal multiple correlation within R. The resulting measure of “nominal correlation” measures the distance between a main diagonal and the off-diagonal elements, and thus is a measure of strong correlation. Cramer’s V measure for pairwise correlation can be generalized in this manner too. It measures the distance between all diagonals (including cross-diagaonals and subdiagonals) and statistical independence, and thus is a measure of weaker correlation. Finally, when also variances are defined then regression coefficients can be determined from the variance-covariance matrix.

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