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. Unpublished.

Abstract

Nominal data currently lack a correlation coefficient, such as has already defined for real data. A measure is possible using the determinant, with the useful interpretation that the determinant gives the ratio between volumes. With M a m × n contingency table and n ≤ m the suggested measure is r = Sqrt[det[A'A]] with A = Normalized[M]. 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. The volume ratio measure can be related to the regression coefficients, not of the variables, but of the categories in the contingency matrix, using the conditional probabilities given the row and column sums.

• Item Type: MPRA Paper
• Additional Information: The March 27 version corrects a formula and introduces the term NominalCorrelation. The April 10 version gives the correct f[R] = Sqrt[1 - det[R]] and explains this measure also for real data. But advisable is the maximal multiple correlation. The May 1 version resolves the issue of positive semi-definiteness and extends the approach to Cramer's V. The May 14 version adds regression. The June 5 version corrects the variance and links up the volume ratio interpretation with the conditional probabilities within the contingency table.
• Institution: Thomas Cool Consultancy & Econometrics
• Language: English
• Keywords: association; correlation; contingency table; volume ratio; determinant; nonparametric methods; nominal data; nominal scale; categorical data; Fisher’s exact test; odds ratio; tetrachoric correlation coefficient; phi; Cramer’s V; Pearson; contingency coefficient; uncertainty coefficient; Theil’s U; eta; meta-analysis; Simpson’s paradox; causality; statistical independence; regression
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods: General > C10 - General

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