Correction for chance and correction for maximum value

Abstract

In data analysis and classification association coefficients are used to quantify the association in contingency tables. Various coefficients are chance-corrected, that is, they have value zero under statistical independence. Examples are the phi coefficient, Cohen's kappa, and the adjusted Rand index. Other coefficients are corrected for maximum value. Correction for chance and correction for maximum value are studied as functions on a space of association coefficients for contingency tables. Both functions are idempotent, and the two functions commute under composition. Furthermore, the composed function maps a coefficient and all its linear transformations given the marginal totals to the same coefficient. The algebraic structure of the two functions, the their composition, and the identity function, turns out to be an idempotent commutative monoid.