Size invariant measures of association: Characterization and difficulties

A measure of association is row-size invariant if it is unaffected by the mutliplication of all entries in a row of a cross-classification table by a same positive number. It is class-size invariant if it is unaffected by the mutliplication of all entries in a class (i.e., a row or a column). We prove that every class-size invariant measure of association assigns to each m×n cross-classification table a number which depends only on the cross-product ratios of its 2× 2 subtables. We propose a monotonicity axiom requiring that the degree of association should increase after shifting mass from cells of a table where this mass is below its expected value to cells where it is above —provided that total mass in each class remains constant. We prove that no continuous row-size invariant measure of association is monotonic if m ≥ 4.