Size invariant measures of association: Characterization and difficulties

• We study measures of association in cross-classification tables.
• We focus on association between categorical variables.
• We characterize the set of measures invariant to rescaling of rows or columns.
• We formulate a monotonicity axiom satisfied by most popular association measures.
• No continuous row-size invariant measure is monotonic if there are at least 4 rows.

A measure of association on cross-classification tables is row-size invariant if it is unaffected by the multiplication of all entries in a row by the same positive number. It is class-size invariant if it is unaffected by the multiplication 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 cross-classification table a number which depends only on the cross-product ratios of its 2×22×2 subtables. We submit that the degree of association should increase when mass is shifted from cells containing a proportion of observations lower than what is expected under statistical independence to cells containing a proportion higher than expected–provided that total mass in each class remains unchanged. We prove that no continuous row-size invariant measure of association satisfies this monotonicity axiom if there are at least four rows.