Measure of departure from symmetry of cumulative marginal probabilities for square contingency tables with ordered categories

Abstract

For the analysis of square contingency tables with ordered categories, Tomizawa, Miyamoto and Ashihara (2003) considered the measure which represents the degree of departure from the marginal homogeneity (MH) model and does not depend on the diagonal probabilities in the table. This paper proposes another measure which represents the degree of departure from MH and depends on the diagonal probabilities. The measure proposed is expressed by using the Cressie-Read power-divergence or Patil-Taillie diversity index, which is applied for the cumulative marginal probabilities that an observation will fall in row (column) category $i$ or below [or in row (column) category $i+1$ or above]. The measure is useful for seeing how far the cumulative marginal probabilities are distant from those with a MH structure, and for comparing the degree of departure from MH in several tables. Examples are given.