A contrast-ratio representation of directional departure from marginal mean equality in ordinal square contingency tables

Summary For ordinal square contingency tables with equally spaced category scores, we give an explicit contrast-ratio representation of the angular measure of departure from marginal mean equality introduced by Ando (2024). The bounded contrast D and the angular measure Ψ satisfy Ψ = (4 /π ) arctan( D ) and therefore contain the same directional information. Ratio and half-log-ratio identities clarify the interpretation of the aggregated cumulative marginal quantities. A ratio-of-means formulation gives a compact asymptotic variance and two simple confidence intervals: a naive Wald interval and an atanh-transformed Wald interval. The transformed interval respects the parameter range whenever defined, but is unavailable at boundary estimates. Simulations examine departures at a fixed denominator and null scenarios with smaller denominators; performance for non-null departures combined with a small denominator is not established. Paired insomnia data illustrate reporting on equivalent scales and the coding-dependent interpretation of between-group comparisons. The contribution is improved representation, interpretation, and formulation of large-sample inference, rather than additional information or a general finite-sample guarantee.