Separation of the refined estimator of the measure for symmetry in square contingency tables

Measures that represent the degree of departure from symmetry, global symmetry, and conditional symmetry have been proposed to analyze square contingency tables. Additionally, a refined estimator of the symmetry measure has been suggested. This study provides refined estimators of the measures for global symmetry and conditional symmetry, and shows the relationship between the proposed estimators and the refined estimator. The proposed estimators can be obtained by the second-order term in the Taylor series expansion. Simulations studies demonstrate that as the sample size increases, the proposed estimators approach the true values faster than the former estimators.