Relative exponential sum-symmetry model and orthogonal decomposition of the sum-symmetry model for ordinal square contingency tables

Summary This study proposes a new exponential sum-symmetry model for square contingency tables with same row and column ordinal classifications. In the existing exponential sum-symmetry (ESS) model, the probability that the sum of row and column levels is t, where the row level is less than the column level, is ∆t−2 times higher than the probability that the sum of row and column levels is t, where the row level is greater than the column level. On the other hand, in the proposed ESS model, the ratio of these two probabilities is ∆t/3. In other words, in the existing ESS model, the ratio of the two probabilities varies exponentially depending on the absolute gap between t and 2, while in the proposed ESS model, the ratio of the two probabilities varies exponentially depending on the relative gap between t and 3, although in both ESS models, the ratio of the two probabilities is ∆ when t is the minimum value (i.e., t = 3). Moreover, this study introduces a new decomposition theorem for the sum-symmetry model using the proposed ESS. The proposed decomposition theorem satisfies asymptotic equivalence for the test statistic.