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

<jats:title>Summary</jats:title> <jats:p>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 <jats:italic>t</jats:italic>, where the row level is less than the column level, is ∆<jats:italic> <jats:sup>t−</jats:sup> </jats:italic> <jats:sup>2</jats:sup> times higher than the probability that the sum of row and column levels is <jats:italic>t</jats:italic>, 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 ∆<jats:italic> <jats:sup>t/</jats:sup> </jats:italic> <jats:sup>3</jats:sup>. In other words, in the existing ESS model, the ratio of the two probabilities varies exponentially depending on the absolute gap between <jats:italic>t</jats:italic> and 2, while in the proposed ESS model, the ratio of the two probabilities varies exponentially depending on the relative gap between <jats:italic>t</jats:italic> and 3, although in both ESS models, the ratio of the two probabilities is ∆ when <jats:italic>t</jats:italic> is the minimum value (i.e., <jats:italic>t</jats:italic> = 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.</jats:p>