Orthogonal decomposition of symmetry model using sum-symmetry model for ordinal square contingency tables

This study focuses on R × R ordinal square contingency tables.Ordinal square contingency tables are always obtained by cross-classifying the matched-pair data of the two ordinal categorical variables with the same classifications.A well-known model in square contingency tables is the symmetry model.This study focuses on the relationship between the symmetry and sum-symmetry models.The sum-symmetry model has a symmetric structure between the probability that the sum of row variable X and column variable Y is t, when X < Y , for t = 3, . . ., 2R -1 and the probability that the sum of X and Y is t, when X > Y .The sum-symmetry model inevitably holds when the symmetry model holds, but the converse is not necessarily true.This study proposes a model that must be satisfied in addition to the sum-symmetry model, to satisfy the symmetry model.We also reveal that the value of the likelihood ratio chi-squared statistic of the symmetry model is equal to the sum of chi-squared statistic of the sumand sum-parameter symmetry models.We evaluate the utility of these properties by applying them to real-world vision data.