Asymmetry models and model selection in square contingency tables with ordinal categories

Various types of asymmetry models have been proposed to analyze square contingency tables with ordinal categories. Here, we show that each of these models can be interpreted as a property that it is the closest to the symmetry model in terms of the Kullback-Leibler divergence under some conditions. The relationship between the likelihood ratio chi-square statistic for symmetry and that for asymmetry is discussed. Additionally, an asymmetry model family is given, and models included in it are referred to as nonhierarchical models. Because it is difficult to compare two asymmetric models, we treat this as a model selection problem. To address, we employ the penalized likelihood approach and conduct simulation studies.