Asymmetry Plus Association Models for Ordinal Square Contingency Tables

In contingency table analysis, statistical models have been used to study the linkage between variables. The independent model is used to analyze whether or not there is an association between variables. In particular, a contingency table in which the rows and columns consist of the same classification is called a square contingency table, and in square contingency table analysis, the variables are strongly related to each other and independence is not established. In this case, alternative statistical models are considered instead of the independence model. For instance, symmetric or asymmetric models have been proposed, which show symmetric or asymmetric structure for cell probabilities. A number of association models have also been considered to analyze complex data. In this paper, we propose models that include both asymmetry and association structures. We provide some theorems on the necessary and sufficient conditions of a certain model based on the proposed model. Additionally, we prove that the likelihood ratio statistic for our models are separable into two statistics. These theorems help interpret results.