Asymptotic distribution of inequality-restricted canonical correlation with application to tests for independence in ordered contingency tables

For two-way ordered categorical data, correspondence analysis and the RC association model (the row–column-effect association model) with order-restricted scores have been proposed mainly for descriptive purposes. In this paper, tests for independence in two-way ordered contingency tables based on these models are developed in a general framework of inequality-restricted canonical correlation analysis. The limiting null distributions are characterized as the maxima of Gaussian random fields and asymptotic expansions of their tail probabilities are derived by the tube method, an integral geometric approach. Some numerical techniques for fitting order-restricted models are discussed. An example of data analysis is given to demonstrate the practical usefulness of the proposed method.