Clustering rows and/or columns of a two-way contingency table and a related distribution theory

The row-wise multiple comparison procedure proposed in Hirotsu [Hirotsu, C., 1977. Multiple comparisons and clustering rows in a contingency table. Quality 7, 27–33 (in Japanese); Hirotsu, C., 1983. Defining the pattern of association in two-way contingency tables. Biometrika 70, 579–589] has been verified to be useful for clustering rows and/or columns of a contingency table in several applications. Although the method improved the preceding work there was still a gap between the squared distance between the two clusters of rows and the largest root of a Wishart matrix as a reference statistic for evaluating the significance of the clustering. In this paper we extend the squared distance to a generalized squared distance among any number of rows or clusters of rows and dissolves the loss of power in the process of the clustering procedure. If there is a natural ordering in columns we define an order sensitive squared distance and then the reference distribution becomes that of the largest root of a non-orthogonal Wishart matrix, which is very difficult to handle. We therefore propose a very nice χ2χ2-approximation which improves the usual normal approximation in Anderson [Anderson, T.W., 2003. An Introduction to Multivariate Statistical Analysis. 3rd ed. Wiley Intersciences, New York] and also the first χ2χ2-approximation introduced in Hirotsu [Hirotsu, C., 1991. An approach to comparing treatments based on repeated measures. Biometrika 75, 583–594]. A two-way table reported by Guttman [Guttman, L., 1971. Measurement as structural theory. Psychometrika 36, 329–347] and analyzed by Greenacre [Greenacre, M.J., 1988. Clustering the rows and columns of a contingency table. Journal of Classification 5, 39–51] is reanalyzed and a very nice interpretation of the data has been obtained.