Suppose, given a subject-by-category data matrix, a profile data in Shepard's (1972) sense, we are required to quantify subjects and categories so that those categories that are similarly responded by all subjects are given similar values and also those subjects similar in response patterns are given similar values. To answer such a requirement categorical principal component analysis (CPCA) was presented by Guttman (1941), while Hayashi's third method of quantification (H3MQ) by Hayashi (1956). A method essentially due to Kendall and Stuart (1961), Aoyama (1965), Shiba (1965a, b) and Iwatsubo (1971) among others, which might be called categorical canonical correlation analysis (CCCA) generalizes these two methods.
The purpose of this paper is to examine some basic properties of CCCA; that the maximum eigenvalue is 1, that a necessary and sufficient condition for 1 to be a simple eigenvalue is that the pattern matrix is connected, that the location constraint _??_=_??_=0 is automatically satisfied by any eigenvector associated with any eigenvalue smaller than 1, that deletion of the location constraint does not change the solution, and that the optimum solution is invariant under pooling of any proportional rows and/or columns of the pattern matrix. A discrepancy of the solutions of CPCA and H3MQ is shown by a numerical example which implies that there should be respective circumstances of appropriate applications for each of them.

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