The correspondence analysis of ordered categorical variables
Eric Beh and Rosaria Lombardo
For the past 25 years or so we have published extensively on examining the role of orthogonal polynomials on a range of issues concerned with correspondence analysis. These issues include the construction and interpretation of low-dimensional visual depictions of the association, as well as the partition of popular measures of association, and correlation and association models. This is because orthogonal polynomials provide an excellent, simple and a flexible means of incorporating the structure of ordinal categorical variables – all they require is an a priori chosen set of initial scores to reflect the ordinal structure of a variable and a three-term recurrence formula to generate the polynomials. Orthogonal polynomials also enable one to determine “generalised correlations" which include as special cases the traditional linear-by-linear correlation coefficient (that everyone should be familiar with) and sources of non-linear association that may exist between the ordinal variables. Alternative approaches involving scaling categories such that the resulting scores (obtained from reciprocal averaging or by other means) are “forced" to be ordered. Unfortunately, such approaches only considered ordered scores across a single dimension and the resulting visual representation of the association may not properly reflect the nature of the association.
This ongoing project examines the impact of orthogonal polynomials on the structure of the association between two or more categorical variables. Methods of three-way and higher-way decomposition using orthogonal polynomials are very much linked to the Tucker3 decomposition and, more generally, to the suite of decomposition methods that are now part of higher-order singular value decomposition (HOSVD). This project also examines the impact on the interpretation of visual summaries of the association obtained by performing correspondence analysis, where the traditional correspondence plot or biplot may be constructed.