On the construction of biplots for the vis?ualisation of ordered categorical variables

Eric Beh and Rosaria Lombardo 

For more than 20 years, variants of correspondence analysis have been developed that accommodate for the structure of ordinal categorical variables using orthogonal polynomials. When the visual display from this analysis is the biplot, projections linking the origin to the standard coordinate of each category is a common feature. When a column variable, say, consists of ordered categories, the biplot can be constructed so that their standard coordinate is determined using orthogonal polynomials which require a set of a priori scores that reflect the ordered structure of the categories. When the first two polynomials are used to construct the biplot they produce a configuration of standard coordinates that appear to be parabolic in shape. This project explores the exact nature of this parabolic relationship and examines the various features of this configuration of points. In particular, simple formulae can be derived to determine the focus, vertex, intercepts and directrix of this relationship. Since the use of orthogonal polynomials requires choosing a priori scores to reflect the ordinal nature of the categories of a variable, this project also explores the impact of different scores on these features. Ongoing research in this area means that this project includes examining the relationship between the first-order and higher-order polynomials and the impact such a relationship has on the interpretation of the biplot.??