Features of the polynomial biplots for ordered contingency tables

For more than 20 years variants of correspondence analysis have arisen that accommodate for the structure of ordered 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. In the case 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 paper verifies the exact nature of this parabolic relationship and examines the various features of this configuration of points. Particular emphasis is given to the focus, vertex, intercepts and directrix of this relationship and we also briefly examine the impact of choosing different a priori scores on these features. The R function, parabola.exe(), used to perform these calculates is included as supplementary material to this paper.