Abstract

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 article 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 article. Supplementary files for this article are available online.

摘要

20 多年来，对应分析的变体已经出现，它们使用正交多项式来适应有序分类变量的结构。当该分析的视觉显示是双标图时，将原点链接到每个类别的标准坐标的投影是一个共同特征。例如，在列变量由有序类别组成的情况下，可以构建双标图，以便使用正交多项式确定它们的标准坐标，该正交多项式需要一组反映类别有序结构的先验分数。当前两个多项式用于构建双图时，它们会产生一个标准坐标配置，其形状似乎是抛物线。本文验证了这种抛物线关系的确切性质，并检查了这种点配置的各种特征。特别强调了这种关系的焦点、顶点、截距和准线，我们还简要检查了选择不同的先验分数对这些特征的影响。用于执行这些计算的 R 函数 parabola.exe() 作为本文的补充材料包含在内。本文的补充文件可在线获取。