Logistic biplot for nominal data

TL;DR

This paper introduces Nominal Logistic Biplot, a novel graphical method for visualizing nominal data by representing variables as convex regions and individuals by their closest category points, extending existing biplot techniques.

Contribution

It extends logistic biplots to nominal data, using computational geometry to create a new interpretative framework with convex prediction regions.

Findings

Provides a geometric interpretation of nominal data visualization.

Develops algorithms for parameter estimation and prediction regions.

Extends MCA and LTA with a new graphical representation.

Abstract

Classical Biplot Methods allow for the simultaneous representation of individuals (rows) and variables (columns) of a data matrix. For Binary data, Logistic biplots have been recently developed.When data are nominal, linear or even binary logistic biplots are not adequate and techniques as Multiple Correspondence Analysis (MCA), Latent Trait Analysis (LTA) or Item Response Theory for nominal items should be used instead. In this paper we extend the binary logistic biplot to nominal data. The resulting method is termed Nominal Logistic Biplot, although the variables are represented as convex prediction regions rather than vectors. Using the methods from Computational Geometry, the set of prediction regions is converted to a set of points in such a way that the prediction for each individual is established by its closest category point. Then interpretation is based on distances rather than on projections. We study the geometry of such a representation and construct computational algorithms for the estimation of parameters and the calculation of prediction regions. Nominal Logistic Biplots extend both MCA and LTA in the sense that gives a graphical representation for LTA similar to the one obtained in MCA.