Logistic Biplots for Ordinal Variables Based on Alternating Gradient Descent on the Cumulative Probabilities, with an Application to Survey Data

Abstract

[EN]Biplot methods provide a framework for the simultaneous graphical representation of both rows and columns of a data matrix. Classical biplots were originally developed for continuous data in conjunction with principal component analysis (PCA). In recent years, several extensions have been proposed for binary and nominal data. These variants, referred to as logistic biplots (LBs), are based on logistic rather than linear response models. However, existing formulations remain insufficient for analyzing ordinal data, which are common in many social and behavioral research contexts. In this study, we extend the biplot methodology to ordinal data and introduce the ordinal logistic biplot (OLB). The proposed method estimates row scores that generate ordinal logistic responses along latent dimensions, whereas column parameters define logistic response surfaces. When these surfaces are projected onto the space defined by the row scores, they form a linear biplot representation. The model is based on a framework, leading to a multidimensional structure analogous to the graded response model used in Item Response Theory (IRT). We further examine the geometric properties of this representation and develop computational algorithms—based on an alternating gradient descent procedure—for parameter estimation and computation of prediction directions to facilitate visualization. The OLB method can be viewed as an extension of multidimensional IRT models, incorporating a graphical representation that enhances interpretability and exploratory power. Its primary goal is to reveal meaningful patterns and relationships within ordinal datasets. To illustrate its usefulness, we apply the methodology to the analysis of job satisfaction among PhD holders in Spain. The results reveal two dominant latent dimensions: one associated with intellectual satisfaction and another related to job-related aspects such as salary and benefits. Comparative analyses with alternative techniques indicate that the proposed approach achieves superior discriminatory power across variables.