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

Biplot methods allow for the simultaneous representation of rows and columns of a data matrix. Classical biplots were developed for continuous data in connection with Principal Components Analysis (PCA). More recently extensions for binary or nominal have been developed. The techniques have been named "Logistic biplots" (LB) because are based on logistic rather than linear responses. None of them are still adequate for ordinal data. In this paper we extend biplot methodology to ordinal data. The resulting method is termed “ordinal logistic biplot” (OLB). Row scores are computed to have ordinal logistic responses along the dimensions and column parameters produce logistic response surfaces that, projected onto the space spanned by the row scores, define a linear biplot. A proportional odds model is used. We study the geometry of such a representation and construct computational algorithms, based on an alternated gradient descent procedure, for estimating model parameters and calculating prediction directions for visualization. We understand the proposed methods as exploratory to reveal interesting patterns in data sets. The main theoretical results are applied to the study of job satisfaction of doctorate (PhD) holders in Spain.