Multiple Correspondence Analysis & the Multilogit Bilinear Model

TL;DR

This paper reveals that Multiple Correspondence Analysis (MCA) can be viewed as a single step of a bilinear exponential family model for categorical data, providing new insights and practical approximation methods.

Contribution

It establishes a theoretical link between MCA and the multinomial logit bilinear model, suggesting MCA as an efficient approximation for complex model parameter estimation.

Findings

MCA corresponds to a single proximal Newton step of the bilinear model.

MCA can be used to approximate parameters of the multinomial logit bilinear model.

MCA is computationally scalable and easier to implement than direct model estimation.

Abstract

Multiple Correspondence Analysis (MCA) is a dimension reduction method which plays a large role in the analysis of tables with categorical nominal variables such as survey data. Though it is usually motivated and derived using geometric considerations, in fact we prove that it amounts to a single proximal Newtown step of a natural bilinear exponential family model for categorical data the multinomial logit bilinear model. We compare and contrast the behavior of MCA with that of the model on simulations and discuss new insights on the properties of both exploratory multivariate methods and their cognate models. One main conclusion is that we could recommend to approximate the multilogit model parameters using MCA. Indeed, estimating the parameters of the model is not a trivial task whereas MCA has the great advantage of being easily solved by singular value decomposition and scalable to large data.