Correspondence Analysis in Psychology

by

Michael Greenacre

• LAST REVIEWED: 26 May 2023
• LAST MODIFIED: 26 May 2023
• DOI: 10.1093/obo/9780199828340-0311

Introduction

Correspondence analysis (CA) is a statistical method of multivariate analysis, which applies to a rectangular table of categorical data, with a wide range of applications in the social sciences, as well as in ecology, archaeology, and linguistics. This bibliography focuses primarily on its development and applications in the field of psychology, where it is often difficult to grasp the interrelationships between observed variables that are generally on categorical scales. The method has several historical origins and equivalent definitions. One of the earliest, due to the eminent British statistician Ronald A. Fisher, defines the method as a way of quantifying the categories of two categorical variables, that is, assigning scale values to the categories, with the objective of maximizing their discriminatory power, equivalent to maximizing their correlation. This idea was generalized to quantifying more than two categorical variables by the psychologist Louis Guttman. The same idea formed the basis of two research schools, led by Chikio Hayashi in Japan, and Jan de Leeuw in the Netherlands. The French linguist and mathematician Jean-Paul Benzécri realized the geometric interpretation of CA and developed the method as a tool for visualizing categorical data, which is its most popular application today. Simple CA visualizes the rows and columns of a two-way table as points in a spatial map, where the association between the row and column categories can be directly interpreted. When the table is a cross-tabulation called a contingency table, CA thus goes beyond the typical measurement and test of row–column association (e.g., the chi-square test) by explicitly showing what the main features of that association are. The generalization of the method, called multiple correspondence analysis (MCA), analyzes more than two categorical variables simultaneously and is routinely used to understand patterns of response in questionnaire surveys that involve many questions with categorical responses. All forms of CA are variants of principal component analysis (PCA), with two important generalizations of the regular way PCA is defined and used: (i) the distance function used to measure differences between the categories, called the chi-square distance, and (ii) the weighting of the categories proportional to their marginal sums. CA has been developed in a similar way to PCA, for example by introducing linear restrictions, where it is called canonical correspondence analysis (CCA), similarly to redundancy analysis (RDA) for PCA. Categorical PCA (CatPCA) is also a special restricted case of MCA.

General Overviews

The original works by the author of Benzécri 1973 were in French, with a rigorous but complex mathematical notation, and for both these reasons, linguistic and notational, the method remained relatively unknown outside of France until the publication of two English textbooks, with more familiar matrix-vector notation, Greenacre 1984 and Lebart, et al. 1984. A shorter, pedagogical account of correspondence analysis (CA) was published by Greenacre in 1993, which has been republished in two expanded editions, the latest being Greenacre 2016. Other generalist books have appeared, including Weller and Romney 1990, Le Roux and Rouanet 2004, Le Roux and Rouanet 2010, Beh and Lombardo 2021, and Husson, et al. 2017. Gifi 1991 is a multi-authored book that gives a comprehensive treatment of the Dutch school of data analysis, where CA and MCA (multiple correspondence analysis, called homogeneity analysis) are fundamental methods.

• Beh, E. J., and R. Lombardo. 2021. An introduction to correspondence analysis. Chichester, UK: Wiley.

  DOI: 10.1002/9781119044482

  Introductory overview of CA, focusing on the application and interpretation of CA and some of its variants.

• Benzécri, J.-P. 1973. L’Analyse des Données. Tôme 1: La Classification. Tôme 2: L’Analyse des Correspondances. Paris: Dunod.

  Volume 1 of Benzécri’s approach to data analysis, concentrating on cluster analysis, but including many aspects related to CA; for example, the definition of chi-square distance and the weighting of categories in the clustering algorithms. Volume 2 concentrates specifically on CA. Like the first volume, many applications of the method are given, but the theoretical parts involve a complex notation of upper and lower indices rather than matrix-vector notation, which limited the method’s diffusion.

• Gifi, A. 1991. Nonlinear multivariate analysis. Chichester, UK: Wiley.

  Situates CA and MCA in a wide framework of multivariate methods that aim to quantify and visualize categorical data.

• Greenacre, M. 1984. Theory and applications of correspondence analysis. London: Academic Press.

  Encyclopedic treatment of Benzécri’s CA up to the early 1980s, giving all the theory in terms of matrix-vector algebra, in particular using the singular value decomposition. Many applications are given as well as English summaries of the content of papers in the French journal published by Benzécri’s laboratory, Les Cahiers de l’Analyse des Données. Out of print, available for free download.

• Greenacre, M. 2016. Correspondence analysis in practice. 3d ed. Boca Raton, FL: Chapman & Hall/CRC Press.

  The third edition, highly extended, with thirty chapters (first and second editions with twelve and twenty chapters in 1993 and 2008, respectively) of the practical guide to CA, with R code using the package ca.

• Husson, F., S. Lê, and J. Pagès. 2017. Exploratory multivariate analysis by example uing R. Boca Raton, FL: Chapman & Hall.

  DOI: 10.1201/b21874

  A wide treatment of multivariate methods related to and including CA, with R code mostly relying on the package FactoMiner by the same authors.

• Lebart, L., A. Morineau, and K. M. Warwick. 1984. Multivariate descriptive statistical analysis. Chichester, UK: Wiley.

  A wide treatment of CA in the general context of multivariate analysis, discussing and applying the various forms of CA as well as cluster analysis.

• Le Roux, B., and H. Rouanet. 2004. Geometric data analysis: From correspondence analysis to structured data analysis. New York: Kluwer.

  Geometric data analysis is the term these authors use for Benzécri’s Analyse des Données (literally, data analysis) to reflect the inherent visual nature of this approach. Structured data analysis reflects the external conditions imposed on the data, for example an experimental design or most often the multi-category classification of the observed individuals, for example their demographic categories where the principal data consist of responses in a sample survey.

• Le Roux, B., and H. Rouanet. 2010. Multiple correspondence analysis. London: SAGE.

  DOI: 10.4135/9781412993906

  A shorter and lighter version of Le Roux and Rouanet 2004.

• Weller, S. C., and A. K. Romney. 1990. Metric scaling: Correspondence analysis. London: SAGE.

  DOI: 10.4135/9781412985048

  Introduction to CA and MCA aimed at practitioners, written by authors with a social science background.