Correspondence Analysis in Psychology

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.