Correspondence and canonical analysis of relational data

Correspondence analysis, a data analytic technique used to study two-way cross-classifications, is applied to social relational data. Such data are frequently termed “sociometric” or “network” data. The method allows one to model forms of relational data and types of empirical relationships not easily analyzed using either standard social network methods or common scaling or clustering techniques. In particular, correspondence analysis allows one to model: —two-mode networks (rows and columns of a sociomatrix refer to different objects) —valued relations (e.g. counts, ratings, or frequencies). In general, the technique provides scale values for row and column units, visual presentation of relationships among rows and columns, and criteria for assessing “dimensionality” or graphical complexity of the data and goodness-of-fit to particular models. Correspondence analysis has recently been the subject of research by Goodman, Haberman, and Gilula, who have termed their approach to the problem “canonical analysis” to reflect its similarity to canonical correlation analysis of continuous multivariate data. This generalization links the technique to more standard categorical data analysis models, and provides a much-needed statistical justification. We review both correspondence and canonical analysis, and present these ideas by analyzing relational data on the 1980 monetary donations from corporations to nonprofit organizations in the Minneapolis St. Paul metropolitan area. We also show how these techniques are related to dyadic independence models, first introduced by Holland, Leinhardt, Fienberg, and Wasserman in the early 1980's. The highlight of this paper is the relationship between correspondence and canonical analysis, and these dyadic independence models, which are designed specifically for relational data. The paper concludes with a discussion of this relationship, and some data analyses that illustrate the fact that correspondence analysis models can be used as approximate dyadic independence models.