The Analysis of Contingency Tables: From Chi-Squared Tests and Log-Linear Models to Models of Mixed Membership

The roots of the modern log-linear model approach to the analysis of cross-classified data in the form of multi-dimensional contingency tables can found in the work of S. N. Roy and his students in the 1950s at the University of North Carolina. These papers set the stage for two major sets of developments in the analysis of categorical data in the 1960s and 1970s. I describe some of these contributions, where they intersected and where they diverged in focus, and some subsequent advances, including the role of latent variables alternatives, mixed membership models, and methods for very large sparse categorical data arrays.