Log-Linear Models for Frequency Data: Sufficient Statistics and Likelihood Equations

A general model is proposed for analysis of frequency tables. This model includes conventional log-linear models for complete and incomplete factorial tables and logit models for quantal response analysis. By use of coordinate-free methods of linear algebra and differential calculus, complete minimal sufficient statistics and likelihood equations for the maximum likelihood estimate are derived. The maximum likelihood estimate is shown to be unique if it exists, and necessary and sufficient conditions are given for its existence.