Categorical Data Analysis: Some Reflections on the Log Linear Model and Logistic Regression. Part I: Historical and Methodological Overview

Abstract

Summary The literature of log linear models and logistic regression is surveyed from a contemporary point of view. A matrix formulation of the general log linear model for product-multinomial random counts is exploited to study the relationship between maximum likelihood and weighted least squares approaches to model fitting. Maximum likelihood fitted parameters and cell expectations are shown to be stationary solutions of a weighted least squares equation. Matrix expressions for asymptotic covariance matrices of efficient fitted parameters and cell counts are developed. Asymptotic covariance matrices of generalized raked contingency tables are obtained from the matrix formulation. Functional asymptotic regression methodology, an approach combining aspects of maximum likelihood and weighted least squares, is described and examined. In Part II, the several methods and relationships are illustrated by seven examples; extensions applicable to noncentrality problems and complex sample survey designs are also presented.