A NOTE ON MULTIVARIATE LOGISTIC MODELS FOR CONTINGENCY TABLES

Publisher Website
Google Scholar
Abstract
Summary: The log‐linear model is a tool widely accepted for modelling discrete data given in a contingency table. Although its parameters reflect the interaction structure in the joint distribution of all variables, it does not give information about structures appearing in the margins of the table. This is in contrast to multivariate logistic parameters, recently introduced by Glonek & McCullagh (1995), which have as parameters the highest order log odds ratios derived from the joint table and from each marginal table. Glonek & McCullagh give the link between the cell probabilities and the multivariate logistic parameters, in an algebraic fashion. The present paper focuses on this link, showing that it is derived by general parameter transformations in exponential families. In particular, the connection between the natural, the expectation and the mixed parameterization in exponential families (Barndorff‐Nielsen, 1978) is used; this also yields the derivatives of the likelihood equation and shows properties of the Fisher matrix. The paper emphasises the analysis of independence hypotheses in margins of a contingency table.