Specifications of Models for Cross-Classified Counts

Log-linear models are useful for analyzing cross-classifications of counts arising in sociology, but it has been argued that in some cases, an alternative approach for formulating models—one based on simultaneously modeling univariate marginal logits and marginal associations—can lead to models that are more directly relevant for addressing the kinds of questions arising in those cases. In this article, the authors explore some of the similarities and differences between the log-linear models approach to modeling categorical data and a marginal modeling appraoch. It has been noted in past literature that the model of statistical independence is conveniently represented within both approaches to specifying models for cross-classifications of counts. The authors examine further the extent to which the two families of models overlap, as well as some important differences. The authors do not present a complete characterization of the conditions describing the intersection of the two families of models but cover many of the models for bivariate contingency tables and for three-way contingency tables that are routinely used in sociological research.