Relationships Between Full and Layer Models with Applications to Level Merging

Analysis of a large dimensional contingency table is quite involved. Models corresponding to layers of a contingency table are easier to analyze than the full model. Relationships between the interaction parameters of the full log-linear model and that of its corresponding layer models are obtained. These relationships are not only useful to reduce the analysis but also useful to interpret various hierarchical models. We obtain these relationships for layers of one variable, and extend the results for the case when layers of more than one variable are considered. We also establish, under conditional independence, relationships between the interaction parameters of the full model and that of the corresponding marginal models. We discuss the concept of merging of factor levels based on these interaction parameters. Finally, we use the relationships between layer models and full model to obtain conditions for level merging based on layer interaction parameters. Several examples are discussed to illustrate the results.