Collapsibility of marginal models for categorical data
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We consider marginal log-linear models for parameterizing distributions on multidimensional contingency tables. These models generalize ordinary log-linear and multivariate logistic models, besides several others. First, we obtain some characteristic properties of marginal log-linear parameters. Then we define collapsibility and strict collapsibility of these parameters in a general sense. Several necessary and sufficient conditions for collapsibility and strict collapsibility are derived using the technique of Möbius inversion. These include results for an arbitrary set of marginal log-linear parameters having some common effects. The connections of collapsibility and strict collapsibility to various forms of independence of the variables are discussed. Finally, we establish a result on the relationship between parameters with the same effect but different margins, and use it to demonstrate smoothness of marginal log-linear models under collapsibility conditions thereby obtaining a curved exponential family.
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