Impact of zero-patterns in the dependence of three-way binary tables

We study the problem of transforming a multi-way contingency table into an equivalent table with uniform margins and the same dependence structure. This is an old question which relates to recent developments in copula modeling for discrete random vectors. In this work, we focus on the three-way binary tables and show how the zero-patterns affect the transformation as well as its statistical interpretability in terms of corresponding dependence structure.