Based on a revised Lancaster-type representation of the additive interactions associated with a probability measure, a new approach for the analysis of high-dimensional contingency tables is proposed. The approach is essentially model-free because the additive interaction tensor is merely a convenient reparameterization of the given table. Single interaction terms are investigated using the bootstrap method whose first-order asymptotic validity is immediate. The global structure can be investigated by using the multiple $p$-values given by Holm’s sequentially rejecting multiple testing procedure. The procedure is based on a characterization of the Moebius function as a solution of the simultaneous eigenproblem for all intersection operators in a finite lattice.
"Exploring interactions in high-dimensional tables: a bootstrap alternative to log-linear models." Ann. Statist. 27 (1) 405 - 413, February 1999. https://doi.org/10.1214/aos/1018031118