Holes in Semigroups and Their Applications to the Two-way Common Diagonal Effect Model

A two-way subtable sum model is a statistical model for two-way contingency tables such that sufficient statistics for the model are row sums, column sums and an additional constraint that the sum of a subtable is also fixed. From a statistical viewpoint this model is related to a block interaction model, a twoway change-point model proposed by Hirotsu, and the quasi-independence model for incomplete two-way contingency tables which contain some structural zeros. When the set of square-free moves of degree two does not form a Markov basis, we know that the semigroup generated by columns of the design matrix for the sufficient statistics of the model is not normal. One such model is the common diagonal effect model. In this paper, we first summarize the results in [Takemura and Yoshida, 2007] and then we study how the difference between the semigroup generated by columns of the design matrix for a common diagonal effect model and its saturation are distributed. keyword: contingency tables, Hilbert bases, integer linear feasibility problem, semigroup.