Minimal Basis for a Connected Markov Chain over 3 × 3 ×K Contingency Tables with Fixed Two‐Dimensional Marginals

Abstract

This paper considers a connected Markov chain for sampling 3×3×K contingency tables having fixed two-dimensional marginal totals. Such sampling arises in performing various tests of the hypothesis of no three-factor interactions. A Markov chain algorithm is a valuable tool for evaluating P -values, especially for sparse datasets where large-sample theory does not work well. To construct a connected Markov chain over high-dimensional contingency tables with fixed marginals, algebraic algorithms have been proposed. These algorithms involve computations in polynomial rings using Gröbner bases. However, algorithms based on Gröbner bases do not incorporate symmetry among variables and are very time-consuming when the contingency tables are large. We construct a minimal basis for a connected Markov chain over 3×3×K contingency tables. The minimal basis is unique.Some numerical examples illustrate the practicality of our algorithms.

Lectures on Algebraic Statistics

“…This bound is independent of s, which can grow large. [10]. In general, it seems a difficult problem to actually compute the values b(Γ, r 2 , .…”

Convex Discrete Optimization

“…We now discuss a recent result of [54], which originates in [4], and its extension in [38], on the stabilization of Graver bases of n-fold matrices. Consider vectors x = (x 1 , .…”

“…Multiway transportation problems form a very important class of discrete optimization problems and have been used and studied extensively in the operations research and mathematical programming literature, as well as in the statistics literature in the context of secure statistical data disclosure and management by public agencies, see e.g. [4,6,11,18,19,42,43,53,60,62] and the references therein. The feasible points in a transportation problem are the multiway tables ("contingency tables" in statistics) such that the sums of entries over some of their lower dimensional sub-tables such as lines or planes ("margins" in statistics) are specified.…”

Markov chain Monte Carlo tests for designed experiments

“…As noted in Section 1, Markov bases have been mainly investigated in the context of contingency tables. For example, [6] gives an expression of minimal Markov bases of all the hierarchical models for 2 4 contingency tables. This list may be sufficient in application for the analysis of 2 4 contingency tables, since the hierarchical model is the most natural class of models to be considered.…”