Minimal basis for connected Markov chain over 3× 3×K contingency tables with fixed two-dimensional marginals

We consider 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. Markov chain algorithm is a valuable tool for evaluating p values, especially for sparse data sets where large-sample theory does not work well. For constructing a connected Markov chain over high dimensional contingency tables with fixed marginals, algebraic algorithms were proposed by Diaconis and Sturmfels (1998). Their 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 size of contingency tables is large. We construct a minimal basis for connected Markov chain over 3× 3×K contingency tables. Some numerical examples are also given to illustrate the practicality of our algorithms.