Markov chain Monte Carlo exact tests for incomplete two-way contingency tables

We consider testing the quasi-independence hypothesis for two-way contingency tables which contain some structural zero cells. For sparse contingency tables where the large sample approximation is not adequate, the Markov chain Monte Carlo exact tests are powerful tools. To construct a connected chain over the two-way contingency tables with fixed sufficient statistics and an arbitrary configuration of structural zero cells, an algebraic algorithm proposed by Diaconis and Sturmfels (1998) can be used. However their algorithm does not seem to be a satisfactory answer, because the Markov basis produced by the algorithm often contains many redundant elements and is hard to interpret. We derive an explicit characterization of a minimal Markov basis, prove its uniqueness and present an algorithm for obtaining the unique minimal basis. A computational example and the discussion on further basis reduction for the case of positive sufficient statistics are also given.