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Markov chain Monte Carlo exact tests for incomplete two-way contingency tables. (English) Zbl 1076.62060

Summary: 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, Markov chain Monte Carlo exact tests are powerful tools. To construct a connected chain over two-way contingency tables with fixed sufficient statistics and an arbitrary configuration of structural zero cells, an algebraic algorithm proposed by P. Diaconis and B. Sturmfels [Ann. Stat. 26, No. 1, 363–397 (1998; Zbl 0952.62088)] 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.

MSC:

62H17 Contingency tables
65C40 Numerical analysis or methods applied to Markov chains
65C60 Computational problems in statistics (MSC2010)
62F03 Parametric hypothesis testing

Citations:

Zbl 0952.62088
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References:

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