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Register a new userMarkov Bases for Typical Block Effect Models of Two-way Contingency Tables
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Markov basis for statistical model of contingency tables gives a useful tool for performing the conditional test of the model via Markov chain Monte Carlo method. In this paper we derive explicit forms of Markov bases for change point models and block diagonal effect models, which are typical block-wise effect models of two-way contingency tables, and perform conditional tests with some real data sets.
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To evaluate a fitting of a statistical model to given data, calculating a conditional $p$ value by a Markov chain Monte Carlo method is one of the effective approaches. For this purpose, a Markov basis plays an important role because it guarantees the connectivity of the chain for unbiasedness of the estimation, and therefore is investigated in various settings such as incomplete tables or subtable sum constraints. In this paper, we consider the two-way change-point model for the ladder determinantal table, which is an extension of these two previous works. Our main result is based on the theory of Groebner basis for the distributive lattice. We give a numerical example for actual data.
For statistical analysis of multiway contingency tables we propose modeling interaction terms in each maximal compact component of a hierarchical model. By this approach we can search for parsimonious models with smaller degrees of freedom than the usual hierarchical model, while preserving conditional independence structures in the hierarchical model. We discuss estimation and exacts tests of the proposed model and illustrate the advantage of the proposed modeling with some data sets.
It has been well-known that for two-way contingency tables with fixed row sums and column sums the set of square-free moves of degree two forms a Markov basis. However when we impose an additional constraint that the sum of a subtable is also fixed, then these moves do not necessarily form a Markov basis. Thus, in this paper, we show a necessary and sufficient condition on a subtable so that the set of square-free moves of degree two forms a Markov basis.
We consider the three-state toric homogeneous Markov chain model (THMC) without loops and initial parameters. At time $T$, the size of the design matrix is $6 times 3cdot 2^{T-1}$ and the convex hull of its columns is the model polytope. We study the behavior of this polytope for $Tgeq 3$ and we show that it is defined by 24 facets for all $Tge 5$. Moreover, we give a complete description of these facets. From this, we deduce that the toric ideal associated with the design matrix is generated by binomials of degree at most 6. Our proof is based on a result due to Sturmfels, who gave a bound on the degree of the generators of a toric ideal, provided the normality of the corresponding toric variety. In our setting, we established the normality of the toric variety associated to the THMC model by studying the geometric properties of the model polytope.
We discuss connecting tables with zero-one entries by a subset of a Markov basis. In this paper, as a Markov basis we consider the Graver basis, which corresponds to the unique minimal Markov basis for the Lawrence lifting of the original configuration. Since the Graver basis tends to be large, it is of interest to clarify conditions such that a subset of the Graver basis, in particular a minimal Markov basis itself, connects tables with zero-one entries. We give some theoretical results on the connectivity of tables with zero-one entries. We also study some common models, where a minimal Markov basis for tables without the zero-one restriction does not connect tables with zero-one entries.
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