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Register a new userHierarchical subspace models for contingency tables
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Added by
Hisayuki Hara
Publication date
2009
fields
Mathematical Statistics
and research's language is
English
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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.
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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.
In this paper, we explore a connection between binary hierarchical models, their marginal polytopes and codeword polytopes, the convex hulls of linear codes. The class of linear codes that are realizable by hierarchical models is determined. We classify all full dimensional polytopes with the property that their vertices form a linear code and give an algorithm that determines them.
In this work we define log-linear models to compare several square contingency tables under the quasi-independence or the quasi-symmetry model, and the relevant Markov bases are theoretically characterized. Through Markov bases, an exact test to evaluate if two or more tables fit a common model is introduced. Two real-data examples illustrate the use of these models in different fields of applications.
The existence of the maximum likelihood estimate in hierarchical loglinear models is crucial to the reliability of inference for this model. Determining whether the estimate exists is equivalent to finding whether the sufficient statistics vector $t$ belongs to the boundary of the marginal polytope of the model. The dimension of the smallest face $F_t$ containing $t$ determines the dimension of the reduced model which should be considered for correct inference. For higher-dimensional problems, it is not possible to compute $F_{t}$ exactly. Massam and Wang (2015) found an outer approximation to $F_t$ using a collection of sub-models of the original model. This paper refines the methodology to find an outer approximation and devises a new methodology to find an inner approximation. The inner approximation is given not in terms of a face of the marginal polytope, but in terms of a subset of the vertices of $F_t$. Knowing $F_t$ exactly indicates which cell probabilities have maximum likelihood estimates equal to $0$. When $F_t$ cannot be obtained exactly, we can use, first, the outer approximation $F_2$ to reduce the dimension of the problem and, then, the inner approximation $F_1$ to obtain correct estimates of cell probabilities corresponding to elements of $F_1$ and improve the estimates of the remaining probabilities corresponding to elements in $F_2setminus F_1$. Using both real-world and simulated data, we illustrate our results, and show that our methodology scales to high dimensions.
Elliptically contoured distributions can be considered to be the distributions for which the contours of the density functions are proportional ellipsoids. Kamiya, Takemura and Kuriki (2006) generalized the elliptically contoured distributions to star-shaped distributions, for which the contours are allowed to be arbitrary proportional star-shaped sets. This was achieved by considering the so-called orbital decomposition of the sample space in the general framework of group invariance. In the present paper, we extend their results by conducting the orbital decompositions in steps and obtaining a further, hierarchical decomposition of the sample space. This allows us to construct probability models and distributions with further independence structures. The general results are applied to the star-shaped distributions with a certain symmetric structure, the distributions related to the two-sample Wishart problem and the distributions of preference rankings.
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