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Fibers of multi-way contingency tables given conditionals: relation to marginals, cell bounds and Markov bases

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  • Aleksandra Slavković
  • Xiaotian Zhu
  • Sonja Petrović

Abstract

A fiber of a contingency table is the space of all realizations of the table under a given set of constraints such as marginal totals. Understanding the geometry of this space is a key problem in algebraic statistics, important for conducting exact conditional inference, calculating cell bounds, imputing missing cell values, and assessing the risk of disclosure of sensitive information. Motivated by disclosure problems, in this paper we study the space of all possible tables for a given sample size and set of observed conditional frequencies. We show that this space can be decomposed according to different possible marginals, which, in turn, are encoded by the solution set of a linear Diophantine equation. Our decomposition has two important consequences: (1) we derive new cell bounds, some including connections to directed acyclic graphs, and (2) we describe a structure for the Markov bases for the given space that leads to a simplified calculation of Markov bases in this particular setting. Copyright The Institute of Statistical Mathematics, Tokyo 2015

Suggested Citation

  • Aleksandra Slavković & Xiaotian Zhu & Sonja Petrović, 2015. "Fibers of multi-way contingency tables given conditionals: relation to marginals, cell bounds and Markov bases," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(4), pages 621-648, August.
    • Handle: RePEc:spr:aistmt:v:67:y:2015:i:4:p:621-648
    DOI: 10.1007/s10463-014-0471-z
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    References listed on IDEAS

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    1. Alexander I. Barvinok, 1994. "A Polynomial Time Algorithm for Counting Integral Points in Polyhedra When the Dimension is Fixed," Mathematics of Operations Research, INFORMS, vol. 19(4), pages 769-779, November.
    2. Satoshi Aoki & Akimichi Takemura, 2008. "Minimal invariant Markov basis for sampling contingency tables with fixed marginals," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(2), pages 229-256, June.
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    Cited by:

    1. Elizabeth Gross & Sonja Petrović & Despina Stasi, 2017. "Goodness of fit for log-linear network models: dynamic Markov bases using hypergraphs," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(3), pages 673-704, June.
    2. Sage, Andrew J. & Wright, Stephen E., 2016. "Obtaining cell counts for contingency tables from rounded conditional frequencies," European Journal of Operational Research, Elsevier, vol. 250(1), pages 91-100.

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