IDEAS home Printed from https://ideas.repec.org/a/spr/aistmt/v67y2015i4p621-648.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10463-014-0471-z
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10463-014-0471-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Le Breton, Michel & Lepelley, Dominique & Smaoui, Hatem, 2012. "The Probability of Casting a Decisive Vote: From IC to IAC trhough Ehrhart's Polynomials and Strong Mixing," IDEI Working Papers 722, Institut d'Économie Industrielle (IDEI), Toulouse.
    2. Sylvain Béal & Marc Deschamps & Mostapha Diss & Issofa Moyouwou, 2022. "Inconsistent weighting in weighted voting games," Public Choice, Springer, vol. 191(1), pages 75-103, April.
    3. Abdelhalim El Ouafdi & Dominique Lepelley & Hatem Smaoui, 2020. "Probabilities of electoral outcomes: from three-candidate to four-candidate elections," Theory and Decision, Springer, vol. 88(2), pages 205-229, March.
    4. Friedrich Eisenbrand & Gennady Shmonin, 2008. "Parametric Integer Programming in Fixed Dimension," Mathematics of Operations Research, INFORMS, vol. 33(4), pages 839-850, November.
    5. Hatem Smaoui & Dominique Lepelley & Issofa Moyouwou, 2016. "Borda elimination rule and monotonicity paradoxes in three-candidate elections," Economics Bulletin, AccessEcon, vol. 36(3), pages 1722-1728.
    6. Sascha Kurz & Nikolas Tautenhahn, 2013. "On Dedekind’s problem for complete simple games," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(2), pages 411-437, May.
    7. Jesús A. De Loera & Raymond Hemmecke & Matthias Köppe, 2009. "Pareto Optima of Multicriteria Integer Linear Programs," INFORMS Journal on Computing, INFORMS, vol. 21(1), pages 39-48, February.
    8. Ahmad Awde & Mostapha Diss & Eric Kamwa & Julien Yves Rolland & Abdelmonaim Tlidi, 2023. "Social Unacceptability for Simple Voting Procedures," Studies in Choice and Welfare, in: Sascha Kurz & Nicola Maaser & Alexander Mayer (ed.), Advances in Collective Decision Making, pages 25-42, Springer.
    9. Dominique Lepelley & Issofa Moyouwou & Hatem Smaoui, 2018. "Monotonicity paradoxes in three-candidate elections using scoring elimination rules," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 50(1), pages 1-33, January.
    10. Mostapha Diss & Michele Gori, 2022. "Majority properties of positional social preference correspondences," Theory and Decision, Springer, vol. 92(2), pages 319-347, March.
    11. Le Breton, Michel & Lepelley, Dominique & Smaoui, Hatem, 2016. "Correlation, partitioning and the probability of casting a decisive vote under the majority rule," Journal of Mathematical Economics, Elsevier, vol. 64(C), pages 11-22.
    12. Achill Schürmann, 2013. "Exploiting polyhedral symmetries in social choice," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 40(4), pages 1097-1110, April.
    13. Hiroshi Hirai & Ryunosuke Oshiro & Ken’ichiro Tanaka, 2020. "Counting Integral Points in Polytopes via Numerical Analysis of Contour Integration," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 455-464, May.
    14. Diss, Mostapha & Louichi, Ahmed & Merlin, Vincent & Smaoui, Hatem, 2012. "An example of probability computations under the IAC assumption: The stability of scoring rules," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 57-66.
    15. Eric Kamwa & Fabrice Valognes, 2017. "Scoring Rules and Preference Restrictions: The Strong Borda Paradox Revisited," Revue d'économie politique, Dalloz, vol. 127(3), pages 375-395.
    16. Eric Kamwa, 2017. "Stable Rules for Electing Committees and Divergence on Outcomes," Group Decision and Negotiation, Springer, vol. 26(3), pages 547-564, May.
    17. Jean B. Lasserre & Eduardo S. Zeron, 2003. "On Counting Integral Points in a Convex Rational Polytope," Mathematics of Operations Research, INFORMS, vol. 28(4), pages 853-870, November.
    18. Gergely Szlobodnyik & Gábor Szederkényi, 2019. "Reachability Analysis of Low-Order Discrete State Reaction Networks Obeying Conservation Laws," Complexity, Hindawi, vol. 2019, pages 1-13, March.
    19. Ahmad Awde & Mostapha Diss & Eric Kamwa & Julien Yves Rolland & Abdelmonaim Tlidi, 2022. "Social unacceptability for simple voting procedures," Working Papers hal-03614587, HAL.
    20. Moyouwou, Issofa & Tchantcho, Hugue, 2017. "Asymptotic vulnerability of positional voting rules to coalitional manipulation," Mathematical Social Sciences, Elsevier, vol. 89(C), pages 70-82.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:aistmt:v:67:y:2015:i:4:p:621-648. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.