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Algebraic Markov Bases and MCMC for Two‐Way Contingency Tables

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  • Fabio Rapallo

Abstract

ABSTRACT. The Diaconis–Sturmfels algorithm is a method for sampling from conditional distributions, based on the algebraic theory of toric ideals. This algorithm is applied to categorical data analysis through the notion of Markov basis. An application of this algorithm is a non‐parametric Monte Carlo approach to the goodness of fit tests for contingency tables. In this paper, we characterize or compute the Markov bases for some log‐linear models for two‐way contingency tables using techniques from Computational Commutative Algebra, namely Gröbner bases. This applies to a large set of cases including independence, quasi‐independence, symmetry, quasi‐symmetry. Three examples of quasi‐symmetry and quasi‐independence from Fingleton (Models of category counts, Cambridge University Press, Cambridge, 1984) and Agresti (An Introduction to categorical data analysis, Wiley, New York, 1996) illustrate the practical applicability and the relevance of this algebraic methodology.

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  • Fabio Rapallo, 2003. "Algebraic Markov Bases and MCMC for Two‐Way Contingency Tables," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(2), pages 385-397, June.
    • Handle: RePEc:bla:scjsta:v:30:y:2003:i:2:p:385-397
    DOI: 10.1111/1467-9469.00337
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    Cited by:

    1. Fabio Rapallo, 2005. "Algebraic exact inference for rater agreement models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 14(1), pages 45-66, February.
    2. Fabio Rapallo, 2022. "Analysis of the Weighted Kappa and Its Maximum with Markov Moves," Psychometrika, Springer;The Psychometric Society, vol. 87(4), pages 1270-1289, December.
    3. Fabio Rapallo & Maria Piera Rogantin, 2007. "Markov chains on the reference set of contingency tables with upper bounds," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(1), pages 35-51.
    4. Abraham Martín del Campo & Sarah Cepeda & Caroline Uhler, 2017. "Exact Goodness-of-Fit Testing for the Ising Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(2), pages 285-306, June.
    5. Kim, Sung-Ho & Choi, Hyemi & Lee, Sangjin, 2009. "Estimate-based goodness-of-fit test for large sparse multinomial distributions," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1122-1131, February.
    6. Fabio Rapallo, 2007. "Toric statistical models: parametric and binomial representations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(4), pages 727-740, December.
    7. Krampe, Anne & Kuhnt, Sonja, 2007. "Bowker's test for symmetry and modifications within the algebraic framework," Computational Statistics & Data Analysis, Elsevier, vol. 51(9), pages 4124-4142, May.

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