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Counting subsets of contingency tables

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  • George Fishman

Abstract

We describe multistage Markov chain Monte Carlo (MSMCMC) procedures which, in addition to estimating the total number of contingency tables with given positive row and column sums, estimate the number, $$Q$$ Q , and the proportion, $$P$$ P , of those tables that satisfy an additional, possibly, nonlinear constraint. Three Options, A, B, and C, are studied. Options A and B exploit locally optimal statistical properties whereas judicious assignment of a particular parameter of Option C allows estimation with approximately minimal standard error. Ten examples of varying dimensions and total entries illustrate and compare the procedures, where $$Q$$ Q and $$P$$ P denote the number and proportion of chi-squared statistics less than a given value. For both small and large dimensional tables, the comparisons favor Options A and B for moderate $$P$$ P and Option C for small $$P$$ P . Additional comparison with sequential importance sampling estimates favors the latter for small dimensional tables and moderate $$P$$ P but favors Option C for large dimensional tables for both small and moderate $$P$$ P . The proposed options extend an earlier MSMCMC technique for estimating total count and, in principle, can be further extended to incorporate additional constraints. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • George Fishman, 2014. "Counting subsets of contingency tables," Computational Statistics, Springer, vol. 29(1), pages 159-187, February.
    • Handle: RePEc:spr:compst:v:29:y:2014:i:1:p:159-187
    DOI: 10.1007/s00180-013-0442-5
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    References listed on IDEAS

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    1. Kalantari, B. & Lari, I. & Rizzi, A. & Simeone, B., 1993. "Sharp bounds for the maximum of the chi-square index in a class of contingency tables with given marginals," Computational Statistics & Data Analysis, Elsevier, vol. 16(1), pages 19-34, June.
    2. Yuguo Chen & Persi Diaconis & Susan P. Holmes & Jun S. Liu, 2005. "Sequential Monte Carlo Methods for Statistical Analysis of Tables," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 109-120, March.
    3. Sabatti C., 2002. "Measuring Dependency With Volume Tests," The American Statistician, American Statistical Association, vol. 56, pages 191-195, August.
    4. Fabio Rapallo & Ruriko Yoshida, 2010. "Markov bases and subbases for bounded contingency tables," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(4), pages 785-805, August.
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