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Markov basis and Gröbner basis of Segre–Veronese configuration for testing independence in group-wise selections

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Abstract

We consider testing independence in group-wise selections with some restrictions on combinations of choices. We present models for frequency data of selections for which it is easy to perform conditional tests by Markov chain Monte Carlo (MCMC) methods. When the restrictions on the combinations can be described in terms of a Segre–Veronese configuration, an explicit form of a Gröbner basis consisting of binomials of degree two is readily available for performing a Markov chain. We illustrate our setting with the National Center Test for university entrance examinations in Japan. We also apply our method to testing independence hypotheses involving genotypes at more than one locus or haplotypes of alleles on the same chromosome.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Kagoshima University, 1-21-35, Korimoto, Kagoshima, Kagoshima, 890-0065, Japan

    Satoshi Aoki

  2. Graduate School of Information Science and Technology, Osaka University, 1-1, Yamadaoka, Suita, Osaka, 565-0871, Japan

    Takayuki Hibi

  3. Department of Mathematics, Rikkyo University, 3-34-1, Nishi Ikebukuro, Toshima-ku, Tokyo, 171-8501, Japan

    Hidefumi Ohsugi

  4. Graduate School of Information Science and Technology, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan

    Akimichi Takemura

Authors
  1. Satoshi Aoki
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  2. Takayuki Hibi
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  3. Hidefumi Ohsugi
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  4. Akimichi Takemura
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Correspondence to Satoshi Aoki.

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Aoki, S., Hibi, T., Ohsugi, H. et al. Markov basis and Gröbner basis of Segre–Veronese configuration for testing independence in group-wise selections. Ann Inst Stat Math 62, 299–321 (2010). https://doi.org/10.1007/s10463-008-0171-7

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  • DOI: https://doi.org/10.1007/s10463-008-0171-7

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