IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v23y2021i1d10.1007_s11009-020-09783-0.html
   My bibliography  Save this article

Asymptotic Distributions of Empirical Interaction Information

Author

Listed:
  • Mariusz Kubkowski

    (Institute of Computer Science Polish Academy of Sciences
    Warsaw University of Technology)

  • Jan Mielniczuk

    (Institute of Computer Science Polish Academy of Sciences
    Warsaw University of Technology)

Abstract

Interaction Information is one of the most promising interaction strength measures with many desirable properties. However, its use for interaction detection was hindered by the fact that apart from the simple case of overall independence, asymptotic distribution of its estimate has not been known. In the paper we provide asymptotic distributions of its empirical versions which are needed for formal testing of interactions. We prove that for three-dimensional nominal vector normalized empirical interaction information converges to the normal law unless the distribution coincides with its Kirkwood approximation. In the opposite case the convergence is to the distribution of weighted centred chi square random variables. This case is of special importance as it roughly corresponds to interaction information being zero and the asymptotic distribution can be used for construction of formal tests for interaction detection. The result generalizes result in Han (Inf Control 46(1):26–45 1980) for the case when all coordinate random variables are independent. The derivation relies on studying structure of covariance matrix of asymptotic distribution and its eigenvalues. For the case of 3 × 3 × 2 contingency table corresponding to study of two interacting Single Nucleotide Polymorphisms (SNPs) for prediction of binary outcome, we provide complete description of the asymptotic law and construct approximate critical regions for testing of interactions when two SNPs are possibly dependent. We show in numerical experiments that the test based on the derived asymptotic distribution is easy to implement and yields actual significance levels consistently closer to the nominal ones than the test based on chi square reference distribution.

Suggested Citation

  • Mariusz Kubkowski & Jan Mielniczuk, 2021. "Asymptotic Distributions of Empirical Interaction Information," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 291-315, March.
    • Handle: RePEc:spr:metcap:v:23:y:2021:i:1:d:10.1007_s11009-020-09783-0
    DOI: 10.1007/s11009-020-09783-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-020-09783-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-020-09783-0?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. William McGill, 1954. "Multivariate information transmission," Psychometrika, Springer;The Psychometric Society, vol. 19(2), pages 97-116, June.
    2. Jin-Ting Zhang, 2005. "Approximate and Asymptotic Distributions of Chi-Squared-Type Mixtures With Applications," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 273-285, March.
    Full references (including those not matched with items on IDEAS)

    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. repec:hig:wpaper:98sti2019 is not listed on IDEAS
    2. Petersen, Alexander M. & Rotolo, Daniele & Leydesdorff, Loet, 2016. "A triple helix model of medical innovation: Supply, demand, and technological capabilities in terms of Medical Subject Headings," Research Policy, Elsevier, vol. 45(3), pages 666-681.
    3. Park, Han Woo & Leydesdorff, Loet, 2010. "Longitudinal trends in networks of university-industry-government relations in South Korea: The role of programmatic incentives," Research Policy, Elsevier, vol. 39(5), pages 640-649, June.
    4. Songyot Nakariyakul, 2019. "A hybrid gene selection algorithm based on interaction information for microarray-based cancer classification," PLOS ONE, Public Library of Science, vol. 14(2), pages 1-17, February.
    5. Louis Verny & Nadir Sella & Séverine Affeldt & Param Priya Singh & Hervé Isambert, 2017. "Learning causal networks with latent variables from multivariate information in genomic data," PLOS Computational Biology, Public Library of Science, vol. 13(10), pages 1-25, October.
    6. Xiaojun Hu & Xian Li & Ronald Rousseau, 2021. "Mathematical reflections on Triple Helix calculations," Scientometrics, Springer;Akadémiai Kiadó, vol. 126(10), pages 8581-8587, October.
    7. Inga A. Ivanova & Loet Leydesdorff, 2014. "A simulation model of the Triple Helix of university–industry–government relations and the decomposition of the redundancy," Scientometrics, Springer;Akadémiai Kiadó, vol. 99(3), pages 927-948, June.
    8. Loet Leydesdorff & Han Woo Park & Balazs Lengyel, 2014. "A routine for measuring synergy in university–industry–government relations: mutual information as a Triple-Helix and Quadruple-Helix indicator," Scientometrics, Springer;Akadémiai Kiadó, vol. 99(1), pages 27-35, April.
    9. Jin-Ting Zhang & Xuehua Liang, 2014. "One-Way anova for Functional Data via Globalizing the Pointwise F-test," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(1), pages 51-71, March.
    10. Lengyel, Balázs & Leydesdorff, Loet, 2015. "The Effects of FDI on Innovation Systems in Hungarian Regions: Where is the Synergy Generated?," MPRA Paper 73945, University Library of Munich, Germany.
    11. Han Woo Park, 2014. "Mapping election campaigns through negative entropy: Triple and Quadruple Helix approach to South Korea’s 2012 presidential election," Scientometrics, Springer;Akadémiai Kiadó, vol. 99(1), pages 187-197, April.
    12. Loet Leydesdorff & Igone Porto-Gomez, 2019. "Measuring the expected synergy in Spanish regional and national systems of innovation," The Journal of Technology Transfer, Springer, vol. 44(1), pages 189-209, February.
    13. Strand, Øivind & Leydesdorff, Loet, 2013. "Where is synergy indicated in the Norwegian innovation system? Triple-Helix relations among technology, organization, and geography," Technological Forecasting and Social Change, Elsevier, vol. 80(3), pages 471-484.
    14. Dennis Knepp & Doris Entwisle, 1969. "Testing significance of differences between two chi-squares," Psychometrika, Springer;The Psychometric Society, vol. 34(3), pages 331-333, September.
    15. Inga Ivanova & Oivind Strand & Loet Leydesdorff, 2019. "The Synergy and Cycle Values in Regional Innovation Systems: The Case of Norway," Foresight and STI Governance (Foresight-Russia till No. 3/2015), National Research University Higher School of Economics, vol. 13(1), pages 48-61.
    16. Zhang, Jin-Ting & Guo, Jia & Zhou, Bu, 2017. "Linear hypothesis testing in high-dimensional one-way MANOVA," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 200-216.
    17. Zhang, Jin-Ting & Zhu, Tianming, 2024. "A fast and accurate kernel-based independence test with applications to high-dimensional and functional data," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    18. Frank Huettner, & Tamer Boyaci, & Yalcin Akcay, 2016. "Consumer choice under limited attention when alternatives have different information costs," ESMT Research Working Papers ESMT-16-04_R2, ESMT European School of Management and Technology, revised 28 Feb 2018.
    19. Huang, Peng & Gu, Yingkui & Li, He & Yazdi, Mohammad & Qiu, Guangqi, 2023. "An Optimal Tolerance Design Approach of Robot Manipulators for Positioning Accuracy Reliability," Reliability Engineering and System Safety, Elsevier, vol. 237(C).
    20. Frank Gelens & Juho Äijälä & Louis Roberts & Misako Komatsu & Cem Uran & Michael A. Jensen & Kai J. Miller & Robin A. A. Ince & Max Garagnani & Martin Vinck & Andres Canales-Johnson, 2024. "Distributed representations of prediction error signals across the cortical hierarchy are synergistic," Nature Communications, Nature, vol. 15(1), pages 1-18, December.
    21. Huang, Zhendong & Ferrari, Davide & Qian, Guoqi, 2017. "Parsimonious and powerful composite likelihood testing for group difference and genotype–phenotype association," Computational Statistics & Data Analysis, Elsevier, vol. 110(C), pages 37-49.

    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:metcap:v:23:y:2021:i:1:d:10.1007_s11009-020-09783-0. 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.