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Analysis of conditional randomisation and permutation schemes with application to conditional independence testing

Author

Listed:
  • Małgorzata Łazȩcka

    (Warsaw University of Technology
    Polish Academy of Sciences
    University of Warsaw)

  • Bartosz Kołodziejek

    (Warsaw University of Technology)

  • Jan Mielniczuk

    (Warsaw University of Technology
    Polish Academy of Sciences)

Abstract

We study properties of two resampling scenarios: Conditional Randomisation and Conditional Permutation schemes, which are relevant for testing conditional independence of discrete random variables X and Y given a random variable Z. Namely, we investigate asymptotic behaviour of estimates of a vector of probabilities in such settings, establish their asymptotic normality and ordering between asymptotic covariance matrices. The results are used to derive asymptotic distributions of the empirical Conditional Mutual Information in those set-ups. Somewhat unexpectedly, the distributions coincide for the two scenarios, despite differences in the asymptotic distributions of the estimates of probabilities. We also prove validity of permutation p-values for the Conditional Permutation scheme. The above results justify consideration of conditional independence tests based on resampled p-values and on the asymptotic chi-square distribution with an adjusted number of degrees of freedom. We show in numerical experiments that when the ratio of the sample size to the number of possible values of the triple exceeds 0.5, the test based on the asymptotic distribution with the adjustment made on a limited number of permutations is a viable alternative to the exact test for both the Conditional Permutation and the Conditional Randomisation scenarios. Moreover, there is no significant difference between the performance of exact tests for Conditional Permutation and Randomisation schemes, the latter requiring knowledge of conditional distribution of X given Z, and the same conclusion is true for both adaptive tests.

Suggested Citation

  • Małgorzata Łazȩcka & Bartosz Kołodziejek & Jan Mielniczuk, 2023. "Analysis of conditional randomisation and permutation schemes with application to conditional independence testing," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(4), pages 1459-1478, December.
    • Handle: RePEc:spr:testjl:v:32:y:2023:i:4:d:10.1007_s11749-023-00878-7
    DOI: 10.1007/s11749-023-00878-7
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    References listed on IDEAS

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    1. Thomas B. Berrett & Yi Wang & Rina Foygel Barber & Richard J. Samworth, 2020. "The conditional permutation test for independence while controlling for confounders," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(1), pages 175-197, February.
    2. Emmanuel Candès & Yingying Fan & Lucas Janson & Jinchi Lv, 2018. "Panning for gold: ‘model‐X’ knockoffs for high dimensional controlled variable selection," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(3), pages 551-577, June.
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