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On the distribution of isometric log-ratio coordinates under extra-multinomial count data

Author

Listed:
  • Noora Kartiosuo

    (University of Turku
    University of Turku
    University of Turku
    Murdoch Children’s Research Institute)

  • Joni Virta

    (University of Turku)

  • Jaakko Nevalainen

    (Tampere University)

  • Olli Raitakari

    (University of Turku
    University of Turku
    University of Turku and Turku University Hospital)

  • Kari Auranen

    (University of Turku
    University of Turku)

Abstract

Compositional data can be mapped from the simplex to the Euclidean space through the isometric log-ratio (ilr) transformation. When the underlying counts follow a multinomial distribution, the distribution of the ensuing ilr coordinates has been shown to be asymptotically multivariate normal. We derive conditions under which the asymptotic normality of the ilr coordinates holds under a compound multinomial distribution inducing overdispersion in the counts. We derive a normal approximation and investigate its practical applicability under extra-multinomial variation using a simulation study under the Dirichlet-multinomial distribution. The approximation works well, except with a small total count or high amount of overdispersion. Our work is motivated by microbiome data, which exhibit extra-multinomial variation and are increasingly treated as compositions. We conclude that if empirical data analysis relies on the normality of ilr coordinates, it may be advisable to choose a taxonomic level with less sparsity so that the distribution of taxon-specific class probabilities remains unimodal.

Suggested Citation

  • Noora Kartiosuo & Joni Virta & Jaakko Nevalainen & Olli Raitakari & Kari Auranen, 2025. "On the distribution of isometric log-ratio coordinates under extra-multinomial count data," Statistical Papers, Springer, vol. 66(5), pages 1-30, August.
    • Handle: RePEc:spr:stpapr:v:66:y:2025:i:5:d:10.1007_s00362-025-01732-8
    DOI: 10.1007/s00362-025-01732-8
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    References listed on IDEAS

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    1. Jane Fry & Tim Fry & Keith McLaren, 2000. "Compositional data analysis and zeros in micro data," Applied Economics, Taylor & Francis Journals, vol. 32(8), pages 953-959.
    2. Haixiang Zhang & Jun Chen & Zhigang Li & Lei Liu, 2021. "Testing for Mediation Effect with Application to Human Microbiome Data," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 13(2), pages 313-328, July.
    3. Fan Xia & Jun Chen & Wing Kam Fung & Hongzhe Li, 2013. "A Logistic Normal Multinomial Regression Model for Microbiome Compositional Data Analysis," Biometrics, The International Biometric Society, vol. 69(4), pages 1053-1063, December.
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