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The chi-square standardization, combined with Box-Cox transformation, is a valid alternative to transforming to logratios in compositional data analysis

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Abstract

The approach to analysing compositional data with a fixed sum constraint has been dominated by the use of logratio transformations, to ensure exact subcompositional coherence and, in some situations, exact isometry as well. A problem with this approach is that data zeros, found in most applications, have to be replaced to permit the logarithmic transformation. A simpler approach is to use the chi-square standardization that is inherent in correspondence analysis. Combined with the Box-Cox power transformation, this standardization defines chi-square distances that tend to logratio distances for strictly positive data as the power parameter tends to zero, and can thus be considered equivalent to transforming to logratios. For data with zeros, a value of the power can be identified that brings the chi-square standardization as close as possible to transforming by logratios, without having to substitute the zeros. Especially in the field of high-dimensional "omics" data, this alternative presents such a high level of coherence and isometry as to be a valid, and much simpler, approach to the analysis of compositional data.

Suggested Citation

  • Michael Greenacre, 2023. "The chi-square standardization, combined with Box-Cox transformation, is a valid alternative to transforming to logratios in compositional data analysis," Economics Working Papers 1857, Department of Economics and Business, Universitat Pompeu Fabra.
    • Handle: RePEc:upf:upfgen:1857
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    1. Greenacre, Michael, 2009. "Power transformations in correspondence analysis," Computational Statistics & Data Analysis, Elsevier, vol. 53(8), pages 3107-3116, June.
    2. W. J. Krzanowski, 1987. "Selection of Variables to Preserve Multivariate Data Structure, Using Principal Components," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 36(1), pages 22-33, March.
    3. John Aitchison & Michael Greenacre, 2002. "Biplots of compositional data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 51(4), pages 375-392, October.
    4. Michael Greenacre, 2008. "Measuring subcompositional incoherence," Economics Working Papers 1106, Department of Economics and Business, Universitat Pompeu Fabra, revised Jan 2011.
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    More about this item

    Keywords

    Box-Cox transformation; chi-square distance; correspondence analysis; isometry; logratios; Procrustes analysis; subcompositional coherence;
    All these keywords.

    JEL classification:

    • C19 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Other
    • C88 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Other Computer Software

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