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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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.
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