The chiPower transformation: a valid alternative to logratio transformations in compositional data analysis

TL;DR

The paper introduces the chiPower transformation as a zero-tolerant, coherent alternative to logratio transformations for compositional data analysis, enabling improved analysis and prediction without zero replacement.

Contribution

It proposes the chiPower transformation, combining chi-square distance and Box-Cox power transformation, as a valid alternative to logratio methods that handle zeros naturally.

Findings

The chiPower transformation approximates logratio distances for positive data.

It effectively handles zeros without data substitution.

It improves predictive accuracy in supervised models.

Abstract

The approach to analysing compositional data 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 allow the logarithmic transformation. An alternative new approach, called the `chiPower' transformation, which allows data zeros, is to combine the standardization inherent in the chi-square distance in correspondence analysis, with the essential elements of the Box-Cox power transformation. The chiPower transformation is justified because it} defines between-sample distances that tend to logratio distances for strictly positive data as the power parameter tends to zero, and are then equivalent to transforming to logratios. For data with zeros, a value of the power can be identified that brings the chiPower transformation as close as possible to a logratio transformation, without having to substitute the zeros. Especially in the area of high-dimensional data, this alternative approach can present such a high level of coherence and isometry as to be a valid approach to the analysis of compositional data. Furthermore, in a supervised learning context, if the compositional variables serve as predictors of a response in a modelling framework, for example generalized linear models, then the power can be used as a tuning parameter in optimizing the accuracy of prediction through cross-validation. The chiPower-transformed variables have a straightforward interpretation, since they are each identified with single compositional parts, not ratios.