Abstract
The development of John Aitchison’s approach to compositional data analysis is followed since his paper read to the Royal Statistical Society in 1982. Aitchison’s logratio approach, which was proposed to solve the problematic aspects of working with data with a fixed-sum constraint, is summarized and reappraised. It is maintained that the properties on which this approach was originally built, the main one being subcompositional coherence, are not required to be satisfied exactly—quasi-coherence is sufficient, that is near enough to being coherent for all practical purposes. This opens up the field to using simpler data transformations, such as power transformations, that permit zero values in the data. The additional property of exact isometry, which was subsequently introduced and not in Aitchison’s original conception, imposed the use of isometric logratio transformations, but these are complicated and problematic to interpret, involving ratios of geometric means. If this property is regarded as important in certain analytical contexts, for example, unsupervised learning, it can be relaxed by showing that regular pairwise logratios, as well as the alternative quasi-coherent transformations, can also be quasi-isometric, meaning they are close enough to exact isometry for all practical purposes. It is concluded that the isometric and related logratio transformations such as pivot logratios are not a prerequisite for good practice, although many authors insist on their obligatory use. This conclusion is fully supported here by case studies in geochemistry and in genomics, where the good performance is demonstrated of pairwise logratios, as originally proposed by Aitchison, or Box–Cox power transforms of the original compositions where no zero replacements are necessary.
Acknowledgments
We acknowledge the support of the Geological Survey of Northern Ireland for the provision of the Tellus geochemical data set used in this study. The Tellus data is supplied under an open government license. Furthermore, we thank the journal editor and the associate editor for their very professional and streamlined treatment of our paper, as well as the reviewers for their role in improving this paper across two revisions. At the same time, we would like to pay tribute to John Aitchison and acknowledge the foundations of CoDA that he laid, starting in the early 1980s. We believe that this reappraisal of the field will contribute to improving the practice of CoDA.
Citation
Michael Greenacre. Eric Grunsky. John Bacon-Shone. Ionas Erb. Thomas Quinn. "Aitchison’s Compositional Data Analysis 40 Years on: A Reappraisal." Statist. Sci. 38 (3) 386 - 410, August 2023. https://doi.org/10.1214/22-STS880
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