Compositional Data as a Methodological Concept

Abstract

Compositional data were defined traditionally as constrained data, like proportions or percentages, with a fixed constant sum constraint (1 or 100, respectively). Nevertheless, from a practical perspective it is much more intuitive to consider them as observations carrying relative information, where proportions stand just for one possible representation. Equivalently, all relevant information in compositional data is contained in ratios between components (parts). According to this broader definition, the decision whether the data at hand are compositional or not depends primarily on the purpose of the analysis, i.e. if the relative structure of the compositional parts is of interest or not. As a consequence, the use of standard statistical methods for the analysis of compositional data that obey specific geometrical properties leads inevitably to biased results. A reasonable way out is to set up an algebraic-geometrical structure that follows the principles of compositional data analysis (scale invariance, permutation invariance, and subcompositional coherence). Nowadays, this is called the Aitchison geometry and it enables to express compositional data in interpretable real coordinates, where standard statistical procedures can directly be applied. These coordinates are formed by logratios of pairs of compositional parts and their aggregations: the logratio methodology was born.