A comparison of isometric and amalgamation logratio balances in compositional data analysis

The isometric logratio transformation, in the form of what has been called a "balance", has been promoted as a way to contrast two groups of parts in a compositional data set by forming ratios of their respective geometric means. This transformation has attractive theoretical properties and hence provides a useful reference, but geometric means are highly affected by parts with small relative values. When a comparison between two groups of parts is required in practical applications, such as the investigation and construction of models, while making use of substantive domain knowledge, it is demonstrated that the logratio of two amalgamations serves as an alternative, interpretable form of balance. A geochemical data set is considered, which has been analyzed previously by transforming to a set of isometric logratio balances. An alternative approach, using a reduced set of pairwise logratios of parts, optionally involving prescribed amalgamations, is very close to optimal in accounting for the variance in this compositional data set. These simpler transformations also have an exact back-transformation to the original parts. This approach highlights for this dataset which compositional parts are driving the data structure, using variables that are easy to interpret and that map well to research-driven objectives.