Statistical Analysis of Compositional Data

Compositional data refer to data sets where not the values as such but rather the ratios between the variables contain the relevant information. Typical examples are chemical element concentrations, or any data where the unit reflects relative information (mg/kg, %, ppm, etc.). Usually, an inclease of the value in one variable has some effect on the values of other variables.

Compositional data are represented in the Aitchison geometry on the simplex, and for applying statistical methods designed for the Euclidean geometry they need to be transformed first. The isometric logratio (ilr) transformation has the best geometrical properties, but usually the results are difficult to interpret because the ilr coordinates are formed by non-linear combinations of the original variables. We show for different multivariate statistical methods how the ilr transformation can be sucessfully used for interpretation. Based on real data examples we compare results from a standard approach and from a compositional data approach.