Title: Univariate analysis of compositional data using weighted balances
Authors: Karel Hron - Palacky University (Czech Republic) [presenting]
Peter Filzmoser - Vienna University of Technology (Austria)
Alzbeta Gardlo - Palacky University (Czech Republic)
Abstract: Compositional data, observations carrying exclusively relative information (with units like percentages, mg/kg, mg/l, etc.), have specific properties that are not compatible with the Euclidean geometry requirement of most standard statistical methods. In order to represent compositional data in the usual Euclidean geometry, they need to be expressed in orthonormal coordinates prior to statistical processing. As it is not possible to construct standard Cartesian coordinates for compositions that assign a coordinate for each of the parts separately, a choice of interpretable orthonormal coordinates is of particular interest. Although recent experiences show clear advantages of such coordinates where the first coordinate aggregates information from log-ratios for a particular compositional part of interest, their usefulness is limited if there are distortions like rounding errors or other data ``problems'' in the involved parts. The purpose is to introduce a ``robust'' (weighted) version of these coordinates, called weighted balances, where the remaining parts (with respect to the part of interest) in the first coordinate are weighted in a way that is relevant to the aims of the statistical analysis. Such weights can be, e.g., derived according to quality assessment analysis and elements of classical/robust variation matrix of compositions. Methodological outputs are accompanied by a real-world example from metabolomics.