Measuring Subcompositional Incoherence
21 Pages Posted: 19 Oct 2008
Date Written: August 16, 2008
Subcompositional coherence is a fundamental property of Aitchison's approach to compositional data analysis, and is the principal justification for using ratios of components. We maintain, however, that lack of subcompositional coherence, that is incoherence, can be measured in an attempt to evaluate whether any given technique is close enough, for all practical purposes, to being subcompositionally coherent. This opens up the field to alternative methods, which might be better suited to cope with problems such as data zeros and outliers, while being only slightly incoherent. The measure that we propose is based on the distance measure between components. We show that the two-part subcompositions, which are the most sensitive to subcompositional incoherence, can be used to establish a distance matrix which can be directly compared with the pairwise distances in the full composition. The closeness of these two matrices can be quantified using a stress measure that is common in multidimensional scaling, providing a measure of subcompositional incoherence. Furthermore, we strongly advocate introducing weights into this measure, where rarer components are weighted proportionally less than more abundant components. The approach is illustrated using power-transformed correspondence analysis, which has already been shown to converge to logratio analysis as the power transform tends to zero.
Keywords: Chi-square distance, correspondence analysis, logratio distance, multidimensional scaling, stress, subcompositional coherence
JEL Classification: C19, C88
Suggested Citation: Suggested Citation