Compositional Data Analysis: Where Are We and Where Should We Be Heading?

Abstract

We take stock of the present position of compositional data analysis, of what has been achieved in the last 20 years, and then make suggestions as to what may be sensible avenues of future research. We take an uncompromisingly applied mathematical view, that the challenge of solving practical problems should motivate our theoretical research; and that any new theory should be thoroughly investigated to see if it may provide answers to previously abandoned practical considerations. Indeed a main theme of this lecture will be to demonstrate this applied mathematical approach by a number of challenging examples.

Differences in Defensive Volatiles of the Forked Fungus Beetle, Bolitotherus cornutus, Living on Two Species of Fungus

“…Accordingly, a centered log ratio transformation (Aitchison 1984; Pawlosky-Glahn and Egozcue 2006) was employed prior to principal components analysis and its canonical equivalent, redundancy analysis, performed using CANOCO 4.5 (ter Braak and Smilauer 2002). In the case where a component was not detected, "0.00000001" was used in place of "0" as the relative peak area for that component in the calculation of centered log ratios (Aitchison and Egozcue 2005).…”

Thurstonian Scaling of Compositional Questionnaire Data

“…The advantage of the log-ratio transformation is that it places compositional data in the unconstrained "multivariate real space, opening up all available standard multivariate techniques" (Aitchison & Egozcue, 2005;p. 831).…”

Toward an Efficient and Integrative Analysis of Limited-Choice Behavioral Experiments: Figure 1.

“…These have been developed largely in fields like geology and chemistry, but also in the social sciences, where compositional data are common (Aitchison and Egozcue, 2005;Pawlowsky-Glahn and Egozcue, 2006;Pawlowsky-Glahn and Buccianti, 2011 and further references contained therein). We may readily tap and adapt these methods for behavioral analysis, since they are conveniently implemented in the freely available R computational environment (van den Boogaart and Tolosana-Delgado, 2008; Hron et al, 2010;Rizzo and Szekely, 2010;R Development Core Team, 2011).…”

“…This has been pioneered and popularized by Aitchison and Egozcue (2005; and the extensive bibliography therein). The rationale and the formalism of this approach, its technical details, and its benefits have been extensively described (for review, see Billheimer et al, 2001;Weltje, 2002;Egozcue et al, 2003;Aitchison and Egozcue, 2005;Pawlowsky-Glahn and Buccianti, 2011).…”