Exploring the Relationship between two Compositions using Canonical Correlation Analys

Abstract

The aim of this article is to describe a method for relating two compositions which combines compositional data analysis and canonical correlation analysis (CCA), and to examine its main statistical properties. We use additive log-ratio (alr) transformation on both compositions and apply standard CCA to the transformed data. We show that canonical variates are themselves log-ratios and log-contrasts. The first pair of canonical variates can be interpreted as the log-contrast of a composition that has the maximum correlation with a log-contrast of the other composition. The second pair can be interpreted as the log-contrast of a composition that has the maximum correlation with a log-contrast of the other composition, under the restriction that they are uncorrelated with the first pair, and so on. Using properties from changes of basis, we prove that both canonical correlations and canonical variates are invariant to the choice of divisors in alr transformation. We show how to implement the analysis and interpret the results by means of an illustration from the social sciences field using data from Kolb's Learning Style Inventory and Boyatzis' Philosophical Orientation Questionnaire, which distribute a fixed total score among several learning modes and philosophical orientations ; The authors would like to acknowledge the support provided by Spanish Health Ministry Grant CB06/02/1002 funding the research group "Epidemiology and Public Health (CIBERESP)"; Catalan Autonomous Government Consolidated Research Group Grants 2014SGR551 and 2014SGR582 funding the research groups "Compositional and Spatial Data Analysis (COSDA)" and "Leadership Development Research Centre (GLEAD)"; Spanish Economy and Competitiveness Ministry grants MINECO/FEDER-EU MTM2015-65016-C2-1-R and EDU2015-68610-R funding the projects "Compositional Data Analysis and RElated meThOdS (CODA-RETOS)" and "Assessing Individual and Team Entrepreneurial Potential"; and University of Girona grants MPCUdG2016/069 and MPCUdG2016/098