The role of perturbation in compositional data analysis

Abstract

In standard multivariate statistical analysis, common hypotheses of interest concern changes in mean vectors and subvectors. In compositional data analysis it is now well established that compositional change is most readily described in terms of the simplicial operation of perturbation and that subcompositions replace the marginal concept of subvectors. Against the background of two motivating experimental studies in the food industry, involving the compositions of cow’s milk and chicken carcasses, this paper emphasizes the importance of recognizing this fundamental operation of change in the associated simplex sample space. Well-defined hypotheses about the nature of any compositional effect can be expressed, for example, in terms of perturbation values and subcompositional stability and testing procedures developed. These procedures are applied to lattices of such hypotheses in the two practical situations. We identify the two problems as being the counterpart of the analysis of paired comparison or split plot experiments and of separate sample comparative experiments in the jargon of standard multivariate analysis.