Linear association in compositional data analysis

With compositional data, ordinary covariation indices, designed for real random variables, fail to describe dependence. There is a need for compositional alternatives to covariance and correlation. Based on the Euclidean structure of the simplex, called Aitchison geometry, compositional association is identified to a linear restriction of the sample space when a log-contrast is constant. In order to simplify interpretation, a sparse and simple version of compositional association is defined in terms of balances which are constant across the sample. It is called b-association. This kind of association of compositional variables is extended to association between groups of compositional variables. In practice, exact b-association seldom occurs, and measures of degree of b-association are reviewed based on those previously proposed. Also, some techniques for testing b-association are studied. These techniques are applied to available oral microbiome data to illustrate both their advantages and difficulties. Both testing and measurements of b-association appear to be quite sensitive to heterogeneities in the studied populations and to outliers ​
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