Share-ratio interpretations of compositional regression models

The interpretation of regression models with compositional vectors as dependent and/or independent variables has been approached from different perspectives. The first approaches that appeared in the literature are done in coordinate space after some log-ratio transformation of the share vectors. Considering the fact that these models are non-linear with respect to classical operations of the real space, another approach has been proposed based on infinitesimal increments or derivatives understood in a simplex sense, leading to elasticities or semi-elasticities interpretations in the original space that have the advantage of being independent of any log-ratio transformations. After briefly reviewing these two points of view, we show that some functions of elasticities or semi-elasticities are constant throughout the sample observations, which makes them natural parameters for interpreting CoDa models. We derive approximations of share ratio variations and link them to these parameters leading to transformatio-free interpretations in the original shares space. We use a real data set on the French presidential election to illustrate each type of interpretation in detail.