Pairwise share ratio interpretations of compositional regression models

The interpretation of regression models with compositional vectors as response and/or explanatory variables has been approached from different perspectives. The initial approaches are performed in coordinate space subsequent to applying a log-ratio transformation to the compositional vectors. Given that these models exhibit non-linearity concerning classical operations within real space, an alternative approach has been proposed. This approach relies on infinitesimal increments or derivatives, interpreted within a simplex framework. Consequently, it offers interpretations of elasticities or semi-elasticities in the original space of shares which are independent of any log-ratio transformations. Some functions of these elasticities or semi-elasticities turn out to be constant throughout the sample observations, making them natural parameters for interpreting CoDa models. These parameters are linked to relative variations of pairwise share ratios of the response and/or of the explanatory variables. Approximations of share ratio variations are derived and linked to these natural parameters. A real dataset on the French presidential election is utilized to illustrate each type of interpretation in detail.