Morais, Joanna and Thomas-Agnan, Christine (2021) Impact of covariates in compositional models and simplicial derivatives. Austrian Journal of Statistics, vol. 50 (n° 2). pp. 1-15.

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Abstract

In the framework of Compositional Data Analysis, vectors carrying relative information, also called compositional vectors, can appear in regression models either as dependent
or as explanatory variables. In some situations, they can be on both sides of the regression
equation. Measuring the marginal impacts of covariates in these types of models is not
straightforward since a change in one component of a closed composition automatically
affects the rest of the composition.
Previous work by the authors has shown how to measure, compute and interpret these
marginal impacts in the case of linear regression models with compositions on both sides
of the equation. The resulting natural interpretation is in terms of an elasticity, a quantity
commonly used in econometrics and marketing applications. They also demonstrate the
link between these elasticities and simplicial derivatives.
The aim of this contribution is to extend these results to other situations, namely
when the compositional vector is on a single side of the regression equation. In these
cases, the marginal impact is related to a semi-elasticity and also linked to some simplicial derivative. Moreover we consider the possibility that a total variable is used as an
explanatory variable, with several possible interpretations of this total and we derive the
elasticity formulas in that case.