Partial least squares regression with compositional response variables and covariates.

The common approach for regression analysis with compositional variables is to express compositions in log-ratio coordinates (coefficients) and then perform standard statistical processing in real space. Similar to working in real space, the problem is that the standard least squares regression fails when the number of parts of all compositional covariates is higher than the number of observations. The aim of this study is to analyze in detail the partial least squares (PLS) regression which can deal with this problem. In this paper, we focus on the PLS regression between more than one compositional response variable and more than one compositional covariate. First, we give the PLS regression model with log-ratio coordinates of compositional variables, then we express the PLS model directly in the simplex. We also prove that the PLS model is invariant under the change of coordinate system, such as the ilr coordinates with a different contrast matrix or the clr coefficients. Moreover, we give the estimation and inference for parameters in PLS model. Finally, the PLS model with clr coefficients is used to analyze the relationship between the chemical metabolites of Astragali Radix and the plasma metabolites of rat after giving Astragali Radix.