Extension and significance testing of Variable Importance in Projection (VIP) indices in Partial Least Squares regression and Principal Components Analysis

Within the context of Partial Least Squares (PLS) regression, Variable Importance in the Projection (VIP) index is extensively used to highlight the relative importance of a given variable in predicting the response. Very often, it is used as a means to select relevant variables by retaining those variables with a VIP value greater than one. As a starting point of this paper, the VIP index is expressed in matrix notation and extended to (i) a subset of variables instead of one variable at a time and (ii) the framework of Principal Component Analysis (PCA). In a subsequent stage, it is proposed to assess the significance of the VIP values of a given subset of variables based on a permutation test. A fast variable selection procedure is then proposed, taking advantage of the first extension. The theory is illustrated using multiblock spectral and omics data.