Determining the Number of Components in PLS Regression on Incomplete Data

Abstract: Partial least squares regression---or PLS---is a multivariate method in which models are estimated using either the SIMPLS or NIPALS algorithm. PLS regression has been extensively used in applied research because of its effectiveness in analysing relationships between an outcome and one or several components. Note that the NIPALS algorithm is able to provide estimates on incomplete data. Selection of the number of components used to build a representative model in PLS regression is an important problem. However, how to deal with missing data when using PLS regression remains a matter of debate. Several approaches have been proposed in the literature, including the $Q^2$ criterion, and the AIC and BIC criteria. Here we study the behavior of the NIPALS algorithm when used to fit a PLS regression for various proportions of missing data and for different types of missingness. We compare criteria for selecting the number of components for a PLS regression on incomplete data and on imputed datasets using three imputation methods: multiple imputation by chained equations, k-nearest neighbor imputation, and singular value decomposition imputation. Various criteria were tested with different proportions of missing data (ranging from 5% to 50%) under different missingness assumptions. Q2-leave-one-out component selection methods gave more reliable results than AIC and BIC-based ones.