A study on Q and T2 in Partial Least Squares regression models

ABSTRACT A distinctive feature of Partial Least Squares (PLS) as a regression method is the definition of a region in the model predictor space, known as the PLSbox, such that the model should not be applied to objects outside this region. In multivariate regressions where the set of original variables is reduced to a smaller set of latent variables, typically in multivariate data generated by modern instrumentation in chemical analysis, this represents a notable advancement in identifying outlier data. Constructing the PLSbox requires assuming a probability distribution for the Q and T 2 statistics and computing a critical value for each at a desired confidence level. Despite its relevance, the impact of the chosen probability distribution has not been systematically examined. Parametric distributions like Snedecor’s F and chi-square have been widely accepted, even though their validity relies on the multivariate normality of the data, an assumption that cannot be made a priori . This work evaluates the effect of the usual parametric distributions for Q and T 2 on the PLS model in four different case studies, revealing a lack-of-fit issue in many cases, and what is more important, a notable impact on the PLS model structure. This problem disappears when a Gaussian kernel, which is distribution-free, is employed. Furthermore, the study highlights the impact of the interaction between the confidence levels chosen for Q and T 2 on the coefficient of determination of the accuracy line, suggesting that these levels should be chosen separately.