Statistical loadings and latent significance simplify and improve interpretation of multivariate projection models

Abstract

Multivariate projection methods are unique in being both multivariable by combining many variables into stronger predictive features (latent variables), and multivariate for being able to model systematic variation both related and orthogonal to an observed response. Orthogonal partial least squares (OPLS) is a versatile multivariate projection method for analysis of correlation, discrimination and effect changes. However, currently OPLS is not fully using its multivariate potential since orthogonal systematic variation is not considered in model interpretation, resulting in univariate interpretation of variable significance. We present a strategy for improved interpretation of OPLS models based upon a post-hoc linear regression analysis that can be used with or without the orthogonal OPLS score(s) as a covariate to make the interpretation multivariate or univariate respectively. By selecting the observed response y or estimated response yhat as a one of the factors in the linear regression the results are related to either of the OPLS loadings w or p. Furthermore, converting the OPLS loading values to statistical t-values creates a direct link to statistical significance. Finally, by applying three different Boolean loadings W, P and W{wedge}P variable significance can be summarized based on three criteria. W and P reveal if the values in w or p respectively are outside the statistical limits with W{wedge}P being the logical conjunction of W and P (significant if outside limits in both W and P). Two examples are used to verify the proposed strategy. First, a synthetic example, simulating a mix of mass spectra, and second a clinical metabolomics study of a dietary intervention. In the simulated example we show that multivariate interpretation gives higher accuracy for estimation of true differences, mainly due to higher true positive rate. Furthermore, we highlight how application of W{wedge}P for summarizing variable significance leads to higher accuracy. For the metabolomics example, we show that a more detailed interpretation, i.e. larger number of significant metabolites of relevance, is obtained using the multivariate interpretation. In summary, the suggested strategy provides means for facilitated interpretation of OPLS models, beyond univariate statistics, and offers a multivariate tool for discovery of biomarker patterns, i.e. latent biomarkers.