Generalized correlation loadings: Extending correlation loadings to congruence and to multi-way models

Correlation loadings are commonly used in bi-linear models to highlight relationships between the original variables of a dataset and the latent variables resulting from the model. This principle is suggested to be renamed congruence loading and is extended to multi-way models. Congruence is proposed as a more meaningful parameter covering the case of centered data, as originally proposed with the correlation loading principle, but also the cases where data are not centered or centered across a different mode. The idea of congruence loadings is also extended to multi-way models, i.e. parallel factor analysis (PARAFAC), Tucker, and N-way partial least squares (N-PLS). In this paper, the method is applied to three-way models where the scores and/or loadings are not orthogonal. Three real datasets are considered to highlight some applications of congruence loadings. In the first example, a three-way sensory profiling dataset (assessors × products × attributes) is considered to illustrate the use of congruence loadings in exploratory data analysis with principal component analysis (PCA) and PARAFAC models. The second example concerns the 1H NMR spectroscopy of a typical metabonomic dataset and shows how congruence loadings of a PARAFAC model makes it easier to visualize minor features in the data. The last example illustrates the use of congruence loadings for variable selection in PARAFAC-based curve-resolution of fluorescence data.