Generalized, Partial and Canonical Correlation Coefficients

We use a simple example to show that Pearson’s correlation matrix R can underestimate the true dependence between two variables when nonlinearities are present by as much as 83%, compared to the newer and easy to compute $$R^*$$ R ∗ in Vinod (Commun Statist Simul Comput 46(6):4513–4534, 2017, https://doi.org/10.1080/03610918.2015.1122048 ). We include intuitive expository discussion of nonparametric kernel methods needed by $$R^*$$ R ∗ with graphs and examples. We illustrate how partial correlation coefficients based on R can underestimate the nonlinear effect of a confounding variable, compared to those from the newer $$R^*$$ R ∗ . This paper develops an entirely new generalization of Hotelling’s canonical correlations based on nonlinear nonparametric pairwise dependencies of $$R^*$$ R ∗ . An example illustrates how traditional methods can underestimate the joint dependence by 266%.