Multiple linear regression on canonical correlation variables

When the explanatory variables of a linear model are split into two groups, two notions of collinearity are defined: a collinearity between the variables of each group, of which the mean is called residual collinearity, and a collinearity between the two groups called explained collinearity. Canonical correlation analysis provides information about the collinearity: large canonical correlation coefficients correspond to some small eigenvalues and eigenvectors of the correlation matrix and characterise the explained collinearity. Other small eigenvalues of this matrix correspond to the residual collinearity. A selection of predictors can be performed from the canonical correlation variables, according to their partial correlation coefficient with the explained variable. In the proposed application, the results obtained by the selection of canonical variables are better than those given by classical regression and by principal component regression.