The first canonical correlation is at least as large as the multiple correlation between the groups and any of the original variables. If the original variables have high within-group correlations, the first canonical correlation can be large even if all the multiple correlations are small. In other words, the first canonical variable can show substantial differences between the classes, even if none of the original variables do. The variables should have an approximate multivariate normal distribution within each class, with a common covariance matrix in order for the probability levels to be valid. Canonical discriminant analysis is equivalent to canonical correlation analysis between the quantitative variables and a set of dummy variables coded from the class variable.
Compute class means on the transformed variables.
Perform a principal component analysis on the means, weighting each mean by the number of observations in the class. Back-transform the principal components into the space of the original variables, obtaining the canonical variables. An interesting property of the canonical variables is that they are uncorrelated whether the correlation is calculated from the total sample or from the pooled within-class correlations. The canonical coefficients are not orthogonal, however, so the canonical variables do not represent perpendicular directions through the space of the original variables. All rights reserved.