A two-group canonical variate analysis biplot for an optimal display of both means and cases

Canonical variate analysis (CVA) entails a two-sided eigenvalue decomposition. When the number of groups, J, is less than the number of variables, p, at most $$J-1$$ J - 1 eigenvalues are not exactly zero. A CVA biplot is the simultaneous display of the two entities: group means as points and variables as calibrated biplot axes. It follows that with two groups the group means can be exactly represented in a one-dimensional biplot but the individual samples are approximated. We define a criterion to measure the quality of representing the individual samples in a CVA biplot. Then, for the two-group case we propose an additional dimension for constructing an optimal two-dimensional CVA biplot. The proposed novel CVA biplot maintains the exact display of group means and biplot axes, but the individual sample points satisfy the optimality criterion in a unique simultaneous display of group means, calibrated biplot axes for the variables, and within group samples. Although our primary aim is to address two-group CVA, our proposal extends immediately to an optimal three-dimensional biplot when encountering the equally important case of comparing three groups in practice.