In many practical applications, samples will already be grouped into subpopulations (for instance, in medical genetics there are often two populations: cases and controls). It is natural to want to test if our recovered eigenvectors reflect differences among the labeled subpopulations. We therefore fix some eigenvector, and can regard each individual as associated with the corresponding coordinate of the eigenvector. We want to test if the means of these coordinate values in each subpopulation differ significantly. Our motivation is firstly that this is a powerful check on the validity of our (unsupervised) Tracy–Widom statistics, and secondly that the supervised analysis helps in interpretation of the recovered axes of variation.
