A Rotation Test to Verify Latent Structure

In multivariate regression models we have the opportunity to look for hidden structure unrelated to the observed predictors. However, when one fits a model involving such latent variables it is important to be able to tell if the structure is real, or just an artifact of correlation in the regression errors. We develop a new statistical test based on random rotations for verifying the existence of latent variables. The rotations are carefully constructed to rotate orthogonally to the column space of the regression model. We find that only non-Gaussian latent variables are detectable, a finding that parallels a well known phenomenon in independent components analysis. We base our test on a measure of non-Gaussianity in the histogram of the principal eigenvector components instead of on the eigenvalue. The method finds and verifies some latent dichotomies in the microarray data from the AGEMAP consortium.