Eigenvalue shrinkage in principal components based factor analysis

Abstract

The concept of shrinkage, as (1) a statistical phenomenon of estimator bias, and (2) a reduction in explained variance resulting from cross-validation, is explored for statistics based on sample eigenvalues. Analytic solutions and previous research imply that the magnitude of eigenvalue shrinkage is a function of the type of shrinkage, sample size, the number of variables in the correlation matrix, the ordinal root position, the population eigenstructure, and the choice of principal components analysis or principal factors analysis. Hypotheses relating these specific independent variables to the magnitude of shrinkage were tested by means of a monte carlo simulation. In particular, the independent variable of population eigenstructure is shown to have an important effect on shrinkage. Finally, regression equations are derived that describe the linear relation of population and cross-validated eigenvalues to the original eigenvalues, sample size, ordinal position, and the number of variables factored. These equations are a valuable tool that allows researchers to accurately predict eigenvalue shrinkage based on available sample information.