Eigenvalue Shrinkage in Principal Components Based Factor Analysis

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Abstract
The concept of shrinkage, as (1) a statistical phe nomenon of estimator bias, and (2) a reduction in ex plained variance resulting from cross-validation, is ex plored 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 vari ables in the correlation matrix, the ordinal root posi tion, the population eigenstructure, and the choice of principal components analysis or principal factors analysis. Hypotheses relating these specific indepen dent variables to the magnitude of shrinkage were tested by means of a monte carlo simulation. In par ticular, the independent variable of population eigen structure 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 vari ables factored. These equations are a valuable tool that allows researchers to accurately predict eigenvalue shrinkage based on available sample information.