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.
Bobko, Philip; Schemmer, F. Mark.
Eigenvalue shrinkage in principal components based factor analysis.
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