Use of Empirical Estimates of Shrinkage in Multiple Regression: A Caution

Abstract

Empirical techniques to estimate the shrinkage of the sample R2 have been advocated as alternatives to analytical formulae. Although such techniques may be appropriate for estimating the coefficient of cross-validation, they do not provide accurate estimates of the population multiple correlation. The accuracy of four empirical techniques (simple cross-validation, multi-cross-validation, jackknife, and bootstrap) were investigated in a Monte Carlo study. Random samples of size 20 to 200 were drawn from a pseudopopulation of actual field data. Regression models were investigated with population coefficients of determination ranging from .04 to .50 and with numbers of regressors ranging from 2 to 10. Substantial statistical bias was evident when the shrunken R² values were used to estimate the population squared multiple correlation. Researchers are advised to avoid the empirical techniques when the parameter of interest is the population coefficient of determination rather than the coefficient of cross-validation.