Large Sample Estimators for Standard Errors of Functions of Correlation Coefficients

Standard errors of estimators that are functions of correlation coefficients are shown to be quite dif ferent in magnitude than standard errors of the ini tial correlations. A general large-sample methodo logy, based upon Taylor series expansions and asymptotic correlational results, is developed for the computation of such standard errors. Three ex emplary analyses are conducted on a correction for attenuation, a correction for range restriction, and an indirect effect in path analysis. Derived for mulae are consistent with several previously pro posed estimators and provide excellent approxima tions to the standard errors obtained in computer simulations, even for moderate sample size (n = 100). It is shown that functions of correlations can be considerably more variable than product-mo ment correlations. Additionally, appropriate hy pothesis tests are derived for these corrected coeffi cients and the indirect effect. It is shown that in the range restriction situation, the appropriate hypothe sis test based on the corrected coefficient is asymp totically more powerful than the test utilizing the uncorrected coefficient. Bias is also discussed as a by-product of the methodology.