Large Sample Estimators for Standard Errors of Functions of Correlation Coefficients

Abstract

Standard errors of estimators that are functions of correlation coefficients are shown to be quite different in magnitude than standard errors of the initial correlations. A general large-sample methodology, based upon Taylor series expansions and asymptotic correlational results, is developed for the computation of such standard errors. Three exemplary analyses are conducted on a correction for attenuation, a correction for range restriction, and an indirect effect in path analysis. Derived formulae are consistent with several previously proposed estimators and provide excellent approximations 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-moment correlations. Additionally, appropriate hypothesis tests are derived for these corrected coefficients and the indirect effect. It is shown that in the range restriction situation, the appropriate hypothesis test based on the corrected coefficient is asymptotically more powerful than the test utilizing the uncorrected coefficient. Bias is also discussed as a by-product of the methodology.Many estimators in the social sciences are functions of Pearson product-moment correlation coefficients. For example, the correction of a validity coefficient for attenuation due to unreliability is a function of both the original correlation and a reliability estimate. Discussion of the corrected coefficient in its use as a parameter estimate usually revolves around its magnitude and/or practical significance. Therefore, some indication of the standard error of these corrected coefficients is crucial to the interpretability of the findings. That corrected coefficients may be greater than unity (cf. Karren, 1978) also supplies impetus to this investigation.The present paper presents a large-sample methodology for deriving estimates of the standard errors of functions of correlation coefficients. For example, it is shown below that correlations corrected for attenuation are considerably more variable than product-moment correlations. The APA Standards for Educational and Psychological Tests (American Psychological Association, 1974) indicates that &dquo;where correlation coefficients are corrected for attenuation or restriction in range, full information relevant to the correction should be presented&dquo; (Section E8.2.1.). Thus, the derivation of standard errors of these corrected coefficients would represent important progress towards &dquo;full information.&dquo;

Comparison of methods for constructing confidence intervals of standardized indirect effects

“…Most SEM packages, such as LIS-REL, Mx (Neale, Boker, Xie, & Maes, 2005), and Mplus, A correlational approach. Another approach to obtaining the standardized indirect effect is to derive it from the correlation coefficients among X, M, and Y (Bobko & Rieck, 1980 To apply Equation 13, we have to estimate the asymptotic variance of a correlation coefficient and the asymptotic covariance between two correlation coefficients (see, e.g., Olkin & Finn, 1995;Olkin & Siotani, 1976 A structural equation modeling approach. An SEM approach may also be used to obtain the standardized indirect effect and its CI (see Figure 2).…”

“…A more direct measure of the effect size is the standardized indirect effect calculated from the standardized scores (see, e.g., Bobko & Rieck, 1980;Preacher & Hayes, 2008b;Raykov, Brennan, Reinhardt, & Horowitz, 2008). There are several advantages to using a standardized indirect effect.…”

A comparison of methods to test mediation and other intervening variable effects.

“…Two other studies included a small simulation to check standard error formulas for specific methods (Allison, 1995;Bobko & Rieck, 1980). No published study to date has compared all of the available methods within the three general approaches.…”

Standard errors and confidence intervals for correlations corrected for indirect range restriction: A simulation study comparing analytic and bootstrap methods

“…Chan and Chan (2004) developed a bootstrap procedure for estimating SEs and CIs for correlations corrected for DRR. They compared its empirical performance with the performance of two formulas for the SEs (Bobko & Rieck, 1980;Fisher, 1954) and the respective CIs of the correlations in simulation studies involving both normal and nonnormal data. Li, Chan, and Cui (2011) used a bootstrap procedure for estimating the SEs and CIs for correlations corrected for IRR.…”

“…A formula for the standard error of correlations corrected for IRR A formula for a large-sample estimate of the SE of correlations corrected for IRR was derived on the basis of the same principle used to derive the large-sample estimate of the SE of correlations corrected for DRR (Bobko & Rieck, 1980). It should be noted that a similar formula for IRR appeared in Allen and Dunbar (1990).…”