# Correcting the Correction: A Revised Formula to Estimate Partial Correlations between True Scores

## Abstract

**:**

## 1. Introduction

## 2. Bohrnstedt’s Derivation

_{XZ}). Simply, the covariances and standard deviations that appear in Equation (2) become covariances and standard deviations of residuals between these same variables and the potential confounding variable (Z). Bohrnstedt, therefore, defines the corrected partial correlation as the ratio between the covariance of estimated residuals and the square root of the product of the residuals’ variances. These residuals, defined by relevant simple linear regressions, X-b

_{xz}Z and Y-b

_{YZ}Z, become parts of Equation (3).

_{xz}as ${{\displaystyle \rho}}_{XZ}{{\displaystyle \sigma}}_{X}/{{\displaystyle \rho}}_{Z{Z}^{\prime}}{{\displaystyle \sigma}}_{Z}$ and makes the comparable change for b

_{yz}. This manipulation, along with restatement of each covariance as the product of the relevant correlation coefficient and standard deviations (e.g., ${{\displaystyle \sigma}}_{XY}={{\displaystyle \sigma}}_{X}{{\displaystyle \sigma}}_{Y}{{\displaystyle \rho}}_{XY}$), produces

_{ZZ′}from the numerator and denominator, producing Equation (10).

_{ZZ′}appeared in all terms within the numerators of the general numerator and the general denominator, then he would have cancelled correctly. However, because only the first of the terms in these numerators contained ρ

_{ZZ′}, such factoring and, consequently, such cancelling should not have occurred. Without realizing the flaws in Equation (10), Bohrnstedt continued with his derivation. He added components within the numerator and the denominator and then cancelled reliabilities from each. The equations

## 3. Correcting the Derivation

## 4. Example

_{XY}= 0.811, ρ = 0.395, and ρ

_{YZ}= 0.550, and the reliability value of ρ

_{XX′}= 0.650 from these data, fit into Equation (14). Then, a partial correlation coefficient corrected for attenuation of 0.7831 emerges through arithmetic simplification.

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Carrol, R.J.; Ruppert, D.; Stafanski, L.A. Measurement Error in Nonlinear Models; Chapman and Hall: New York, NY, USA, 1995. [Google Scholar]
- Spearman, C. The proof and measurement of association between two things. Am. J. Psychol.
**1904**, 15, 72–101. [Google Scholar] [CrossRef] - Hakstian, A.R.; Schroeder, M.L.; Rogers, W.T. Inferential procedures for correlation coefficients corrected for attenuation. Psychometrika
**1988**, 53, 27–43. [Google Scholar] [CrossRef] - Bohrnstedt, G.W. Observations on the measurement of change. Sociol. Methodol.
**1969**, 1, 113–133. [Google Scholar] [CrossRef] - Carter, A.H. The estimation and comparison of residual regressions where there are two or more related sets of observations. Biometrika
**1949**, 36, 26–46. [Google Scholar] [CrossRef] - Wright, E.M.; Manning, W.H.; Dubois, P.H. Determinants in multivariate correlation. J. Exp.
**1959**, 27, 195–202. [Google Scholar] [CrossRef] - Wetcher-Hendricks, D. Adjustments to the Correction for Attenuation. Psychol. Methods
**2006**, 11, 207–215. [Google Scholar] [CrossRef] [PubMed] - Gustafson, P. Measurement Error and Misclassifications in Statistics and Epidemiology: Impacts and Bayesian Adjustments; Chapman and Hall/CRC Press: Boca Raton, FL, USA, 2003. [Google Scholar]

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**MDPI and ACS Style**

Wetcher-Hendricks, D.
Correcting the Correction: A Revised Formula to Estimate Partial Correlations between True Scores. *Psych* **2021**, *3*, 19-24.
https://doi.org/10.3390/psych3010003

**AMA Style**

Wetcher-Hendricks D.
Correcting the Correction: A Revised Formula to Estimate Partial Correlations between True Scores. *Psych*. 2021; 3(1):19-24.
https://doi.org/10.3390/psych3010003

**Chicago/Turabian Style**

Wetcher-Hendricks, Debra.
2021. "Correcting the Correction: A Revised Formula to Estimate Partial Correlations between True Scores" *Psych* 3, no. 1: 19-24.
https://doi.org/10.3390/psych3010003