Correction for Attenuation

Correction for attenuation (CA) is a method that allows researchers to estimate the relationship between two constructs as if they were measured perfectly reliably and free from random errors that occur in all observed measures. All research seeks to estimate the true relationship among constructs; because all measures of a construct contain random measurement error, the CA is especially important in order to estimate the relationships among constructs free from the effects of this error. It is called the CA because random measurement error attenuates, or makes smaller, the observed relationships between constructs. For correlations, the correction is as follows:

where δxy is the corrected correlation between variables x and y, rxy is the observed correlation between variables x and y, rxx is the reliability estimate for the x variable, and ryy is the reliability estimate for the y variable. For standardized mean differences, the CA is as follows:

where δxy is the corrected standardized mean difference, dxy is the observed standardized mean difference, and ryy is the reliability for the continuous variable. In both equations, the observed effect size (correlation or standardized mean difference) is placed in the numerator of the right half of the equation, and the square root of the reliability estimate(s) is (are) placed in the denominator of the right half of the equation. The outcome from the left half of the equation is the estimate of the relationship between perfectly reliable constructs. For example, using Equation 1, suppose the observed [Page 261]correlation between two variables is .25, the reliability for variable X is .70, and the reliability for variable Y is .80. The estimated true correlation between the two constructs is .25/(.70∗.80) = .33. This entry describes the properties and typical uses of CA, shows how the CA equation is derived, and discusses advanced applications for CA.