 Correction for attenuation

Correction for attenuation is a statistical procedure, due to Spearman (1904), to "rid a correlation coefficient from the weakening effect of measurement error" (Jensen, 1998), a phenomenon also known as regression dilution. In measurement and statistics, it is also called disattenuation. The correlation between two sets of parameters or measurements is the estimated a manner that accounts for measurement error contained within the estimates of those parameters.
Contents
Background
Correlations between parameters are diluted or weakened by measurement error. Disattenuation provides for a more accurate estimate of the correlation between the parameters by accounting for this effect.
Derivation of the formula
Let β and θ be the true values of two attributes of some person or statistical unit. These values are regarded as random variables by virtue of the statistical unit being selected randomly from some population. Let and be estimates of β and θ derived either directly by observationwitherror or from application of a measurement model, such as the Rasch model. Also, let
where and are the measurement errors associated with the estimates and .
The correlation between two sets of estimates is
which, assuming the errors are uncorrelated with each other and with the estimates, gives
where R_{β} is the separation index of the set of estimates of β, which is analogous to Cronbach's alpha; this is, in terms of Classical test theory, R_{β} is analogous to a reliability coefficient. Specifically, the separation index is given as follows:
where the mean squared standard error of person estimate gives an estimate of the variance of the errors, . The standard errors are normally produced as a byproduct of the estimation process (see Rasch model estimation).
The disattenuated estimate of the correlation between two sets of parameters or measures is therefore
That is, the disattenuated correlation is obtained by dividing the correlation between the estimates by the square root of the product of the separation indices of the two sets of estimates. Expressed in terms of Classical test theory, the correlation is divided by the square root of the product of the reliability coefficients of two tests.
Given two random variables X and Y, with correlation r_{xy}, and a known reliability for each variable, r_{xx} and r_{yy}, the correlation between X and Y corrected for attenuation is .
How well the variables are measured affects the correlation of X and Y. The correction for attenuation tells you what the correlation would be if you could measure X and Y with perfect reliability.
If X and Y are taken to be imperfect measurements of underlying variables X' and Y' with independent errors, then r_{x'y'} measures the true correlation between X' and Y'.
See also
References
 Jensen, A.R. (1998). The g Factor: The Science of Mental Ability Praeger, Connecticut, USA. ISBN 0275961036
 Spearman, C. (1904) "The Proof and Measurement of Association between Two Things". The American Journal of Psychology, 15 (1), 72–101 JSTOR 1412159
External links

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