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 estimated in 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.
Definition
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 geometric mean of the separation indices of the two sets of estimates. Expressed in terms of Classical test theory, the correlation is divided by the geometric mean of the reliability coefficients of two tests.
Given two random variables
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
Derivation of the formula
Let
where
The correlation between two sets of estimates is
which, assuming the errors are uncorrelated with each other and with the estimates, gives
where
where the mean squared standard error of person estimate gives an estimate of the variance of the errors,