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Correction for attenuation

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Correction for attenuation is a statistical procedure, due to Spearman, to "rid a correlation coefficient from the weakening effect of measurement error" (Jensen, 1998).

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 r_{x'y'} = \frac{r_{xy}}{\sqrt{r_{xx}r_{yy}}}.

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'.

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