Correction for attenuation

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 $$Y'$$.