Definitions

# 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 $r_\left\{xy\right\}$, and a known reliability for each variable, $r_\left\{xx\right\}$ and $r_\left\{yy\right\}$, the correlation between $X$ and $Y$ corrected for attenuation is $r_\left\{x\text{'}y\text{'}\right\} = frac\left\{r_\left\{xy\right\}\right\}\left\{sqrt\left\{r_\left\{xx\right\}r_\left\{yy\right\}\right\}\right\}$.

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 $X\text{'}$ and $Y\text{'}$ with independent errors, then $r_\left\{x\text{'}y\text{'}\right\}$ measures the true correlation between $X\text{'}$ and $Y\text{'}$.