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 $r\_\{x\text{'}y\text{'}\}\; =\; frac\{r\_\{xy\}\}\{sqrt\{r\_\{xx\}r\_\{yy\}\}\}$.

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

Derivation

See also

• Errors-in-variables model

References

• Jensen, A.R. (1998). The g Factor. Praeger, Connecticut, USA.