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 $Y'$.

References

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