= Attenuation bias from classical measurement error
= Regression dilution
{synonym}
If $Y=\beta Z+\varepsilon$ but the predictor is observed as $X=Z+\eta$, with independent centred errors, the population regression slope is
$$
\frac{\operatorname{Cov}(X,Y)}{\operatorname{Var}(X)}
=\beta\frac{\operatorname{Var}(Z)}
{\operatorname{Var}(Z)+\operatorname{Var}(\eta)}.
$$
Measurement error in the predictor therefore shrinks the slope toward zero, while independent response noise changes its uncertainty but not its population value.
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