 Measurement error

A measurement error is the difference between a recorded measurement $W$ and the quantity $U$ intended to be measured. The additive model $W=U+e$ separates that error $e$ from genuine variation in $U$. A shared offset is a systematic error; fluctuations between repeated measurements are random measurement error. Repeating a measurement can reduce independent Random measurement error, but does not remove a shared systematic error.

measurement error in a predictor can bias a regression slope even when it has mean zero. Suppose $Y=a+bU+\epsilon$, where $e$ is uncorrelated with $U$ and $\epsilon$, and $\epsilon$ is uncorrelated with $U$. A regression of $Y$ on the recorded $W$ has population slopeThus equal predictor and error variances halve the slope: this is attenuation bias from classical measurement error. In contrast, independent mean-zero error added only to $Y$ increases residual variance without changing that population slope. An errors-in-variables model describes the measurement mechanism explicitly rather than treating noisy predictors as exact.