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.

A component of measurement error fluctuating between repetitions under specified conditions, as distinguished from a shared systematic error. Independent zero-mean fluctuations of variance ${\sigma }^2$ give a sample-mean variance ${\sigma }^2/N$; correlated fluctuations need not follow this averaging law.

Measurement noise is a fluctuating contribution to measurement error. An additive model for a measured output is $y_m(t)=y(t)+n(t)$, where $y$ is the actual plant output. A controller using feedback control can react to $n$ as though it were a change in the plant output. This effect is absent from the ideal noiseless examples under control theory; an errors-in-variables model represents a related distinction between latent quantities and their noisy observations.

An Errors-in-variables model distinguishes a latent true predictor from its noisy measurement and models both the measurement and the response. For example, $W_i\mid U_i\sim N(U_i,s^2)$ and $Y_i\mid U_i\sim N(a+b{U}_i,v^2)$, with a population distribution for $U_i$, propagate predictor uncertainty into inference on $b$. Treating $W_i$ as exact can create attenuation bias from classical measurement error. Identifiability requires appropriate information about the measurement process or latent predictor distribution.