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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.

The baseline assay is a noisy measurement just like the follow-up assays. On the log scale, a normal distribution for its error is a reasonable approximation if variability is approximately multiplicative on the original titre scale. The latent ${\mu }_{0i}$ represents the true baseline log-titre, withUsing a common variance is a simplifying assumption justified if baseline and follow-up measurements have comparable assay variability; it should not be treated as a fact guaranteed by the study design. Treating $y_{0i}$ as exact ignores measurement error in a predictor and can lead to attenuation bias from classical measurement error and understated uncertainty.

This is attenuation bias from classical measurement error, also called regression dilution. The latent predictor is $Z$, but $X_1$ and $X_2$ are noisy proxies. The noisier proxy produces a slope closer to zero.

For a regression with an intercept, the population slope isHere all variables are centred and the noises are independent. Thereforeso regressing $Y$ on $Z$ has slope $\beta =2$. Forwe haveThusWith the stated values, the slopes arematching the simulation.

Under this independent additive measurement-error model, the magnitude of the $Y$-on-$X_1$ slope is generally smaller than the magnitude of the $Y$-on-$Z$ slope whenever ${\sigma }_{x_1}^2>0$. Increasing ${\sigma }_y$ does not change the population slope, because response noise contributes neither to $\mathrm{Cov}(X_1,Y)$ nor to $\mathrm{Var}(X_1)$. It increases residual variance and the sampling variability of the estimate; with one million observations, doubling it should leave the displayed slope close to $1.6$ while increasing its standard error.