Additive measurement error and categorical misclassification models

Let $X,W,U$ be real random variables on one probability space, meaning measurable maps into $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ (Random elements and real random variables). The variable $X$ is the latent true value, $W$ is its measured value, and $U$ is an error term. We use the following two additive models:

• Classical additive error: $W=X+U$, with $\mathbb{E}[U]=0$ and $U$ independent of $X$ in the sense of Independent random elements.
• Berkson additive error: $X=W+U$, with $\mathbb{E}[U]=0$ and $U$ independent of $W$ in that same sense.

The equations have opposite directions and their independence conditions refer to different variables. They are distinct model assumptions.

For nonempty finite sets $\mathcal{X}$ and $\mathcal{W}$, a categorical misclassification mechanism is a row-stochastic matrix $K(w\mid x)\in [0,1]$, indexed by $x\in \mathcal{X}$ and $w\in \mathcal{W}$: for every $x$, ${\sum }_{w\in \mathcal{W}}K(w\mid x)=1$. When categorical random variables $X,W$ have $\mathbb{P}(X=x)>0$, the entry represents $K(w\mid x)=\frac{\mathbb{P}(W=w,X=x)}{\mathbb{P}(X=x)}\mathrm{.}$ If $\mathbb{P}(X=x)=0$, the observed joint law determines no conditional row at that $x$; any row there is an additional part of a specified model, not an empirical consequence.

The matrix $K(w\mid x)$ describes the conditional law given $X$ alone. If an outcome or covariate is also named, whether the error is nondifferential must be stated relative to those variables; the marginal matrix alone does not settle that question.

Validation data that observe $X$ alongside $W$, replicate measurements, and a known error law supply different additional information. Any correction using such information needs its own design and model assumptions; none follows from the observed marginal law of $W$ alone.