Measurement error and misclassification

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.

Let $X,W,Y,Z$ be real random variables on a common probability space. Let $\mathcal{G}$ be the smallest sub-sigma-algebra of the ambient sigma-algebra that makes both $X$ and $Z$ measurable; when no $Z$ is named, take $\mathcal{G}=\sigma (X)$. The error in measuring $X$ by $W$ is nondifferential with respect to $Y$ relative to $(X,Z)$ exactly when $W\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp Y\mid \mathcal{G},$ using the library definition Conditional independence given a sigma-algebra. It is differential with respect to $Y$ relative to $(X,Z)$ when this conditional independence fails.

The named variables are part of the definition: changing $Y$ or $Z$ can change the classification. This condition alone fixes no direction of bias for a specified estimator or target. Such a direction requires additional model and target assumptions.

Axiom: This definition assumes the Axiom of Choice (The Axiom of Choice) because the cited conditional-independence definition is stated using the library's conditional-expectation classes.

Let $X,U,ϵ,W,Y$ be square-integrable real random variables on one probability space with $\mathbb{E}[X]=\mathbb{E}[U]=\mathbb{E}[ϵ]=0$, $U$ independent of the random element $(X,ϵ)$, and $ϵ$ independent of $X$. Set $W=X+U,Y=\alpha +\beta X+ϵ,\alpha ,\beta \in \mathbb{R},$ so $W$ has classical additive error. If $\mathrm{Var}(X)+\mathrm{Var}(U)>0$, the coefficient $\gamma$ of the best affine $L^2$ predictor of $Y$ from $W$ is $\gamma =\frac{\mathrm{Cov}(Y,W)}{\mathrm{Var}(W)}=\beta \frac{\mathrm{Var}(X)}{\mathrm{Var}(X)+\mathrm{Var}(U)}\mathrm{.}$ The attenuation factor lies in $[0,1]$. When both variances are positive, it is strictly between $0$ and $1$, so for $\beta \mathrm{\ne }0$ the coefficient has the same sign as $\beta$ and smaller absolute value; when $\beta =0$, $\gamma =0$. The boundary cases are: if $\mathrm{Var}(X)=0$, the factor is $0$; if $\mathrm{Var}(U)=0$ and $\mathrm{Var}(X)>0$, it is $1$.

Let $W,U,ϵ,X,Y$ be square-integrable real random variables on one probability space with $\mathbb{E}[W]=\mathbb{E}[U]=\mathbb{E}[ϵ]=0$, $U$ independent of $W$, and $ϵ$ independent of $W$. Put $X=W+U,Y=\alpha +\beta X+ϵ,\alpha ,\beta \in \mathbb{R},$ so $X$ has Berkson additive error relative to $W$. If $\mathrm{Var}(W)>0$, the coefficient of the best affine $L^2$ predictor of $Y$ from $W$ is $\beta$. Independence between $U$ and $ϵ$ is not assumed or needed.

Within the class of centered, square-integrable classical additive error models with $U$ independent of $X$, the observed law of $W=X+U$ need not identify the target $\psi (\theta )=({\mathrm{Var}}_{\theta }(X_{\theta }),{\mathrm{Var}}_{\theta }(U_{\theta }))\in [0,+\mathrm{\infty }{)}^2\mathrm{.}$ In particular, there is a two-point parametric model whose two parameter values give the same law of $W$ but different latent-error variance pairs.