Batch and measurement-process variation

Fix nonempty finite index sets $I$ and $J_i$ for $i\in I$, and positive integers $m_{ij}$ for $i\in I$ and $j\in J_i$. The index $i$ names a fixed process condition, such as a fixed instrument configuration or an operator held fixed by design. The index $j$ names a batch nested within that condition (for example, a day, or a specified day–operator combination); $k\in \{1,\dots ,m_{ij}\}$ names a repeat reading within that batch. If days and operators are crossed, or their separate contributions are needed, the model must use additional indices and terms: this nested model does not separate such effects.

On one probability space, let the real random variables $B_{ij}$ and ${ϵ}_{ijk}$ be jointly independent and square-integrable, with $\mathbb{E}[B_{ij}]=0,\mathrm{Var}(B_{ij})={\sigma }_B^2<\mathrm{\infty },\mathbb{E}[{ϵ}_{ijk}]=0,\mathrm{Var}({ϵ}_{ijk})={\sigma }_{ϵ}^2<\mathrm{\infty }\mathrm{.}$ The variances are common over the indicated indices and may be zero. Let $\mu$ and each ${\tau }_i$ be fixed real constants. The nested batch measurement model is $Y_{ijk}=\mu +{\tau }_i+B_{ij}+{ϵ}_{ijk}\mathrm{.}$ Here ${\tau }_i$ represents the fixed process-condition offset, $B_{ij}$ is the random batch offset, and ${ϵ}_{ijk}$ is the repeat error. The replicate mean is ${\bar{Y}}_{ij\cdot }=m_{ij}^{-1}{\sum }_{k=1}^{m_{ij}}{Y}_{ijk}$. This is a scalar random-intercept specialization of the finite-moment mixed-model form in Finite-dimensional linear mixed model with explicit longitudinal roles; it adds mutual independence across the finite batch and repeat families and assumes a common repeat-error variance.

In this article, repeatability means variation among readings made under the same stated short-term conditions, represented by the repeat-error term in this model. Reproducibility means variation across the explicitly named conditions being compared, such as days or operators; the parameter ${\sigma }_B^2$ describes only the random batch factor represented by $j$. A design-based interpretation as variation over a target population requires batches sampled from that population; under a working random-effects model, its interpretation instead depends on the stated model and inferential target. Stability means constancy over the stated time interval of the measurement process law, including its calibration relation and relevant error behavior. These words are used with these declared scopes; terminology in measurement standards can vary.

Choose the batch-effect specification to match the model and inferential target. A design-based interpretation treats $B_{ij}$ as random when observed batches are sampled from a named population and the target includes variation over it. A random-effects distribution may also be used as a working model for selected batches that are not a probability sample, but that choice alone does not justify design-based extrapolation. If the selected batches are the entire target and no distributional model is intended, represent their offsets as fixed batch parameters; the random-effect variance formula below then does not describe extrapolation to unobserved batches. Pool calibration curves only under an explicit common-curve or equivalence model over the stated calibration range. Without that model, report batch-specific calibration descriptions. This definition and its variance consequences assert no empirical adequacy or stability of a particular measurement process. The measurement occasion called a batch and the calibration-curve terminology agree with Measurands, calibration, and measurement uncertainty.

Let $I$ and each $J_i$ be nonempty finite sets, and let $m_{ij}\ge 1$ be an integer for each $i\in I$ and $j\in J_i$. On one probability space, suppose the real random variables $B_{ij}$ and ${ϵ}_{ijk}$, for $1\le k\le m_{ij}$, are jointly independent and square-integrable, with $\mathbb{E}[B_{ij}]=0,\mathrm{Var}(B_{ij})={\sigma }_B^2<\mathrm{\infty },\mathbb{E}[{ϵ}_{ijk}]=0,\mathrm{Var}({ϵ}_{ijk})={\sigma }_{ϵ}^2<\mathrm{\infty },$ where these two variances are common over their indices. For fixed real constants $\mu$ and ${\tau }_i$, set $Y_{ijk}=\mu +{\tau }_i+B_{ij}+{ϵ}_{ijk},{\bar{Y}}_{ij\cdot }=\frac{1}{m_{ij}}{\sum }_{k=1}^{m_{ij}}{Y}_{ijk}\mathrm{.}$ Then for every $i,j$ and distinct $k,\mathrm{\ell }\in \{1,\dots ,m_{ij}\}$, $\mathrm{Cov}(Y_{ijk},Y_{ij\mathrm{\ell }})={\sigma }_B^2,$ and for every $i,j$, $\mathrm{Var}({\bar{Y}}_{ij\cdot })={\sigma }_B^2+\frac{{\sigma }_{ϵ}^2}{m_{ij}}\mathrm{.}$ No Gaussian assumption is needed.

The result is a model identity, not an estimate or an empirical claim about a particular instrument. Its proof uses finite sums and no choice axiom.