= Item heterogeneity contribution to proficiency-test scores
{title2=$\sigma_h^2/\sigma_{\mathrm{pt}}^2$}
Differences between supplied <proficiency-test items> can contribute to a <proficiency-test z-score> even when the laboratory's own <measurement procedure> has zero <systematic error>. Model the result from laboratory $i$ as
$$
X_i=\theta+h_i+b_i+\varepsilon_i,
$$
where $\theta$ is an independently established batch target, $h_i$ is the supplied unit's deviation, $b_i$ is a fixed laboratory <systematic error>, and $\varepsilon_i$ is analytical variation. Assume centered independent $h_i$ and $\varepsilon_i$, with <variances> $\sigma_h^2$ and $\sigma_a^2$, and treat the <assigned value in proficiency testing> as exactly $\theta$ for this model. Then the <proficiency-test z-score> satisfies
$$
\mathbb E[Z_i]=\frac{b_i}{\sigma_{\mathrm{pt}}},\qquad
\boxed{\operatorname{Var}(Z_i)=\frac{\sigma_h^2+\sigma_a^2}{\sigma_{\mathrm{pt}}^2}},
$$
where $\sigma_{\mathrm{pt}}$ is the <standard deviation for proficiency assessment>. A large deviation cannot be attributed to $b_i$ from the score alone. The primary guide requires item <reference material homogeneity> and <reference material stability> to remain appropriate when a stricter assessment scale is selected.<Selection, Use and Interpretation of Proficiency Testing (PT) Schemes by Brian Brookman and Ian Mann>[§8.2.3, printed p. 21, smaller assessment scales and item suitability]{ref}
A <homogeneity study with replicated test portions> can separate two <variance components>. Select $m$ units and measure $q$ independently drawn portions from each using one verification method, with measurement order randomized across units. Under the <random effect> model $Y_{jk}=\mu+H_j+E_{jk}$, independent equal-variance within-unit errors give the <one-way analysis of variance> mean squares
$$
MS_W=\frac{\sum_{j=1}^m\sum_{k=1}^q(Y_{jk}-\bar Y_j)^2}{m(q-1)},\qquad
MS_B=\frac{q\sum_{j=1}^m(\bar Y_j-\bar Y)^2}{m-1}.
$$
Their expectations are $\mathbb E[MS_W]=\sigma_a^2$ and $\mathbb E[MS_B]=\sigma_a^2+q\sigma_h^2$. A <method-of-moments variance component estimate> is therefore $(MS_B-MS_W)/q$. Here the within-unit component is treated as analytical variation; variation between portions of the same unit would also enter it unless separated by further measurements. Clipping a negative estimate to zero enforces a nonnegative <variance component>; it introduces boundary bias and does not establish perfect material uniformity.
For an original arithmetic example, four units yield the following paired <mass concentration> readings, in $\mathrm{mg/L}$:
$$
\begin{pmatrix}
99.4&99.6\\
100.0&100.2\\
99.8&100.0\\
100.6&100.8
\end{pmatrix}.
$$
The overall <arithmetic mean> is $100.05$. These chosen data give $MS_W=0.02$, $MS_B=0.50$ and $\widehat\sigma_h^2=0.24$, with squared concentration units. Suppose a participant also has analytical variance $0.02$ and the assessment scale is $0.20\,\mathrm{mg/L}$. The stipulated unbiased score then has <standard deviation> $\sqrt{0.26}/0.20\approx2.55$, despite the absence of a laboratory <systematic error> in the model.
\Image[/proficiency-item-heterogeneity-and-averaging.png]
{title=Item differences remain when repeated analyses reduce measurement noise}
{description=Original paired readings and a conditional score-variance calculation. The right panel assumes all repeated analyses use the same supplied unit, independent analytical noise and the displayed estimated variance components.}
Averaging $r$ independent analyses from the same unit changes the analytical contribution to $\sigma_a^2/r$ but retains its unit effect:
$$
\operatorname{Var}(Z_i^{(r)})=\frac{\sigma_h^2+\sigma_a^2/r}{\sigma_{\mathrm{pt}}^2}.
$$
For the chosen numbers, the score's <standard deviation> approaches $\sqrt{0.24}/0.20\approx2.45$ as $r$ increases. Repeatedly reading one aliquot may reduce an even narrower instrumental contribution. Sampling independently supplied units addresses a different source of variation.
This is an explanatory <measurement model>, not a prescribed proficiency-testing acceptance rule. Its <variance components> require evidence about portion size, preparation and measurement conditions. <Instrumental drift> confounded with unit order, changing material, correlated errors or an uncertain consensus benchmark require additional terms and an appropriate design. The provider's homogeneity evidence is essential to interpreting the score; a smaller numerical scoring denominator does not improve the items.
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