= Homogeneity study with replicated test portions
A \b[homogeneity study with replicated test portions] samples several containers of a <reference material> and measures separately prepared portions from each. The replication separates variation between containers from variation remaining within a container and the analytical procedure. Repeated detector readings of one prepared portion do not supply the same information.
The 2006 IUPAC protocol recommends randomly selecting at least ten distribution containers, homogenizing each selected container separately, and analyzing two portions from each in randomized order under a specified <repeatability condition of measurement>. It also requires adequate analytical precision and checks for trends, excessive rounding and discordant duplicate pairs.<The International Harmonized Protocol for the proficiency testing of analytical chemistry laboratories by Michael Thompson and others>[Appendix 1, A1.1, printed p. 177; §3.11.4]{ref}
An original balanced <random effect> model is
$$
y_{ij}=\mu+b_i+e_{ij},\qquad i=1,\ldots,m,\quad j=1,\ldots,r.
$$
Assume independent zero-mean container effects with <variance> $\sigma_b^2$ and independent equal-variance residuals with variance $\sigma_e^2$. Residuals include analytical scatter and whatever within-container variation remains at the chosen portion size. <One-way analysis of variance> then gives
$$
\mathbb E[MS_W]=\sigma_e^2,\qquad
\mathbb E[MS_B]=\sigma_e^2+r\sigma_b^2,
$$
so a method-of-moments estimate is $\widehat\sigma_b^2=(MS_B-MS_W)/r$. A negative estimate can occur through sampling variation; it is not negative physical variance or proof of perfect <reference material homogeneity>.
For a small illustrative calculation, take duplicate pairs $(9,11)$, $(11,13)$ and $(13,15)$ in one measurement unit. The container means are 10, 12 and 14. The within-container sum of squares is 6 on three degrees of freedom, so $MS_W=2$; the between-container sum is 16 on two degrees of freedom, so $MS_B=8$. Thus $\widehat\sigma_b^2=(8-2)/2=3$. These three invented containers explain the arithmetic; they do not meet the protocol's recommended ten-container study.
Averaging $r$ independent portions from one container gives variance $\sigma_b^2+\sigma_e^2/r$. More replication reduces the residual term without removing the common container effect. Randomizing the measurement order helps prevent <instrumental drift> from masquerading as between-container variation.
Fitness for purpose requires comparing material variation with an appropriate tolerance, not merely failing to reject zero variation. The protocol explicitly distinguishes sufficient homogeneity from the stricter hypothesis that sampling variance is exactly zero.<The International Harmonized Protocol for the proficiency testing of analytical chemistry laboratories by Michael Thompson and others>[§3.11.2, printed p. 171]{ref} Keep the portion size, sampling design and uncertainty with the conclusion.
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