= Unbalanced sampling-uncertainty design
An \b[unbalanced sampling-uncertainty design] estimates collection and analytical contributions while performing different numbers of analyses on the collected samples. One useful duplicate design collects two samples independently at each target, analyses one twice and the other once. The Eurachem guide presents this three-result arrangement as a less costly alternative to analysing both samples twice.<Measurement uncertainty arising from sampling by Michael H. Ramsey and others>[§9.4.2, Note 2 and Figure 2; Appendix D.2, Figure D.2(b)]{ref}
An original variance calculation shows why both components remain identifiable. At target $i$, let
$$
Y_{ijr}=\mu_i+S_{ij}+\varepsilon_{ijr},\qquad
\operatorname{Var}S_{ij}=\sigma_s^2,\quad
\operatorname{Var}\varepsilon_{ijr}=\sigma_a^2.
$$
Here $j=1,2$ labels independently collected samples; only $j=1$ has analytical replicates $r=1,2$. Assume centered independent random effects, common <variance> components and the same analytical precision for each result. Define within-target contrasts
$$
D_i=Y_{i11}-Y_{i12},\qquad
Q_i=\frac{Y_{i11}+Y_{i12}}2-Y_{i21}.
$$
The target mean cancels. The first contrast also cancels the sample effect; the second compares independent collections with unequal analytical averaging. Consequently,
$$
\operatorname{Var}D=2\sigma_a^2,\qquad
\operatorname{Var}Q=2\sigma_s^2+\tfrac32\sigma_a^2.
$$
The coefficient $3/2$ matters: one sample mean contains half the analytical variance, while the other result contains all of it. Applying the balanced-design coefficient would misallocate uncertainty.
For stipulated contrast variances $s_D^2=4$ and $s_Q^2=25$ in squared concentration units, the corresponding method-of-moments estimates are
$$
\widehat\sigma_a^2=2,\qquad
\widehat\sigma_s^2=\frac{25-3}2=11.
$$
A single analysis of a single collected sample then has modeled <standard uncertainty> $\sqrt{11+2}\approx3.61$ in concentration units. These are invented values illustrating the model, not measurements from the guide.
Across eight targets, this design requires sixteen collections and twenty-four analyses, compared with thirty-two analyses for two analytical replicates on each collection. The saving is in analytical runs; equal cost does not imply equal statistical precision. Repeat the nested design across targets to estimate the contrast variances. One triple of results does not reliably estimate the collection variance, and duplicate collection by one unchanged protocol does not reveal a bias shared by every sample. A negative estimated variance component signals finite-data or model issues requiring an appropriate estimation method, rather than a physically negative uncertainty.
Back to article page
