 Batch-specific blank allocation

Batch-specific blank allocation assigns background controls to the specimen groups whose preparation conditions they actually represent. In an experiment on blank-corrected carbon recovery in polymer tests, a batch may share inoculum, medium, collection time or reagent preparation. A precisely measured blank from another batch can still be the wrong correction. OECD's test framework uses parallel inoculum controls and warns that endogenous activity with a test substance need not equal the control's activity.^[ref]

For an original independent-batch model, estimate a weighted net mean byThe weights specify the scientific target; they are not automatically the observed fractions of specimens. Within a batch, reusing its blank estimate creates the shared-blank covariance in carbon recovery. Common corrections across batches require additional covariance terms rather than the independence assumed here.^[ref]

If specimen counts are fixed and each blank costs the same, minimizing the remaining variance under a total blank-count budget gives $n_{B,b}\propto |w_b|{\sigma }_{B,b}$. Consider two batches with weights $0.8,0.2$, ten specimen readings each, specimen standard deviation 1 mg, blank standard deviation 2 mg and twelve blank readings available. Ten blanks in the first batch and two in the second give variance $0.64(0.1+4/10)+0.04(0.1+4/2)=0.404\mathrm{m}{\mathrm{g}}^2$. Splitting blanks six and six gives approximately $0.5213\mathrm{m}{\mathrm{g}}^2$. These stipulated numbers illustrate allocation; integer and minimum-replication constraints still apply.

Batch-matched blank weighting also controls bias. If batch backgrounds are 1 and 5 mg, their scientifically weighted background is $0.8(1)+0.2(5)=1.8$ mg. Subtracting an equally pooled blank mean of 3 mg instead biases that weighted net result downward by 1.2 mg. Extra replication of the pooled estimate does not remove this mismatch.

Record batch membership and independent preparations before deciding which blanks to pool. Allocation can reduce sampling variance; it cannot prove that a background correction is chemically additive.