 Batch-matched blank weighting

Batch-matched blank weighting applies background corrections in the same proportions as the specimen results they correct. It concerns the definition of the target mean as well as the precision of the blank measurements. An experiment on blank-corrected carbon recovery in polymer tests can have different backgrounds across preparation batches even when every batch follows the same nominal procedure.

For an original additive model, let $T_b={\mu }_b+{\beta }_b+{ϵ}_b$ be a batch specimen mean and $B_b={\beta }_b+{\eta }_b$ its blank mean, with zero-mean measurement errors. The desired scientific target is $\mu ={\sum }_b{w}_b{\mu }_b$, where ${\sum }_b{w}_b=1$. If specimen means use $w_b$ but blanks use different weights $a_b$, the estimatorhas bias ${\sum }_b(w_b-a_b){\beta }_b$. Matching $a_b=w_b$ removes that bias for arbitrary batch backgrounds in this model. Pooling can remain unbiased under an additional justified common-background assumption, but that is a different model.

For stipulated backgrounds 2 and 8 mg and target weights 0.75 and 0.25, the relevant correction is 3.5 mg. An equally weighted pooled correction is 5 mg and biases the net mean downward by 1.5 mg. Dividing that net result by a fixed 50 mg initial-carbon amount shifts the recovery fraction by $-1.5/50=-0.03$, or three percentage points. These are constructed examples, not OECD test results.

With independent specimen and blank estimates, variance propagation gives $\mathrm{Var}(\hat{\mu })={\sum }_b{w}_b^2\mathrm{Var}(T_b)+{\sum }_b{a}_b^2\mathrm{Var}(B_b)$. Common reference measurements require covariance terms as well.^[ref] Changing weights to reduce variance can change the target or introduce bias; it is not automatically an improvement.

Transporting blank corrections between batches requires evidence that their backgrounds are exchangeable for the intended correction. OECD's general principle already cautions that endogenous activity with a test chemical need not match the inoculum control.^[ref] Matching batch weights repairs an aggregation error; it does not establish chemical additivity. State the target population, batch labels, normalization amounts and pooling assumptions before deciding which measurements share a correction.