= Tetrazolium contrast uncertainty
\b[<Tetrazolium contrast uncertainty>] quantifies uncertainty in the four-reading <cell-free tetrazolium interaction contrast>, including correlations between its inputs. In the order reagent/test, no-reagent/test, reagent/vehicle and no-reagent/vehicle, write
$$
I=A_1-A_2-A_3+A_4=w^{\mathsf T}\mathbf A,\qquad
w=(1,-1,-1,1)^{\mathsf T}.
$$
For finite input variances and known <covariance matrix> $\Sigma$, the <variance> is exactly
$$
u_I^2=w^{\mathsf T}\Sigma w
=\sum_iw_i^2u_i^2+2\sum_{i<j}w_iw_j\operatorname{Cov}(A_i,A_j).
$$
The exactness here follows from linearity of the contrast. JCGM's general <uncertainty propagation> framework uses <sensitivity coefficients> and correlated input quantities; this contrast has coefficients $w_i$ independent of the readings.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[Sections5.1–5.2, especially the correlated-input expression in5.2.2]{ref}
An original example gives each reading an independent residual with <standard uncertainty> $0.010$ and all four readings the same additive offset with <standard uncertainty> $0.030$. Then
$$
\Sigma=(0.010)^2I_4+(0.030)^2\mathbf1\mathbf1^{\mathsf T}.
$$
Since $w^{\mathsf T}\mathbf1=0$, the common offset cancels, giving $u_I=0.020$. Treating those four appearances of one shared offset as independent would instead give $\sqrt{4(0.010^2+0.030^2)}\simeq0.0632$. The distinction concerns the physical source of variation, not the number of numbers in the subtraction.
A shared multiplicative gain does not cancel. If $A_i=(1+g)\mu_i+b+e_i$, with independent zero-mean gain error $g$, common offset $b$ and independent residuals $e_i$, then
$$
I=(1+g)I_0+w^{\mathsf T}\mathbf e,\qquad
\operatorname{Var}(I)=I_0^2u_g^2+4u_e^2.
$$
For chosen $I_0=0.17$, $u_g=0.05$ and $u_e=0.010$, the combined <standard uncertainty> is approximately $0.0217$. The gain contribution is a relative uncertainty, whereas the residual contribution is in <absorbance> units.
The input model should distinguish repeated readings of one preparation from independently prepared controls. A small propagated uncertainty does not test whether controls reproduce the sample's chemistry or whether a nonzero optical contrast identifies a particular <formazan>. Those are separate questions about the <measurement model>.
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