Tetrazolium contrast uncertainty

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, writeFor finite input variances and known covariance matrix $\Sigma$, the variance is exactlyThe 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.^[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 $0.030$. ThenSince $w^{\mathsf{T}}1=0$, the common offset cancels, giving $u_I=0.020$. Treating those four appearances of one shared offset as independent would instead give $\sqrt{4(0.010^2+0.030^2)}\simeq 0.0632$. 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 $e_i$, thenFor 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.