 Reference-sensitivity cancellation check

A reference-sensitivity cancellation check determines whether a proposed comparison really removes the influence of a shared calibration reference. It checks the measurand, the reference's sensitivity coefficients and the range over which the claimed cancellation applies.

For estimates $X=f_X(R,\dots )$ and $Y=f_Y(R,\dots )$, a contrast $D=c_{X}X+c_{Y}Y$ has local reference sensitivityIts first-order reference variance is $s_R^2{u}^2(R)$. Checking the contrast directly and checking the full covariance propagation should agree. The GUM defines the sensitivities and retains correlations between inputs.^[ref]

An original linear model is $X=R+d_X$ and $Y=1.02R+d_Y$. For $D=X-Y$, the sensitivity is $-0.02$. Recomputing the unchanged comparison with the assigned reference value increased by $1\mu \mathrm{m}$ changes $D$ by $-0.02\mu \mathrm{m}$. With reference standard uncertainty 20 nm, its contribution to this contrast is 0.4 nm. Treating the two uses of $R$ as independent would miss this suppression.

A useful computational check perturbs one shared input consistently in every place it occurs. Perturbing its occurrences independently tests a different model. Finite-perturbation reference sensitivity also reveals whether a derivative evaluated at one point adequately describes the actual uncertainty range.

Zero first derivative is weaker than exact invariance. For an illustrative nonlinear model, take $X=R+d_X$ and $Y=R+\beta (R-R_0{)}^2+d_Y$. The derivative of $X-Y$ vanishes at $R_0$, but a reference perturbation $\delta R$ leaves a term $-\beta \delta R^2$. If $\delta R$ is zero-mean normal with standard deviation $s$, that term has mean $-\beta s^2$ and standard deviation $\sqrt{2}|\beta |s^2$. The first-order cancellation has not removed the nonlinear contribution.

With stipulated $\beta =1\mathrm{m}{\mathrm{m}}^{-1}$ and $s=0.001\mathrm{mm}$, these are $-1$ nm and about 1.41 nm. These are model coefficients, not measured comparator behaviour.

Finally, verify that the chosen contrast is the quantity the experiment intends to determine. Replacing $X-Y$ by another combination solely to cancel a reference changes the question. Retain calibration lineage and independent checks on comparator response; NIST's physical uncertainty budget distinguishes the master history from comparison-process variability.^[ref]