= Reference-sensitivity cancellation check
A \b[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 $X=f_X(R,\ldots)$ and $Y=f_Y(R,\ldots)$, a contrast $D=c_XX+c_YY$ has local reference sensitivity
$$
s_R=c_X\frac{\partial f_X}{\partial R}+c_Y\frac{\partial f_Y}{\partial R}.
$$
Its first-order reference variance is $s_R^2u^2(R)$. Checking the contrast directly and checking the full covariance propagation should agree. The GUM defines the sensitivities and retains correlations between inputs.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.1.3–5.1.4 and 5.2]{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 $1\,\mathrm{\mu m}$ changes $D$ by $-0.02\,\mathrm{\mu m}$. 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 $\sqrt2|\beta|s^2$. The first-order cancellation has not removed the nonlinear contribution.
With stipulated $\beta=1\,\mathrm{mm^{-1}}$ 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.<The Gauge Block Handbook by Ted Doiron and John S. Beers>[§4.4.3]{ref}
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