Recovery-correction uncertainty

Recovery-correction uncertainty quantifies uncertainty in an amount inferred through a recovery correction. If $\hat{q}=y/r$ uses measured recovered amount $y$ and recovery estimate $r>0$, both inputs contribute. The uncertainty of the corrected result is not obtained merely by copying the uncertainty attached to $y$.

The first-order sensitivity coefficients are $\partial \hat{q}/\partial y=1/r$ and $\partial \hat{q}/\partial r=-y/r^2$. Combining them givesThis is the linear propagation of uncertainty law with correlated inputs.^[ref] With positive nominal inputs and correlation coefficient $\rho$, the relative form isRecovery-estimate covariance can matter because an amount and its estimated recovery may share measurements or common response changes.

For original values $y=6\mathrm{nmol}$, $r=0.60$, $u_y=0.30\mathrm{nmol}$ and $u_r=0.06$, the corrected amount is $10\mathrm{nmol}$. Independence gives relative variance $0.05^2+0.10^2=0.0125$ and standard uncertainty about $1.12\mathrm{nmol}$. With $\rho =0.8$, the variance becomes $0.0045$ and uncertainty about $0.67\mathrm{nmol}$. Correlation cannot be chosen to improve the answer; it must describe the actual joint measurements.

The formula is an approximation near the nominal values. A recovery near zero or with large relative uncertainty can make a quotient strongly asymmetric and poorly represented by a symmetric standard uncertainty. Model inadequacy—such as applying a recovery from an unrepresentative sample matrix—is a separate issue that careful numerical propagation alone does not resolve.

Recovery-estimate covariance describes joint variation between an estimated extraction recovery and another input used with it, such as the measured recovered amount. It matters when a recovery correction divides one estimate by the other: shared data can make their uncertainties correlated.

For corrected amount $q=y/r$, first-order uncertainty propagation givesThe sign of the covariance contribution follows from the opposite signs of the quotient's sensitivity coefficients. A positive covariance reduces the propagated variance relative to a calculation assuming independence; a negative covariance increases it. Neither sign is a discretionary choice.^[ref]

An original shared-data example sets $r=y/z$, where independent measured amounts have nominal values $y=6\mathrm{nmol}$ and $z=10\mathrm{nmol}$, with standard uncertainties $u_y=0.30\mathrm{nmol}$ and $u_z=0.40\mathrm{nmol}$. Then $r=0.60$ andSubstitution gives $u_q^2=0.25+0.41-0.50=0.16\mathrm{nmo}{\mathrm{l}}^2$, hence $u_q=0.40\mathrm{nmol}$. This agrees with the exact algebraic cancellation $y/(y/z)=z$. Ignoring the shared numerator would instead give $\sqrt{0.66}\approx 0.81\mathrm{nmol}$.

The cancellation also reveals a validation trap: using the same recovered amount both to estimate its recovery and to correct itself does not independently demonstrate accurate recovery of an unknown sample. Build the joint measurement model from the underlying observations. Shared calibration standards, preparations or instrument drift may create other covariance structures; matched replicate data or an explicit common-input model can estimate them. Do not assume independence merely because a table lists two separate estimates.