 Reference-standard uncertainty in relative potency

Reference-standard uncertainty in relative potency is the contribution of a shared activity reference to uncertainty in results calibrated against it. It may arise through preparation, dilution, stability or transfer of the reference material. The contribution can remain common to many specimen results even when their biological readouts are independently repeated.

For an original calibration model, let $A_i=R{\rho }_i$, where $R$ is reference activity on a specified concentration basis and ${\rho }_i$ is measured relative biological potency. First-order propagation givesThe covariance term matters when reference preparation or calibration information also enters the fitted ratio. The GUM explicitly retains correlations arising from common standards and reference data.^[ref]

Suppose $R$ is independent of all specimen ratio estimates, but is shared unchanged between them. For independent ${\rho }_i$ and ${\rho }_j$,This identity is exact under the stated independence assumptions. The reference contribution is not independent just because the two specimens occupy different wells or are assayed on different days.

An original numerical model uses reference mean 100 activity units per millilitre with standard uncertainty 5, and independent ratio estimates with mean 0.8 and standard deviation 0.04. The first-order standard uncertainty of one calibrated result near 80 units per millilitre is $\sqrt{16+16}\approx 5.66$. Averaging four independent ratio estimates against the same reference gives approximately $\sqrt{16+16/4}=4.47$, not $5.66/2$.

For this model the exact variance of the average of $n$ calibrated results isThe small extra 0.04 term comes from multiplying two uncertain quantities. As $n$ grows, standard uncertainty approaches 4 activity units per millilitre. This common-reference floor in potency averaging is a consequence of the shared calibration, rather than an unexplained failure of repeatability.

A reference's assigned unitage can itself be conventional. A 2017 WHO collaborative-study report proposed separate assigned bioactivity units and explained that first-standard unitage is set arbitrarily, while replacement standards are calibrated against the first.^[ref] Uncertainty in realizing, preparing or transferring that convention is distinct from uncertainty in a stipulated number. Specify which quantity the uncertainty budget represents.

A ratio of two results using exactly the same multiplicative reference has $A_1/A_2={\rho }_1/{\rho }_2$, so that common factor cancels algebraically. A comparison involving different reference lots need not have the same cancellation. Retain the calibration lineage and shared terms before combining results or interpreting a change in calibrated potency.