 Common-reference floor in potency averaging

A common-reference floor in potency averaging is the nonzero limiting uncertainty left when many independently measured relative biological potency ratios are multiplied by the same uncertain reference activity. Repetition can reduce specimen-readout variation while preserving the shared reference-standard uncertainty in relative potency.

For an original probability model, let $A_i=R{\rho }_i$. Assume the reference $R$ is independent of all ratio estimates, and that the ${\rho }_i$ are mutually independent with common mean ${\mu }_{\rho }$ and variance ${\sigma }_{\rho }^2$. Write $\mathbb{E}[R]={\mu }_R$ and $\mathrm{Var}(R)={\sigma }_R^2$. Then $\overline{A}=R\overline{\rho }$ and independence gives the exact identityThus the limiting standard uncertainty is $|{\mu }_{\rho }|{\sigma }_R$. The product term ${\sigma }_R^2{\sigma }_{\rho }^2/n$ distinguishes the exact result from first-order uncertainty propagation. The GUM's treatment of shared standards and input correlations explains why a common reference must not be counted as independent for every specimen.^[ref]

Take hypothetical ${\mu }_R=100$, ${\sigma }_R=5$ activity units per millilitre, ${\mu }_{\rho }=0.6$ and ${\sigma }_{\rho }=0.03$. The mean calibrated activity is 60 units per millilitre, whileis approximately 4.245 for one ratio estimate, 3.355 for four and 3.015 for one hundred. The asymptotic floor is 3. Repeating the biological readout cannot turn an uncertain shared calibration into an exact one.

The floor is conditional on which uncertainty is shared. Fresh independent reference preparations can change the covariance structure, while a common stock or assigned calibration can preserve it across separate plates. If results share exactly the same nonzero multiplier, their ratio obeys $A_i/A_j={\rho }_i/{\rho }_j$: that reference term cancels in a ratio even though it remains in the average. Conversely, a conventionally assigned first-standard unitage is not itself a random measurement; distinguish the definition of the scale from uncertainty in realizing it.^[ref]