Repeat-count targets under shared model error

Repeat-count targets under shared model error choose how many independent readings are worth averaging when part of the uncertainty is common to every repeat. In the local measurement model $X_i=S+D+{ϵ}_i$, let the shared discrepancy have variance ${\tau }^2$, and the independent repeat noise have variance ${\sigma }^2$. Assuming the two components are independent,The common term produces covariance between readings and survives averaging. The GUM's correlated-input rule retains precisely such shared contributions.^[ref]

For a desired total standard uncertainty $u_{\ast }>\tau$, solve the inequality before choosing the integer repeat count:If $\sigma >0$, a target equal to $\tau$ cannot be attained with any finite repeat count, and a target below $\tau$ is unattainable within this model. This is the feasibility threshold for averaging under shared uncertainty.

In an original body surface area example, take $\sigma =0.04{\mathrm{m}}^2$ and $\tau =0.08{\mathrm{m}}^2$. A target $u_{\ast }=0.081{\mathrm{m}}^2$ needs ten repeats; $u_{\ast }=0.0801{\mathrm{m}}^2$ needs one hundred. A target of $0.079{\mathrm{m}}^2$ instead requires reducing the shared component or changing the model. Merely collecting more readings is insufficient.

Two relative criteria illustrate why the target must be explicit. Requiring repeat-noise variance to be at most $\eta {\tau }^2$ gives $n\ge {\sigma }^2/(\eta {\tau }^2)$: at $\eta =0.01$, 25 repeats suffice. Requiring total standard uncertainty within 1% of its floor instead giveswhich needs 13 repeats with the same stipulated numbers. A 1% variance criterion and a 1% standard-uncertainty criterion are different requirements.

The calculation assumes independent repeat noise and a justified shared-error model. Repeated observations alone cannot estimate an individual fixed discrepancy; independent reference measurements or a suitable validation population are needed to constrain it.

A feasibility threshold for averaging under shared uncertainty is the smallest uncertainty that repetition can approach while a common uncertain contribution remains unchanged. In the measurement model $X_i=S+D+{ϵ}_i$, suppose the shared term has variance ${\tau }^2$ and independent repeat noise has variance ${\sigma }^2$. ThenFor $\sigma >0$, every finite repeat count has $u_n>\tau$, although $u_n$ tends to $\tau$ as $n$ grows. A target below the floor is infeasible within this model, and a target exactly at it needs an infinite count. The GUM's correlated-input propagation explains why a common contribution is retained rather than divided by the repeat count.^[ref]

Approaching the floor has a quantifiable cost. To obtain $u_n\le \tau (1+\delta )$ with $\tau >0$, solveFor small positive $\delta$, the required count grows approximately as $1/\delta$. With stipulated $\sigma =0.04{\mathrm{m}}^2$ and $\tau =0.08{\mathrm{m}}^2$, getting within 1% of the floor needs 13 repeats, within 0.1% needs 125, and within 0.01% needs 1,250. These are planning examples for body surface area prediction, not observed clinical performance limits.

Precision limits of weighting correlated repeats show why a clever average cannot remove the common term for free. A linear estimate $\hat{S}={\sum }_i{w}_i{X}_i$ that preserves the unknown $S$ requires ${\sum }_i{w}_i=1$. The shared discrepancy therefore enters with coefficient one as well, givingThe inequality follows from ${\sum }_i{w}_i^2\ge 1/n$; equal weights attain it under the stipulated equal independent repeat-noise variances. Negative weights do not evade the floor while preserving the signal coefficient.

Reaching a lower uncertainty requires additional information or a changed measurement model that reduces the common contribution. Repeating a validation method with genuinely independent reference errors can supply different information, whereas repeating one person's height and mass measurements does not redraw that person's model discrepancy. A known bias calls for a correction, not a claim that averaging will make it disappear.^[ref]

Precision limits of weighting correlated repeats describes what a weighted mean can achieve when repeated observations share an uncertain contribution. Supposewith independent, zero-mean repeat errors ${ϵ}_i$ independent of $D$. An estimate $\hat{S}={\sum }_i{w}_i{X}_i$ must satisfy ${\sum }_i{w}_i=1$ to preserve the unknown signal $S$. The shared contribution then has coefficient one as well:This is a special case of correlated-input measurement uncertainty propagation. The covariance matrix is $\Sigma ={\tau }^{2}11^{\mathsf{T}}+\mathrm{diag}({\sigma }_i^2)$; omitting its off-diagonal terms would incorrectly treat the common uncertainty as independently redrawn.^[ref]

The Cauchy-Schwarz inequality gives a sharp bound:Equality occurs for inverse-variance weighting,For equal repeat variances this reduces to equal weights and $u_{\min }^2={\tau }^2+{\sigma }^2/n$. Allowing negative weights cannot improve this bound within the stated model.

For an original body surface area prediction example, take shared standard uncertainty $\tau =0.050{\mathrm{m}}^2$ and two repeat-noise values ${\sigma }_1=0.040{\mathrm{m}}^2$ and ${\sigma }_2=0.080{\mathrm{m}}^2$. The optimal weights are $0.8$ and $0.2$, giving $u_{\min }=\sqrt{0.0025+0.00128}\approx 0.0615{\mathrm{m}}^2$. Equal weights give approximately $0.0671{\mathrm{m}}^2$. Weighting helps, but it does not erase the shared $0.050{\mathrm{m}}^2$ contribution. These values are stipulated, not clinical performance data.

For a fixed individual, $D$ can represent an unknown constant model discrepancy rather than a fluctuation between readings. No weights summing to one can distinguish that discrepancy from $S$, because both enter every observation identically. Independent reference information or a better anatomical measurement model is required to constrain it. If the residual errors themselves have further covariance, their full matrix replaces the diagonal repeat-noise model; the displayed inverse-variance weights no longer follow automatically.