Accounting for bias in measurement uncertainty estimation

Abstract

Generic issues and current approaches in the evaluation of bias studies with respect to estimation of measurement uncertainty are discussed, focusing on two main scenarios. In the first, for a within-laboratory assessment of a fully developed uncertainty budget, bias studies are carried out to verify the pre-established performance of a measurement procedure, and design corrective actions if necessary. In the second scenario, for an estimation of measurement uncertainty from within-laboratory validation data, bias studies are carried out to “calibrate” the whole measurement procedure and evaluate its performance, more specifically to estimate the uncertainty of measurement by combination of bias and precision estimates.

Appendix 1 Accounting for correlation between repeated measurements

Let y and y′ be measurement results obtained using the same measurement procedure. Then y and y′ may be viewed as depending on the same input variables, taking partly the same and partly different values. For example, if the ambient temperature in the laboratory (monitored on a regular basis) is an input variable, then y and y′ could either depend on the same temperature measurement or on different temperature measurements. Assuming that all input quantities were determined independently, the covariance between y and y′ is given by

$$u(y,{y}') = {\sum {c_{y} (z)c_{{{y}'}} (z)u^{2} (z)} }$$

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where the sum is over all input quantities z shared by y and y′, and the cs are the sensitivity coefficients concerned. Let u _com(y) and u _com(y′) be the part of the combined standard uncertainty of y and y′, respectively, which is due to the shared (common) input values, i.e.

$$u^{2}_{{{\text{com}}}} (y) = {\sum {[c_{y} (z)u(z)]^{2} } }$$

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Then $u(y,{y}') \leqslant u_{{{\text{com}}}} (y)u_{{{\text{com}}}} ({y}')$, and the product on the right-hand side of this inequality is in fact a reasonable approximation if $c_{y} (z) \approx c_{{{y}'}} (z)$ for all z, or if the ratio $c_{y} (z)/c_{{{y}'}} (z)$ is approximately constant. This may be assumed if y and y′ are obtained using the same measurement procedure on similar objects/samples, as done in the derivation of Eq. 6. An estimate of the covariance between y and y′ is required for proper evaluation of the uncertainty for combinations of measurement results such as

$$u^{2} (y \pm {y}') = u^{2} (y) + u^{2} ({y}') \pm 2u(y,{y}')$$

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and analogous expressions for products and quotients, then involving relative uncertainties and covariances.

The estimate $u(y,{y}') = u_{{{\text{com}}}} (y)u_{{{\text{com}}}} ({y}')$ is a fortiori applicable to replicate measurements on the same object/sample. This may be used to derive Eq. 2 as follows. Let x ₁, x ₂,... x _n be the results of replicate measurements on the same object or sample, and let <x> denotes the mean value of the x _i . The standard uncertainty of this mean is given by

$$u^{2} (\langle x\rangle ) = \frac{1}{{n^{2} }}{\left[ {{\sum\limits_i {u^{2} (x_{i} )} } + {\sum\limits_{j \ne k} {u(x_{j} ,x_{k} )} }} \right]}$$

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Since we are dealing with replicates, we may assume that all the u(x _i ) are the same and replace them by a common estimate u(x). Now consider a decomposition of the combined standard uncertainty u(x) of a single measurement according to

$$u^{2} (x) = u^{2}_{{{\text{var}}}} (x) + u^{2}_{{{\text{inv}}}} (x)$$

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where u _var(x) is the combined standard uncertainty accounting for all influence quantities which are effectively varied between replications, while u _inv(x) is the combined standard uncertainty accounting for all influence quantities which are effectively invariant under replication conditions and are thus shared by all replicates. Then the covariance between any two replicates is given by $u(x,{x}') = u_{{{\text{inv}}}} (x)u_{{{\text{inv}}}} ({x}') = u^{2}_{{{\text{inv}}}} (x)$. Using this estimate and Eq. 19, the standard uncertainty of a mean value 〈x〉 is obtained as follows:

$$u^{2} (\langle x\rangle ) = \frac{1}{{n^{2} }}[nu^{2} (x) + n(n - 1)u^{2}_{{{\text{inv}}}} (x)] = \frac{{u^{2}_{{{\text{var}}}} (x)}}{n} + u^{2}_{{{\text{inv}}}} (x)$$

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