The ISO Guide to the Expression of Uncertainty in Measurement (GUM) recommends the use of a first-order Taylor series expansion for propagating errors and uncertainties. The GUM also endorses the use of 'other analytical or numerical methods' when the conditions for using the Taylor expansion do not apply. In this paper we propose an alternative approach for evaluating measurement uncertainty based on the principle of generalized inference. The proposed approach can be applied to measurement models having any number of input quantities and a vector-valued measurand. We use several examples from the GUM to illustrate the implementation of the proposed approach for the calculation of uncertainties in measurement results.
Results of International Key Comparisons of National MeasurementStandards provide the technical basis for the Mutual Recognition Arrangement (MRA) formulated by Le Comité International des Poids et Mesures (CIPM). With many key comparisons already completed and a number of new key comparison experiments currently under way, we now have a better understanding of the statistical issues that need to be addressed for successfully analysing data from key comparisons and making proper interpretations of the results. There is clearly a need for a systematic approach to statistical analyses of key comparison data that can be implemented routinely by all participating laboratories.The determination of a key comparison reference value (KCRV) and its associated uncertainty and the degrees of equivalence are the central tasks in the evaluation of key comparison data. A satisfactory definition of a KCRV, however, is based on the assumption that all laboratories are estimating the same unknown quantity of the common circulating artefact, that is, the results from the different laboratories are mutually consistent. In this paper, we compare a number of statistical procedures for testing the consistency assumption.
A measurand θ of interest is the ratio of two other quantities, μp and μq. A measurement experiment is conducted and results P and Q are obtained as estimates of μp and μq. The ratio Y = P/Q is generally reported as the result for the measurand θ. In this paper we consider the problem of computing an uncertainty interval for θ having a prescribed confidence level of 1−α. Although an exact procedure, based on an approach due to Fieller, is available for this problem, it is well known that this procedure can lead to unbounded confidence regions in certain situations. As a result, practitioners often use various non-exact methods. One such non-exact method is based on the propagation-of-errors approach described in the ISO Guide to the Expression of Uncertainty in Measurement to calculate a standard uncertainty uy for Y. A confidence interval for θ with a presumed confidence level of 95% is obtained as [Y−2uy, Y+2uy]. In this paper we develop a highly accurate approximation for the coverage probability associated with the interval [Y−kuy, Y+kuy]. In particular, we demonstrate that, using n−1 degrees of freedom for uy, and the corresponding Student's t coverage factor k = t1−α/2 : n−1 rather than k = 2, leads to uncertainty intervals [Y−t1−α/2 : n−1uy, Y+t1−α/2 : n−1uy], that are nearly identical to Fieller's exact intervals whenever the measurement relative uncertainties are small, as is the case in most metrological applications. In addition, they are easy to compute and may be recommended for routine use in metrological applications. Improved coverage factors k can be derived based on the results of this paper for those exceptional situations where the t-interval may not have coverage probability sufficiently close to the desired value.
This paper proposes a measure for assessing the equivalence between the results of two laboratories. The measure is called asymmetric interchangeability. It is asymmetric since, based on this measure, one laboratory may be considered interchangeable with another laboratory but the converse may not be true. Such a situation can arise when the accuracy and precision of one laboratory are noticeably greater than that of the other. The proposed measure of interchangeability depends on the parameters of the measurement models, which include means, variances and the correlation of the two laboratories being compared. Since the level of correlation is often difficult to assess, it is assumed to be zero in this paper. Fiducial procedures are presented for testing the hypothesis that a laboratory is directionally interchangeable with another. The procedure is based on comparing a probability measure, which is shown to be closely related to asymmetric interchangeability, with an agreed threshold. Computer programs for calculating the p-value of the tests, written by use of open-source software, are listed.
We consider an estimation problem described in the Guide to the Expression of Uncertainty in Measurement (GUM). The problem is concerned with estimating a measurand that is a non-linear function of input quantities. The GUM describes two methods for estimating the measurand—method 1 is based on the same non-linear function of input-quantity estimates and method 2 is based on the mean of that function of individual measurements. We use several examples to compare the two methods based on their mean-squared errors and to demonstrate that a uniformly preferred method may not be available except for the simplest cases. We also consider an approach based on the Monte Carlo method in Supplement 1 of the GUM for the problem and compare it with the two methods.
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