The MRA defines the degree of equivalence of national measurement standards as a deviation of a NMI measurement result from a reference value of key comparisons.
Correspondingly, the supreme task is the evaluation of the reference value of key com- parisons. In publications one may come across various estimates of the reference value, but the most wide-spread ones are weighted mean, mean, and median.
A weighted mean is the best one, provided the the uncertainties presented are reli- able and that there is no correlation of the results obtained by different NMIs. In cases where these conditions are met, in [126, 127, 129] it is recommended using as the KCRV a weighted mean with weights inversely proportional to the squares of com- bined standard uncertainties.
In practice, judging by the reports on key comparisons contained in the BIPM data- base, a mean successfully competes with a weighted mean, since there exists a certain caution with respect to the measurement uncertainties declared by laboratories.
Let us consider the models used in evaluating key comparison data. The basic model corresponds to the situation of a traveling transfer measurement standard (of an invari- able measurand) which is time stable and has the appearance
Xi DX, (1.1)
where X is the measurand andXi is the measurand obtained in thei-th laboratory.
Participants in comparisons present the measured value and corresponding standard uncertaintyxi,ui.
To process the key comparison data the least-squares method (LSM) is applied. The KCRV is calculated as a weighted mean of NMIs measurement results, using inverse squares of the corresponding values of the standard uncertainty:
xw D x1=u2.x1/C Cxn=u2.xn/ 1=u2.x1/C C1=u2.xn/ . The corresponding standard uncertainty is equal to
u2.xref/D 1
u2.x1/C C 1 u2.xn/
1
.
The weighted mean can be taken as the KCRV only in the case where the data presented by the laboratories are consistent (agree with the model accepted), which can be checked using the criterion2. For this an observed value of statistics 2 is calculated:
2obs D .x1y/2
u2.x1/ C C.xny/2 u2.xn/ .
It is believed that a check of the data consistency does not “go through” (the data consistency is not confirmed), ifPạ2.n1/ > 2obsº<0.05. Here the normal prob- ability distributions are assumed.
In the case of positive check results the degree of equivalence is calculated as a pair of values – deviation of a measurement result from the reference value and uncertainty of this deviation:
di Dxi xref
u2.di/Du2.xi/u2.xref/.
The degree of equivalence between the measurement standards of two NMIs, the results of which, as in the case given, have been obtained in the same comparison, is calculated by the formula
dij Ddidj Dxixref.xj xref/Dxixj u2.dij/Du2.xi/Cu2.xj/.
There can be a number of reasons for a check by the criterion2not “going through”.
Among them there are two main reasons: failure of the transfer standard stability, and underestimation of the existing uncertainty of measurements by some of the compari- son participants. In both cases we are dealing with inconsistent comparison data. There is no unified strictly grounded method for evaluating inconsistent data [158, 250, 309, 521, 548].
These methods of data evaluation can be conditionally divided into two groups.
The first one realizes various procedures for removing the inconsistency of data. If the reason for data inconsistency is a traveling standard drift, then a correction can be introduced into the results obtained by the participants [116, 490, 559, 560]. After that, the traditional procedure of data evaluation can be applied.
If the reason for inconsistency is that discrepant data has turned up, then the form a consistent subset of measurement results is attempted [128, 159, 367, and 546]. There are different strategies for revealing and removing “outliers”, i.e. the results that are not consistent with the remaining ones within the limits of the declared uncertainties.
The criterionEn, at which the normalized deviation of each result from the reference value is calculated, has become the most used:
EnD jxixrefj 2u .xixref/.
In case of data consistency theEnvalues should not exceed the unit.
Another approach to forming a group of consistent results is based on the use of the criterion2. The largest subset, the results of which are checked against the criterion 2, is selected [128]. If there are several subsets with a similar number of elements, then the most probable one is chosen, i.e. a subset for which a sample value of statistics 2obsis the closest to the mathematical expectationk1 of the distribution2(k1), wherekis the number of subset elements.
On the basis of the subset formed of consistent measurement results, a reference value is determined as the weighted mean of these results. In case of several consistent
Section 1.3 Basic approaches to evaluating key comparison data 11 subsets with a similar number of elements, the Bayesian model averaging procedure can be applied [158].
The second group of methods of inconsistent data evaluation is based on a more complicated model. As an alternative to model (1.1), the model including a bias of laboratory results is
Xi DXCBi, (1.2)
whereBiis the bias of a result in thei-th laboratory.
Various interpretations of model (1.2) connected with additional assumptions rela- tive toBi are possible [118, 308, 498, 542, 547, 548]. The introduction of additional assumptions in this case is needed to obtain a single solution, since the model param- eter (nC1) is evaluated on the basis ofnmeasurement results.
In the “hidden error” [548] and “random effect” [521] models it is supposed thatBi is the sample of one distribution with zero mathematical expectation and variance. An estimate of the measurandXand variance is obtained using the maximum likeli- hood method.
Model (1.2) can be applied, for example, when there are grounds to believe that, in the process of comparisons, insignificant, as compared to measurement, uncertainties ( < ui/, changes of the values of a traveling standard have taken place. In this case an additional source of the KCRV uncertainty arises, which is not related to the initial data uncertainties.
Moreover, model (1.2) is used in the situation when the data inconsistency is caused by underestimating the measurement uncertainty by comparison participants. In our view, this is possible only in the case when there is reason to believe that all par- ticipants have underestimated a certain common random factor when evaluating the uncertainty of measurements. For example, in [548] it is stated that an underestimate of the instrumental uncertainty can take place under the condition that all participants use measuring instruments of one and the same type.
In the “fix effect model”Bidescribes systematic biases that have not been taken into account in the uncertainty budget. An additional assumption allowing the estimates Bi,X to be evaluated is required [309, 498, 521]. For example, it can be a condition under which the sum of systematic biases is equal to zero:P
Bi D0.
In Section 1.6 model (1.2) is treated for the case of consistent measurements, when information about uncertainty budget components, evaluated according to type A and type B, is used instead of the combined uncertainty of measurements [109, 110, 311].
This makes it possible to obtain estimates of the systematic biases of laboratory re- sults without additional constrains, thanks to a joint analysis of all data obtained by comparison participants.
The base procedure of evaluating consistent comparison data can be extended to the case with several transfer measurement standards, using the generalized method of least squares [366]. In the general case the initial model for applying the least-squares
method can be represented in the matrix form YE DX Ea, where
YE is the vector of measurement results obtained in different laboratories for different traveling standards;
Xis the experiment plan matrix the elements of which are the units and zeros depend- ing on the fact, whether the measurements of a standard given have been performed by a certain particular laboratory or not;
E
ais the vector of the values of different measurement standards.
The least-squares method allows the following estimate for the vector aEto be ob- tained:
a*refD.XT†X /1XT†1YE,
where†is the covariance matrix of measurement results:.†/ij Dcov.yi,yj/.
Correspondingly, we have for the vector of equivalence degrees dED EY X Earef,
with the respective matrix
W D†X.XT†1X /1XT.
An advantage of this approach is its generality and applicability to a great variety of schemes for performing key comparisons.