Coverage factor uncertainty, expanded

Pauwels (1999) argues that the certified values of CRMs should be presented in the form of an expanded combined uncertainty according to the ISO Guide on the expression of uncertainty in measurement, so that coverage factor should always be clearly mentioned in order to allow an easy recalculation of the combined standard uncertainty. This is needed for uncertainty propagation when the CRM is used for calibration and the ISO Guide should be revised accordingly. The use of the expanded uncertainty has been pohcy in certification by NIST since 1993 (Taylor and Kuyatt 1994). 

Example the concentration of a reference solution is 1000 3 mg I 1. where the reported uncertainty is an expanded uncertainty, calculated using a coverage factor of k = 2, which gives a level of confidence of approximately 95%. 

A coverage factor (usually denoted by the letter k) is used to increase (expand) a standard uncertainty to give the required level of confidence (usually 95%). Expanded uncertainties are discussed in more detail in Section 6.3.6. To convert an expanded uncertainty back to a standard uncertainty, simply divide by the stated coverage factor. In this example, k = 2, so the standard uncertainty is 1.5 mg l-1. 

Consider the previous example of calculating the concentration of a standard solution. The combined standard uncertainty of 2.69 mg l-1 would be multiplied by a coverage factor of 2 to give an expanded uncertainty of 5.38 mg l-1. We can now report the result as follows concentration of solution = (1004 5) mg 1 1, where the reported uncertainty is an expanded uncertainty calculated using a coverage factor of 2, which gives a level of confidence of approximately 95%. Note that the coverage factor is applied only to the final combined uncertainty. 

En numbers are used when the assigned value has been produced by a reference laboratory, which has provided an estimate of the expanded uncertainty. This scoring method also requires a valid estimate of the expanded uncertainty for each participant s result. A score of En < 1 is considered satisfactory. The acceptability criterion is different from that used for z-, z - or zeta-scores as En numbers are calculated using expanded uncertainties. However, the En number is equal to zeta/2 if a coverage factor of 2 is used to calculate the expanded uncertainties (see Chapter 6, Section 6.3.6). En numbers are not normally used by proficiency testing scheme providers but are often used in calibration studies. 

Expanded Uncertainty is the combined standard uncertainty multiphed with the coverage factor k. Often k is chosen to be 2 or sometimes 3. With k=2 about 95%, and with k=3 about 99% of all likely values are encompassed. 

The last step is the calculation of the uncertainty with a higher level of confidence, the expanded rmcertainty. As described above this is done by multiplying with a coverage factor k. 

Of importance to the traceability of the results is a proper estimate of the measurement uncertainty of each participant. The error bars in figure 5.7 are the expanded uncertainties reported by the laboratories, in some cases with a coverage factor relating to more appropriate degrees of freedom. Table... 

There are several terms used in measurement uncertainty that must be defined. An uncertainty arising from a particular source, expressed as a standard deviation, is known as the standard measurement uncertainty (u). When several of these are combined to give an overall uncertainty for a particular measurement result, the uncertainty is known as the combined standard measurement uncertainty (uc), and when this figure is multiplied by a coverage factor ( ) to give an interval containing a specified fraction of the distribution attributable to the measurand (e.g., 95%) it is called an expanded measurement uncertainty [U). I discuss these types of uncertainties later in the chapter. 

The term izbias must be included even if the bias is considered insignificant. The expanded uncertainty is obtained by multiplying izsample by an appropriate coverage factor. Note that sr should be obtained from a suitably large (at least 10) number of repeats taken over a few days and batches of samples. A similar approach is taken for recovery, defined as... 

When assessing measurement uncertainty as part of a method validation, enough experiments are done to have degrees of freedom that do not adversely affect the coverage factor, and usually is taken as 2. As long as subsequent field measurements followed the validated method, a measurement uncertainty can be then quoted with = 2. For the most part, therefore, the expanded uncertainty should be calculated from the combined standard uncertainty by... 

For a measurement result to be metrologically traceable, the measurement uncertainty at each level of the calibration hierarchy must be known. Therefore, a calibration standard must have a known uncertainty concerning the quantity value. For a CRM this is included in the certificate. The uncertainty is usually in the form of a confidence interval (expanded uncertainty see chapter 6), which is a range about the certified value that contains the value of the measurand witha particular degree of confidence (usually 95%). There should be sufficient information to convert this confidence interval to a standard uncertainty. Usually the coverage factor ( see chapter 6) is 2, corresponding to infinite degrees of freedom in the calculation of measurement uncertainty, and so the confidence interval can be divided by 2 to obtain uc, the combined standard uncertainty. Suppose this CRM is used to calibrate... 

The major components of uncertainty are combined according to the rules of propagation of uncertainty, often with the assumption of independence of effects, to give the combined uncertainty. If the measurement uncertainty is to be quoted as a confidence interval, for example, a 95% confidence interval, an appropriate coverage factor is chosen by which to multiply the combined uncertainty and thus yield the expanded uncertainty. The coverage factor should be justified, and any assumptions about degrees of freedom stated. 

Fig 5 Traceability chain for pH in Germany. The uncertainty stated is the expanded uncertainty with a coverage factor k- 2. The uncertainty due to the Bates-Guggenheim convention is not taken into account... 

The expanded uncertainty U is obtained by multiplying crx by an appropriate coverage factor (k) as specified in the Guide to the expression of Uncertainty in Measurements (GUM). Thus U=kax and X. U. 

Calculating combined (overall) uncertainty as a root of a sum of variances associated with the uncertainty components, and then calculating expanded uncertainty multiplying the combined uncertainty by an appropriate coverage factor corresponding to the necessary level of confidence. 

Precision %RSD< 15 % at>3 levels n>5 at each level %RSD<20 % at LLOQ %RSD<20% n=4-7 CV<2.2 % Uncertainty evaluation Coefficient of variation (CV) 1 % Uncertainty of volumetric error 0.3 % Uncertainty of reference standard 0.1 % Uncertainty of weighing 0.5 % Uncertainty of other systematic errors Combined standard uncertainty Coverage factor Expanded uncertainty Relative expanded uncertainty (%) Intraassay CV=2-9 % Interassay CV=4-12%... 

Use of expanded uncertainty and coverage factor A report to a client contains a statement as follows ... 

The reported uncertainty is an expanded uncertainty calculated using a coverage factor of 2 which gives a level of confidence of 95%. 

Coverage factor Expanded uncertainty Method of least squares Multivariate statistics Null hypothesis Power of a test Probability levels Regression of j on jc... 

Value of the factor k, for which the expanded uncertainty has been calculated (where k is the coverage factor)... 

The expanded uncertainty can then be estimated after the calculation of the effective number of degrees of freedom, df (Eq. 7). Therefore the coverage factor used was the Student t defined for that number and a 95% significance level (/(df, 95%). The estimated confidence interval is defined by... 

The estimated uncertainties were compared with the experimental ones by an F-test for the 95% confidence level [10]. Figure 2 represents the obtained experimental values associated with the estimated expanded uncertainty (95% confidence level). The coverage factor used was 1.96 for the average effective number of degrees of freedom, df, of 57500. The Eurachem [1] proposal of a coverage factor of 2 is adequate for this case. 

To determine whether Rm is significantly different from 1, the calculated value of t is compared with the coverage factor, k = 2, which will be used to calculate the expanded uncertainty [19]. A t value greater than 2 suggests that Rm is significantly different from 1. However, if in the normal application of the method, no correction is made to take account of the fact that the method recovery is significantly different from 1, the... 

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