Uncertainty budgeting

To assess homogeneity, the distribution of chemical constituents in a matrix is at the core of the investigation. This distribution can range from a random temporal and spatial occurrence at atomic or molecular levels over well defined patterns in crystalline structures to clusters of a chemical of microscopic to macroscopic scale. Although many physical and optical methods as well as analytical chemistry methods are used to visualize and quantify such spatial distributions, the determination of chemical homogeneity in a CRM must be treated as part of the uncertainty budget affecting analytical chemistry measurements. 

As the uncertainty of the mean is now related to the mass used for analysis and it is based on a careful evaluation of the element distribution in the material, this approach could help to evaluate further the total uncertainty budget for various analytical techniques as is required in accreditation and certification campaigns. 

INAA is well suited to study homogeneity of small samples because of its dynamic range of elemental sensitivity. The technique allows for the use of small solid samples, with the smallest usable sample size in the range of 0.5 mg to i mg as determined by handling and blank considerations. The INAA analytical procedure is well understood and characterized with mathematical relationships. Its analytical uncertainties can be sufficiently controlled and can be well determined for a particular procedure. This allows the calculation of the contribution of material heterogeneity to the uncertainty budget based on experimental data. 

Certified Reference Materials for Microanalytical Methods 1135 4.3.4.3 Uncertainty Budget of INAA... 

In this Chapter we highlight the practical considerations that must be understood by all users of RMs and CRMs we look at some of the issues of traceability and make the CRM user aware of the uncertainty budgets that need to be considered with the use of CRMs. No attempt will be made to advise CRM users on the proper use of statistics in the analytical measurement process and no statistical approaches on the establishment of measurement uncertainty will be given. There are a number of good texts on the subject which should be consulted. These are listed in the Further Reading section of the references at the end of this Chapter and include Miller and Miller (1993) and Taylor s work for NIST (Taylor 1985). 

The proper application of the ISO Guide to the expression of uncertainty in measurement requires that aU sources of rmcertainty are included. In practical terms this means an uncertainty budget has to be developed also by the user. The development of an rmcertainty budget, and the consequences for both analysts and producers is described later in Section 7.2. 

In this Section we aim to make the CRM user aware of the uncertainty budgets that need to be considered with the use of CRMs. Certified values in CRMs are the property values (mass fraction, concentration, or amount of substance) and their uncertainty, the uncertainty being in many instances a specified confidence interval for the certified property. As we discussed before, this uncertainty value is not always a complete uncertainty budget for an analytical process from sampling to production of data. But even when disregarding the subtle differences in the certificates, the way a CRM is used has serious consequences on the uncertainty budget that has to be applied to a user s result. This is summarized in Table 7.2. These uses may affect accuracy claims as well as traceability claims. It is the user s obligation to establish com-... 

On most occasions CRMs are used as Quality Control materials, rather than as calibrations . As outlined above, this common application adds significantly to the user s uncertainty budget, since at a minimum it is necessary to consider at least two independent measurement events (Um). so increasing the combined uncertainty of the results. Again this process rapidly increases the combined uncertainty with increasing complexity of the analytical system and so the usefulness of a control analysis may be downgraded when a correct uncertainty budget is formulated. 

In their broadest application, CRMs are used as controls to verify in a direct comparison the accuracy of the results of a particular measurement parallel with this verification, traceability may be demonstrated. Under conditions demonstrated to be equal for sample and CRM, agreement of results, e.g. as defined above, is proof. Since such possibilities for a direct comparison between samples and a CRM are rare, the user s claims for accuracy and traceability have to be made by inference. Naturally, the use of several CRMs of similar matrix but different analyte content will strengthen the user s inference. Even so, the user stiU has to assess and account for all uncertainties in this comparison of results. These imcertainty calculations must include beyond the common analytical uncertainty budget (i) a component that reflects material matrix effects, (2) a component that reflects differences in the amount of substance determined, (3) the uncertainty of the certified or reference value(s) used, and 4) the uncertainty of the comparison itself AU this information certainly supports the assertion of accuracy in relation to the CRM. However, the requirement of the imbroken chain of comparisons wiU not be formally fulfilled. 

Assess suitability of methods used for dietary intake estimations of different types of survey data by using Uncertainty Budget to quantify error incurred for each step. 

The calculation of a complete uncertainty budget for all routine tests often is impossible for chemical laboratories... 

Each of these error components adds its own uncertainty to the total uncertainty budget of the analytical procedure. Therefore, the different error components are referred to as sources of uncertainty. Depending on the sources of uncertainty taken into account and thus on the conditions of the measurement, the overall MU will be different and another definition of MU will apply. This means that there is no single, straighforward definition of MU. It is rather a concept the interpretation of... 

Table 5.3. Measurement uncertainty budget for the measurement of cholesterol in serum by isotope dilution/GCMS...

When making a chemical measurement by taking a certain amount of the test material, working it up in a form that can be analyzed, calibrating the instrument, and performing the measurement, analysts understand that there will be some doubt about the result. Contributions to uncertainty derive from each step in the analysis, and even from the basis on which the analysis is carried out. An uncertainty budget documents the history of the assessment... 

Once the uncertainty components have been identified and quantified as standard uncertainties, the remainder of the procedure to estimate uncertainty is a somewhat complicated but mostly straightforward. Most software products on the market will perform this task. Otherwise, some spreadsheet manipulation or mathematics must be done to reach the uncertainty. The combined standard uncertainty of a result is obtained by mathematical manipulation of the standard uncertainties as part of the uncertainty budget. These standard uncertainties may also be combinations of other uncertainties, and so on, as the branches and sub-branches of the cause-and-effect diagram are worked through. A combined standard uncertainty of a quantity is written uc(y). 

If an estimate of the measurement uncertainty has been made and quoted for a single determination, it is possible to infer the measurement uncertainty for repeated determinations by going through the uncertainty budget and... 

It might be useful to have a picture of the contributions of effects, and this can be done by a Pareto or other form of bar graph. Although a client is unlikely to need these graphs, they are a useful tool when assessing the uncertainty budget and can be used in summaries for quality managers and accreditation bodies. 

Consideration of factors that could contribute to measurement uncertainty from the use in the analysts laboratory revision of the uncertainty budget accordingly... 

The protocol must present an uncertainty budget. Its components should be carefully estimated, and may be stated in standard uncertainties, but expanded uncertainties can have great utility, provided the k factor is carefully chosen and indicated [2, 4, 6]13. All supposa-ble uncertainty sources (of types A and B)14, must be considered. Uncertainty components are concerned with contaminations, matrix effects, corrections, lack of stability or of stoichiometry, impurities in reagents, instrument non-linearities and calibrations, inherent uncertainties in standard methods, and uncertainties from subsample selection. Explicitly excluded may have to be sample selection in the field before submission to the laboratory and contamination prior to sample submission to the laboratory. The responsibility for adhering to the protocol s procedures, for which the planned complete uncertainty budget applies, rests with the laboratory and the analyst in charge of the measurement. 

Although every form of training includes teaching the avoidance of blunders, sometimes called spurious errors [8], these errors are unfortunately not always prevented. Nevertheless, the uncertainty budgets of protocols should not include components for possible blunders , because the magnitude of their effects on measurements is completely unpredictable. Such uncertainties are clearly not of a scientific nature. 

The relative uncertainties of the difference or ratio between two values, for example, are often far smaller than the relative uncertainty of either value which would not appear in the uncertainty budget... 

In some instances chemists will prepare in-laboratory RMs, generally by dissolution of a pure compound in a solvent to predetermined concentrations or even by dilution of commercially prepared stock solutions. The uncertainties connected with such RMs and relevant procedures must be understood by the analyst and entered into the uncertainty budget for the measurement of the unknown. The analyst then assumes responsibility for the uncertainties of the link of the RM value to the SI, all expressed in the relevant SI units, even when relative uncertainties are used. 

The naive prehistoric (in the traceability sense) method devised long ago is quite appropriate for providing reliable quantitative results and should be selected as a suitable candidate for any certification exercise where it is applicable, on condition that the analytical chemists applying it realise the inherent limitations and pitfalls and make an adequate uncertainty budget. 

In conclusion traceability concepts as they are now defined are of little practical use. On the other hand, methods of analysis such as these are certainly needed in science and society and so are RMs and methods. Work must go on in the absence of a full uncertainty budget and lack of clarity concerning the traceability reference [9], The BCR was right to develop a number of CRMs for speciation analysis as a diagnostic aid in the further development of this particular methodology. 

The definition of metrological traceability (see above) stipulates that each link in the chain has a known uncertainty. Nowadays, this concept and its application have been reformulated by the BIPM and recently detailed in the Guide to the expression of uncertainty in measurement (GUM) [26] parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand . Useful explanations are provided in several other guides [26-30] as well as commentaries [e.g. 31-33], The philosophy is to apply a bot-tom-up approach by formulating a function of all input quantities giving the measurand as output. An uncertainty budget of all sources of uncertainty is established. Important items to consider are ... 

The evaluated uncertainty of this substitution, i.e. the uncertainty of K, will limit the uncertainty of the measurement result of the unknown sample and must be taken into account as a contribution to the total uncertainty budget of the result obtained on the unknown sample since ... 

A traceability chain is therefore a prerequisite, i.e. an a priori requirement for the establishment of an uncertainty budget (the uncertainties are associated with the links in the chain) it follows that ... 

Traceability chains can exist with either large or small uncertainties in their links and values, and therefore with large or small overall uncertainty budgets from which it follows that ... 

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