Uncertainty functions, a compact way of summarising or specifying the behaviour of analytical systems

Analytical chemists can advantageously use an uncertainty function to describe the performance of an analytical system in terms of the standard uncertainty or standard deviation as a function of the concentration of the analyte. This “characteristic function” is useful for estimating uncertainty at a new concentration. A similar function can be used to prescribe the uncertainty that is regarded as fit for purpose for a particular application. This “fitness function” is useful in setting standards of accuracy in proficiency tests and similar exercises without revealing the concentration of the analyte. In combination, these two functions provide a rational basis for method selection.     

Effect of the sample matrix on measurement uncertainty in X-ray fluorescence analysis

The estimation of measurement uncertainty, with reference to univariate calibration functions, is discussed in detail in the Eurachem Guide “Quantifying Uncertainty in Analytical Measurement”. The adoption of these recommendations to quantitative X-ray fluorescence analysis (XRF) involves basic problems which are above all due to the strong influence of the sample matrix on the analytical response. In XRF-analysis, the proposed recommendations are consequently applicable only to the matrix corrected response. The application is also restricted with regard to both the matrices and analyte concentrations.In this context the present studies are aimed at the problems to predict measurement uncertainty also with reference to more variable sample compositions. The corresponding investigations are focused on the use of the intensity of the Compton scattered tube line as an internal standard to assess the effect of the individual sample matrix on the analytical response relatively to a reference matrix. Based on this concept the estimation of the measurement uncertainty of an analyte presented in an unknown specimen can be predicted in consideration of the data obtained under defined matrix conditions.     

Factor transformation to produce statistics describing the uncertainty of analytical data.

Analytical uncertainty produced by random error has a positively skewed distribution and accuracy and precision have non-linear scales. Compared with conventional statistics, factor transformation of the data allows more appropriate interpretation of results and facilitates graphical inspection of data. Statistics are compared for practical examples of performance in proficiency tests and of repeatability and reproducibility in collaborative studies. Factor transformation is shown to be applicable to wide ranges of analyte concentration and measurement precision.     

Estimation of uncertainty for the determination of mercury in food by CVAAS

This paper is aimed at reporting a full validation of a Cold Vapour Atomic Absorption Spectrometry (CVAAS) method for mercury determination in fishery products. Method precision, trueness, limit of detection and limit of quantification are evaluated. The uncertainty of measurement is estimated following the bottom-up approach in conjunction with the validation data as suggested for complex analysis. The precision is evaluated using a pooled relative standard deviation and the recovery is added to the measurement uncertainty budget. Uncertainties of weight of sample, dilution factor, calibration, work solution and analyte amount are also assessed. The most relevant uncertainty sources result those associated to amount of mercury determined in the final sample solution, to method precision and to recovery. The relative contributions have different weights according to the analyst decision regarding recovery correction of results. In conclusion, the adopted CVAAS method fully complies with EU requirements. Presented at AOAC Europe/Eurachem Symposium March 2005, Brussels, Belgium     

A model to set measurement quality objectives and to establish measurement uncertainty expectations in analytical chemistry laboratories using ASTM proficiency test data

A model is presented that correlates historical proficiency test data as the log of interlaboratory standard deviations versus the log of analyte concentrations, independent of analyte (measurand) or matrix. Analytical chemistry laboratories can use this model to set their internal measurement quality objectives and to apply the uncertainty budget process to assign the maximum allowable variation in each major step in their bias-free measurement systems. Laboratories that are compliant with this model are able to pass future proficiency tests and demonstrate competence to laboratory clients and ISO 17025 accreditation bodies. Electronic supplementary material to this paper can be obtained by using the Springer LINK server located at http://dx.doi.org/ 10.1007/s007690100398-y. Received: 31 March 2001 Accepted: 11 September 2001     

Assessment of limits of detection and quantitation using calculation of uncertainty in a new method for water determination

 An approach to the assessment of the limit of detection and the limit of quantitation using uncertainty calculation is discussed. The approach is based on the known evaluation of the limits of detection and quantitation as concentrations of the analyte equal to three and ten standard deviations of the blank response, respectively. It is shown that these values can be calculated as the analyte concentrations, for which relative expanded uncertainty achieves 66% and 20% of possible results of the analyte determination, correspondingly. For example, the calculation is performed for the validation of a new method for water determination in the presence of ene-diols or thiols, developed for analysis of chemical products, drugs or other materials which are unsuitable for direct Karl Fischer titration. A good conformity between calculated values and experimental validation data is observed. Received: 27 July 1998 · Accepted: 29 November 1998     

Quantities and units in analytical chemistry

Analytical chemistry is largely concerned with the determination of the composition of mixtures. The result of the analysis of a component in a mixture should comprise the product of a ‘numerical value’ and a ‘unit’ in order to express the value of the ‘quantity’ being measured (and an associated statement of uncertainty). The quantities and units which can be used to express these results are subtly different and can often be confused and misused. This article clarifies their meaning, presents a novel method of demonstrating the relationship between them, and discusses the advantages and drawbacks of their usage in analytical chemistry, particularly with respect to environmental analysis. Suggestions for best practice for use in analytical chemistry are also made.     

Measurement uncertainty of food carotenoid determination

For consistent interpretation of an analytical method result it is necessary to evaluate the confidence that can be placed in it, in the form of a measurement uncertainty estimate. The Guide to the expression of Uncertainty in Measurement issued by ISO establishes rules for evaluating and expressing uncertainty. Carotenoid determination in food is a complex analytical process involving several mass transfer steps (extraction, evaporation, saponification, etc.), making difficult the application of these guidelines. The ISO guide was interpreted for analytical chemistry by EURACHEM, which includes the possibility of using intra- and interlaboratory information. Measurement uncertainty was estimated based on laboratory validation data, including precision and method performance studies, and also, based on laboratory participation in proficiency tests. These methods of uncertainty estimation were applied to analytical results of different food matrices of fruits and vegetables. Measurement uncertainty of food carotenoid determination was 10–30% of the composition value in the great majority of cases. Higher values were found for measurements near instrumental quantification limits (e.g. 75% for β-cryptoxanthin, and 99% for lutein, in pear) or when sample chromatograms presented interferences with the analyte peak (e.g. 44% for α-carotene in orange). Lower relative expanded measurement uncertainty values (3–13%) were obtained for food matrices/analytes not requiring the saponification step. Based on these results, the saponification step should be avoided if food carotenoids are not present in the ester form. Food carotenoid content should be expressed taking into account the measurement uncertainty; therefore the maximum number of significant figures of a result should be 2.     

Interlaboratory study of a liquid chromatography method for erythromycin: determination of uncertainty

Erythromycin is a mixture of macrolide antibiotics produced by Saccharopolyspora erythreas during fermentation. A new method for the analysis of erythromycin by liquid chromatography has previously been developed. It makes use of an Astec C18 polymeric column. After validation in one laboratory, the method was now validated in an interlaboratory study. Validation studies are commonly used to test the fitness of the analytical method prior to its use for routine quality testing. The data derived in the interlaboratory study can be used to make an uncertainty statement as well. The relationship between validation and uncertainty statement is not clear for many analysts and there is a need to show how the existing data, derived during validation, can be used in practice. Eight laboratories participated in this interlaboratory study. The set-up allowed the determination of the repeatability variance, s(2)r and the between-laboratory variance, s(2)L. Combination of s(2)r and s(2)L results in the reproducibility variance s(2)R. It has been shown how these data can be used in future by a single laboratory that wants to make an uncertainty statement concerning the same analysis.     

Capabilities of laboratories to determine melamine in food—results of an international proficiency test

A proficiency test to assess the capabilities of laboratories to determine melamine in a milk powder and a baking mix, representing starch-containing foods like bread and biscuits, was carried out in January 2009. The need for such an interlaboratory comparison arose from a health scare in China about melamine-tainted powdered milk in the second half of 2008. Laboratories in 31 countries, including Australia, China, India, Japan, New Zealand and the USA, and 21 of the 27 Member States of the European Union participated and reported back 114 results for the milk powder and 112 for the baking mix test materials. The reported results were compared to reference values determined by exact-matching double isotope dilution mass spectrometry. The so-determined assigned values were 10.0 ± 0.6 mg/kg melamine in the milk powder and 3.18 ± 0.17 mg/kg melamine in the baking mix. A coverage factor k of 2 was applied to calculate the expanded uncertainties. Three quarters of all reported results for both materials had associated z scores which were satisfactory (z ≤ |2|). Of the reported results, 90% was accompanied by a measurement uncertainty statement, and the majority of the measurement uncertainty ranges were reasonable. A number of laboratories were found to underestimate their measurement uncertainties. Methods that involved the use of stable-isotope-labelled melamine were shown to be clearly advantageous with regard to the accuracy of the results. However, no significant influence by other method parameters could be identified.     

Areas of application and characteristics of quantitative chromatographic analysis by the consecutive standard addition method

Quantitative chromatographic analysis by the standard addition method was recommended for determining analytes in the samples whose matrices possess sorption properties. It was found that the use of several consecutive additives increased the accuracy of results, and it is applicable under the conditions of the nonlinear dependence of detector signals on the analyte amount, in particular, in the HPLC analysis with electrospray ionization and mass-spectrometric detection. Different dependences of the analyte concentrations to be determined on the total amounts of additives were established; because of this, the results should be extrapolated to either zero or infinitely large amounts of additives. A criterion was proposed for predicting a method for the extrapolation of experimental data.     

Empirical versus modelling approaches to the estimation of measurement uncertainty caused by primary sampling

Measurement uncertainty is a vital issue within analytical science. There are strong arguments that primary sampling should be considered the first and perhaps the most influential step in the measurement process. Increasingly, analytical laboratories are required to report measurement results to clients together with estimates of the uncertainty. Furthermore, these estimates can be used when pursuing regulation enforcement to decide whether a measured analyte concentration is above a threshold value. With its recognised importance in analytical measurement, the question arises of 'what is the most appropriate method to estimate the measurement uncertainty?'. Two broad methods for uncertainty estimation are identified, the modelling method and the empirical method. In modelling, the estimation of uncertainty involves the identification, quantification and summation (as variances) of each potential source of uncertainty. This approach has been applied to purely analytical systems, but becomes increasingly problematic in identifying all of such sources when it is applied to primary sampling. Applications of this methodology to sampling often utilise long-established theoretical models of sampling and adopt the assumption that a 'correct' sampling protocol will ensure a representative sample. The empirical approach to uncertainty estimation involves replicated measurements from either inter-organisational trials and/or internal method validation and quality control. A more simple method involves duplicating sampling and analysis, by one organisation, for a small proportion of the total number of samples. This has proven to be a suitable alternative to these often expensive and time-consuming trials, in routine surveillance and one-off surveys, especially where heterogeneity is the main source of uncertainty. A case study of aflatoxins in pistachio nuts is used to broadly demonstrate the strengths and weakness of the two methods of uncertainty estimation. The estimate of sampling uncertainty made using the modelling approach (136%, at 68% confidence) is six times larger than that found using the empirical approach (22.5%). The difficulty in establishing reliable estimates for the input variable for the modelling approach is thought to be the main cause of the discrepancy. The empirical approach to uncertainty estimation, with the automatic inclusion of sampling within the uncertainty statement, is recognised as generally the most practical procedure, providing the more reliable estimates. The modelling approach is also shown to have a useful role, especially in choosing strategies to change the sampling uncertainty, when required.     

Methods of calibration in the direct analysis of solid samples by electrothermal atomic absorption spectrometry

One of the major problems involved in the direct analysis of solid samples by electrothermal atomic absorption spectrometry (ETAAS) lies in the calibration step because non-spectral interference effects are often pronounced. Three standardization techniques have been described and used in solid sampling-ETAAS: (i) standard additions method; (ii) calibration relative to a certified reference material; and (iii) calibration curve technique. However, an adequate statistical evaluation of the uncertainty in the analyte concentration in the solid sample is most frequently neglected, and reported errors may be seriously underestimated. This can be attributed directly to the complexity of the statistical expressions required to accurately account for errors in each of the calibration techniques mentioned above, and the general lack of relevant reference literature. The object of this work has been to develop a computer package which will perform the necessary statistical analyses of solid sampling-ETAAS data; the result is the program “SOLIDS” described here in the form of an electronic publication in Spectrochimica Acta Electronica, the electronic section of Spectrochimica Acta Part B. The program could also be useful in other analytical fields where similar calibration methods are used. The hard copy text, outlining the calibration models and their associated errors, is accompanied by a diskette containing the program, some data files, and a manual. Use of the program is exemplified in the text, with some of the data files discussed included on the diskette which, together with the manual, should enable the reader to become familiarized with the operation of the program, and the results generated.     

Determination of coumarin, vanillin, and ethyl vanillin in vanilla extract products: liquid chromatography mass spectrometry method development and validation studies

A LC-MS method was developed for the determination of coumarin, vanillin, and ethyl vanillin in vanilla products. Samples were analyzed using LC-electrospray ionization (ESI)-MS in the positive ionization mode. Limits of detection for the method ranged from 0.051 to 0.073 microg mL(-1). Using the optimized method, 24 vanilla products were analyzed. All samples tested negative for coumarin. Concentrations ranged from 0.38 to 8.59 mg mL(-1) (x =3.73) for vanillin and 0.33 to 2.27 mg mL(-1) (x =1.03) for ethyl vanillin. The measured concentrations are compared to values calculated using UV monitoring and to results reported in a similar survey in 1988. Analytical results, method precision, and accuracy data are presented.     

Quantifying the measurement uncertainty of results from environmental analytical methods

The Eurachem-CITAC Guide Quantifying Uncertainty in Analytical Measurement was put into practice in a public laboratory devoted to environmental analytical measurements. In doing so due regard was given to the provisions of ISO 17025 and an attempt was made to base the entire estimation of measurement uncertainty on available data from the literature or from previously performed validation studies. Most environmental analytical procedures laid down in national or international standards are the result of cooperative efforts and put into effect as part of a compromise between all parties involved, public and private, that also encompasses environmental standards and statutory limits. Central to many procedures is the focus on the measurement of environmental effects rather than on individual chemical species. In this situation it is particularly important to understand the measurement process well enough to produce a realistic uncertainty statement. Environmental analytical methods will be examined as far as necessary, but reference will also be made to analytical methods in general and to physical measurement methods where appropriate. This paper describes ways and means of quantifying uncertainty for frequently practised methods of environmental analysis. It will be shown that operationally defined measurands are no obstacle to the estimation process as described in the Eurachem/CITAC Guide if it is accepted that the dominating component of uncertainty comes from the actual practice of the method as a reproducibility standard deviation.     

Quantifying the measurement uncertainty of results from environmental analytical methods

The Eurachem–CITAC Guide Quantifying Uncertainty in Analytical Measurement was put into practice in a public laboratory devoted to environmental analytical measurements. In doing so due regard was given to the provisions of ISO 17025 and an attempt was made to base the entire estimation of measurement uncertainty on available data from the literature or from previously performed validation studies. Most environmental analytical procedures laid down in national or international standards are the result of cooperative efforts and put into effect as part of a compromise between all parties involved, public and private, that also encompasses environmental standards and statutory limits. Central to many procedures is the focus on the measurement of environmental effects rather than on individual chemical species. In this situation it is particularly important to understand the measurement process well enough to produce a realistic uncertainty statement. Environmental analytical methods will be examined as far as necessary, but reference will also be made to analytical methods in general and to physical measurement methods where appropriate. This paper describes ways and means of quantifying uncertainty for frequently practised methods of environmental analysis. It will be shown that operationally defined measurands are no obstacle to the estimation process as described in the Eurachem/CITAC Guide if it is accepted that the dominating component of uncertainty comes from the actual practice of the method as a reproducibility standard deviation.     

Current perspectives and recommendations for the development of mass spectrometry methods for the determination of allergens in foods

Allergen detection and quantification is an essential part of allergen management as practiced by food manufacturers. Recently, protein MS methods (in particular, multiple reaction monitoring experiments) have begun to be adopted by the allergen detection community to provide an alternative technique to ELISA and PCR methods. MS analysis of proteins in foods provides additional challenges to the analyst, both in terms of experimental design and methodology: (1) choice of analyte, including multiplexing to simultaneously detect several biologically relevant molecules able to trigger allergic reactions; (2) choice of processing stable peptide markers for different target analytes that should be placed in publicly available databases; (3) markers allowing quantification (e.g., through standard addition or isotopically labeled peptide standards); (4) optimization of protease digestion protocols to ensure reproducible and robust method development; and (5) effective validation of methods and harmonization of results through the use of naturally incurred reference materials spanning several types of food matrix.     

Comments to EURACHEM/CITAC guide “Measurement uncertainty arising from sampling”

The EURACHEM/CITAC Guide “Measurement uncertainty arising from sampling” deals with the design and analysis of experiments for the evaluation of the sampling and analytical standard deviation when a defined sampling and analytical method is used for the determination of the concentration, expressed as mass fraction (mg/kg), of an analyte in a specified material. The Guide recommends reporting the relative expanded uncertainty and using it directly, i.e. it implicitly assumes that the standard deviation is proportional to the mass fraction even in case the experimental data do not support this assumption. Example A1 (and some of the other examples of the Guide) demonstrates that this can result in extreme levels of underestimation or overestimation of the uncertainty of measurement results. Hence, such recommendations should be avoided!     

Validation of the uncertainty evaluation for the determination of metals in solid samples by atomic spectrometry

 Every analytical result should be expressed with some indication of its quality. The uncertainty as defined by Eurachem ("parameter associated with the result of a measurement that characterises the dispersion of the values that could reasonably be attributed to the, . . ., quantity subjected to measurement") is a good tool to accomplish this goal in quantitative analysis. Eurachem has produced a guide to the estimation of the uncertainty attached to an analytical result. Indeed, the estimation of the total uncertainty by using uncertainty propagation laws is components-dependent. The estimation of some of those components is based on subjective criteria. The identification of the uncertainty sources and of their importance, for the same method, can vary from analyst to analyst. It is important to develop tools which will support each choice and approximation. In this work, the comparison of an estimated uncertainty with an experimentally assessed one, through a variance test, is performed. This approach is applied to the determination by atomic absorption of manganese in digested samples of lettuce leaves. The total uncertainty estimation is calculated assuming 100% digestion efficiency with negligible uncertainty. This assumption was tested. Received: 3 November 1997 · Accepted: 2 January 1998     

Measurement uncertainty in analytical methods in which trueness is assessed from recovery assays

We propose a new procedure for estimating the uncertainty in quantitative routine analysis. This procedure uses the information generated when the trueness of the analytical method is assessed from recovery assays. In this paper, we assess trueness by estimating proportional bias (in terms of recovery) and constant bias separately. The advantage of the procedure is that little extra work needs to be done to estimate the measurement uncertainty associated to routine samples. This uncertainty is considered to be correct whenever the samples used in the recovery assays are representative of the future routine samples (in terms of matrix and analyte concentration). Moreover, these samples should be analysed by varying all the factors that can affect the analytical method. If they are analysed in this fashion, the precision estimates generated in the recovery assays take into account the variability of the routine samples and also all the sources of variability of the analytical method. Other terms related to the sample heterogeneity, sample pretreatments or factors not representatively varied in the recovery assays should only be subsequently included when necessary. The ideas presented are applied to calculate the uncertainty of results obtained when analysing sulphides in wine by HS-SPME-GC.