Uncertainty in photometric analysis: a case study

A procedure for estimation of measurement uncertainty of photometric analysis based on the ISO GUM method is presented. Two variations of the procedure—for the calibration graph and the standard addition method, respectively—are discussed. The variations are based on mathematical models involving 64 and 80 input quantities, respectively. The uncertainty of the result strongly depends on changes in experimental details. These dependencies are explored for a practical example of determination of the iron content of aluminum. The importance of taking uncertainty from sample preparation into account in uncertainty estimation is stressed. The number of effective degrees of freedom is calculated and discussed. The examples are available as GUM Workbench files in the Electronic Supplementary Material.Electronic Supplementary Material  Supplementary material is available for this article at .

Comparison of ISO-GUM and Monte Carlo methods for the evaluation of measurement uncertainty: application to direct cadmium measurement in water by GFAAS

The propagation stage of uncertainty evaluation, known as the propagation of distributions, is in most cases approached by the GUM (Guide to the Expression of Uncertainty in Measurement) uncertainty framework which is based on the law of propagation of uncertainty assigned to various input quantities and the characterization of the measurand (output quantity) by a Gaussian or a t-distribution. Recently, a Supplement to the ISO-GUM was prepared by the JCGM (Joint Committee for Guides in Metrology). This Guide gives guidance on propagating probability distributions assigned to various input quantities through a numerical simulation (Monte Carlo Method) and determining a probability distribution for the measurand.In the present work the two approaches were used to estimate the uncertainty of the direct determination of cadmium in water by graphite furnace atomic absorption spectrometry (GFAAS). The expanded uncertainty results (at 95% confidence levels) obtained with the GUM Uncertainty Framework and the Monte Carlo Method at the concentration level of 3.01 μg/L were ±0.20 μg/L and ±0.18 μg/L, respectively. Thus, the GUM Uncertainty Framework slightly overestimates the overall uncertainty by 10%. Even after taking into account additional sources of uncertainty that the GUM Uncertainty Framework considers as negligible, the Monte Carlo gives again the same uncertainty result (±0.18 μg/L). The main source of this difference is the approximation used by the GUM Uncertainty Framework in estimating the standard uncertainty of the calibration curve produced by least squares regression. Although the GUM Uncertainty Framework proves to be adequate in this particular case, generally the Monte Carlo Method has features that avoid the assumptions and the limitations of the GUM Uncertainty Framework.     

Determination of measurement uncertainty for the determination of triazines in groundwater from validation data

Laboratories are increasingly urged to submit full uncertainties of their analytical results rather than only standard deviations. The determination of measurement uncertainties in compliance with the Guide to the Expression of Uncertainty in Measurement (GUM) is demonstrated using the validation approach explicitly endorsed by the recent edition of the EURACHEM guide for the determination of measurement uncertainty. Measurement uncertainty was split into uncertainty of the sample mass, uncertainty of the concentration of the stock standard solution, uncertainty of the calibration and uncertainty connected to within- and between-series precision. Uncertainties of sample mass and of the concentration of the stock standard solution were 0.26 and 1.14% for all analytes, which is negligible compared with the contributions of precision and calibration. Uncertainty of calibration was estimated from the calibration graph. Relative uncertainty of calibration was found to be strongly concentration dependent and to be the main uncertainty contribution below 0.2 microgram L-1. Precision was split into within-series and between-series standard deviation, which dominate the combined standard uncertainty at higher concentrations. The results obtained from these calculations are compared with results for a certified reference material and with the performance in an interlaboratory comparison. It was found that all results agreed within their uncertainty with the target values, showing that the estimated uncertainties are realistic.     

Uncertainty in liquid chromatographic analysis of pharmaceutical product: influence of various uncertainty sources

An ISO GUM measurement uncertainty estimation procedure was developed for a liquid-chromatographic drug quality control method-assay of simvastatin in drug formulation. In quantification of uncertainty components several practical approaches for including difficult-to-estimate uncertainty sources (such as uncertainty due to peak integration, uncertainty due to nonlinearity of the calibration curve, etc.) have been presented. Detailed analysis of contributions of the various uncertainty sources was carried out. The results were calculated based on different definitions of the measurand and it was demonstrated that unequivocal definition of the measurand is essential in order to get rigorous uncertainty estimate. Two different calibration methods - single-point (1P) and five-point (5P) - were used and the obtained uncertainties and uncertainty budgets were compared. Results calculated using 1P and 5P calibrations agree very well. The uncertainty estimate for 1P is only slightly larger than with 5P calibration.     

Modelling the calibration of the industrial platinum-resistance thermometers according to GUM

According to the Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100: 2008), the calibration process and its uncertainty evaluation should be expressed in terms of mathematical function(s) of input quantities. However, in practice, expressing measurement or calibration in a way that is fully compliant with GUM might be unrealistic and require a clear definition of the calibration process itself. Depending on the applied calibration process, different modelling equations with various complexities can be written. In this paper, four different approaches are given to model the calibration process of industrial platinum-resistance thermometers.     

Approaches to uncertainty evaluation based on proficiency testing schemes in chemical measurements

Since the advent of the Guide to the Expression of Uncertainty in Measurement (GUM) founding the principles of uncertainty evaluation, numerous projects have been carried out to develop alternative practical methods when it is impossible to model technical or economical aspects of the measurement process. These methods can use all the experimental data available to the laboratories, such as repeatability, reproducibility, quality-control charts, etc. The studies presented in this paper compare the results obtained by the modelling method from GUM with the uncertainties found by applying alternative methods. They show two examples, one in the field of environmental monitoring, the other in the biomedical field, based on the exploitation of PT schemes results. Presented at BERM-11, October 2007, Tsukuba, Japan.     

Estimation of the standard uncertainty of a calibration curve: application to sulfur mass concentration determination in fuels

The construction of a calibration curve using least square linear regression is common in many analytical measurements, and it comprises an important uncertainty component of the whole analytical procedure uncertainty. In the present work, various methodologies are applied concerning the estimation of the standard uncertainty of a calibration curve used for the determination of sulfur mass concentration in fuels. The methodologies applied include the GUM uncertainty framework, the Kragten numerical method, the Monte Carlo method (MCM) as well as the approximate equation calculating the standard error of prediction. The standard uncertainty results obtained by all methodologies agree well (0.172?C0.175?ng???L^?1). Aspects of inappropriate use of the approximate equation of the standard error of prediction, which leads to overestimation or underestimation of calculated uncertainty, are discussed. Moreover, the importance of the correlation between calibration curve parameters (slope and intercept) within GUM, MCM and Kragten approaches is examined.     

Reference measurement procedure for Δ9-tetrahydrocannabinol in serum

An isotope dilution gas chromatography/mass spectrometry (ID-GC/MS) reference measurement procedure for Δ9-tetrahydrocannabinol (THC) in serum was developed and validated. The method complies with the concept of a ratio primary reference measurement procedure. The uncertainty was determined for two concentrations of THC in serum (1 ng/mL and 2.4 ng/mL). The calculation procedure is based on the Guide to the Expression of Uncertainty in Measurement (GUM). The relative expanded uncertainty was found to be less than 2% for both concentration levels, corresponding to a 95% confidence interval. For the reference method, it was shown that the measurement of THC within the concentration range covered by the current threshold values is very accurate. The method has the potential to provide traceability for the methods used in practical forensics.     

A model for the evaluation of uncertainty in routine multi-element analysis

A model for calculating uncertainty in routine multi-element analysis is described. The model is constructed according to the principles of GUM/EURACHEM. Control chart results are combined with other existing data and results from the actual measurement into a concentration-dependent estimate of combined standard uncertainty. Since possible sources of bias are included in the calculation, overall bias as estimated from the data is used only as a control to identify needs for modification of the model and/or the analytical procedure. For each individual sample, uncertainty can be calculated automatically based on two pre-calculated parameters together with measured concentration and instrumental standard deviation. As an example, the model is demonstrated for inductively coupled plasma-mass spectrometry (ICP-MS) analysis of sewage sludge including laboratory sub-sampling, sample preparation, and instrumental determination.     

Estimation of uncertainty in p<Emphasis Type="Italic">K</Emphasis><Subscript>a</Subscript> values determined by potentiometric titration

A procedure is presented for estimation of uncertainty in measurement of the pK(a) of a weak acid by potentiometric titration. The procedure is based on the ISO GUM. The core of the procedure is a mathematical model that involves 40 input parameters. A novel approach is used for taking into account the purity of the acid, the impurities are not treated as inert compounds only, their possible acidic dissociation is also taken into account. Application to an example of practical pK(a) determination is presented. Altogether 67 different sources of uncertainty are identified and quantified within the example. The relative importance of different uncertainty sources is discussed. The most important source of uncertainty (with the experimental set-up of the example) is the uncertainty of pH measurement followed by the accuracy of the burette and the uncertainty of weighing. The procedure gives uncertainty separately for each point of the titration curve. The uncertainty depends on the amount of titrant added, being lowest in the central part of the titration curve. The possibilities of reducing the uncertainty and interpreting the drift of the pK(a) values obtained from the same curve are discussed.     

From GUM to alternative methods for measurement uncertainty evaluation

Since the advent of the Guide to the expression of Uncertainty in Measurement (GUM) in 1995 laying the principles of uncertainty evaluation numerous projects have been carried out to develop alternative practical methods that are easier to implement namely when it is impossible to model the measurement process for technical or economical aspects. In this paper, the author presents the recent evolution of measurement uncertainty evaluation methods. The evaluation of measurement uncertainty can be presented according to two axes based on intralaboratory and interlaboratory approaches. The intralaboratory approach includes “the modelling approach” (application of the procedure described in section 8 of the GUM, known as GUM uncertainty framework) and “the single laboratory validation approach”. The interlaboratory approaches are based on collaborative studies and they are respectively named “interlaboratory validation approach” and “proficiency testing approach”.     

Impact of dissolution on the uncertainty of spent fuel analysis

One of the objectives of the French Alternative Energies and Atomic Energy Commission in the Marcoule Centre is to accurately quantify the composition of nuclear spent fuel, i.e. to determine the concentration of each isotope with suitable measurement uncertainty. These analysis results are essential for the validation of calculation codes used for the simulation of fuel behaviour in nuclear reactors and for nuclear matter accountancy. The different experimental steps are first the reception of a piece of spent fuel rod at the laboratory of dissolution studies, and then dissolution in a hot cell of a sample of the spent fuel rod received. Several steps are necessary to obtain a complete dissolution. Taking into account these process steps, and not only those of analysis for the evaluation of measurement uncertainties, is new, and is described in this paper. The uncertainty estimation incorporating the process has been developed following the approach proposed by the Guide to the Expression of Uncertainty in Measurement (GUM). The mathematical model of measurement was established by examining the dissolution process step by step. The law of propagation of uncertainty was applied to this model. A point by point examination of each step of the process permitted the identification of all sources of uncertainties considered in this propagation for each input variable. The measurement process presented involves the process and the analysis. The contents of this document show the importance of taking the process into account in order to give a more reliable uncertainty assessment to the result of a concentration or isotope ratio of two isotopes in spent fuel.     

Evaluation of measurement uncertainty in analytical assays by means of Monte-Carlo simulation

The main limitations of the Guide to the expression of Uncertainty Measurement (GUM) approach for evaluating the measurement uncertainty of analytical assays are presented and explained. The advantages of using Monte-Carlo simulation against the GUM approach are outlined and discussed and the principle of propagation of distributions is explained. The procedure of Monte-Carlo analysis is illustrated by two case studies. A first simple example quoted from the EURACHEM Guide and dealing with the preparation of a calibration standard is used to present the technique with detail in a step-by-step way. In this case the results obtained by both approaches are very similar. A second example deals with the calibration of mass according to a strong non-linear model. In this case, the Monte-Carlo analysis leads to better results.     

Estimation of uncertainty in routine pH measurement

A procedure for estimation of measurement uncertainty of routine pH measurement (pH meter with two-point calibration, with or without automatic temperature compensation, combination glass electrode) based on the ISO method is presented. It is based on a mathematical model of pH measurement that involves nine input parameters. Altogether 14 components of uncertainty are identified and quantified. No single uncertainty estimate can be ascribed to a pH measurement procedure: the uncertainty of pH strongly depends on changes in experimental details and on the pH value itself. The uncertainty is the lowest near the isopotential point and in the center of the calibration line and can increase by a factor of 2 (depending on the details of the measurement procedure) when moving from around pH 7 to around pH 2 or 11. Therefore it is necessary to estimate the uncertainty separately for each measurement. For routine pH measurement the uncertainty cannot be significantly reduced by using more accurate standard solutions than ±0.02 pH units – the uncertainty improvement is small. A major problem in estimating the uncertainty of pH is the residual junction potential, which is almost impossible to take rigorously into account in the framework of a routine pH measurement.¹ Received: 11 August 2001 Accepted: 22 February 2002     

Measurement uncertainty as a tool for evaluating the ‘grey zone’ to reduce the false negatives in immunochemical screening of blood donors for infectious diseases

The risk of misclassifying infected individuals as healthy constitutes a crucial challenge when screening blood donors by means of immunoassays. This risk is especially challenging when the numerical results are close to the clinical decision level, i.e. in the ‘grey zone’. The concept of using measurement uncertainty for evaluating the ‘grey zone’ has previously not been systematically applied in this context. This article explains methods, models and empirical (top-down) approaches for the calculation of measurement uncertainty using results from a blood bank according to the internationally accepted GUM principles, focusing on uncertainty sources in the analytical phase. Of the different approaches available, the intralaboratory empirical approaches are emphasised since modelling (bottom-up) approaches are impracticable due to the lack of reliable model equations for immunoassays. Different methods are applied to estimate the measurement uncertainty for the Abbott Prism^® HCV immunoassay. The expanded uncertainty obtained at the clinical decision level from the intralaboratory empirical approach was 36 %. The estimated uncertainty was used to set acceptance and rejection zones following the procedure set in the Eurachem guideline, emphasising the need to minimise the occurrence of false negatives.     

Systematic errors in analytical measurement results

Definitions of the concepts of bias and recovery are discussed and approaches to dealing with them described. The Guide To Uncertainty in Measurement (GUM) recommends correction for all significant systematic effects, but it is also possible to expand measurement uncertainty to take account of uncorrected bias. Run, laboratory and method bias can be defined as components of the bias of a particular measurement result, and can be useful as concepts used in method validation. Estimation of run bias allows a simplification of the estimation of measurement uncertainty. Multivariate calibration brings its own biases that must be quantified and minimised.     

SI-traceable certification of the IMEP-12 water sample combining contributions from different reference laboratories

In the International Measurement Evaluation Programme (IMEP-12) comparison, a synthetically prepared water sample was offered to the analytical laboratories to perform measurements of As, B, Cd, Cr, Cu, Fe, Mg, Mn, Ni and Pb. The choice of elements to be measured was based on EU legislation, which the comparison was aiming to support. As to the IMEP policy, the laboratories’ results were presented according to the certified/assigned reference values established by several reference laboratories all around the world. The performed certification campaign is described in detail in this paper. Isotope dilution mass spectrometry (IDMS) was applied as a primary method of measurement (PMM), whenever possible, to achieve SI-traceable results. Apart from IDMS for reference measurements of some elements, k _o-neutron activation analysis (k _o-NAA) and external calibration (Ext-Calib) using inductively coupled plasma-mass spectrometry (ICP-MS) were applied. The reference values were characterised as “certified” (for B, Cd, Cr, Cu, Fe, Mg, Ni and Pb) or “assigned” (for As and Mn) according to the IMEP policy. Measurement uncertainty of the certified/assigned reference values was calculated according to the ISO/BIPM guide using the specialised software GUM Workbench.

     

离子色谱法测定地下水中硝酸盐氮含量的不确定度评定

采用不确定度连续传递模式,讨论了不同回归方式对离子色谱法测定岩溶地下水中硝酸根分析结果不确定度评定的差异.结果表明：（1）测定结果的不确定度主要来源于标准溶液配制过程引入的不确定度、校准曲线拟合及回归过程产生的不确定度和仪器测定过程引入的不确定度三部分;（2）不同校准曲线回归方式对测定结果的不确定度评定在不同测量水平上有不同的影响,当地下水中硝酸根含量越低,差异越大.此外,对不确定度评定过程中的各个分量进行量化,通过合成得到硝酸根测定结果的不确定度评定模型.     

From total allowable error via metrological traceability to uncertainty of measurement of the unbiased result

The concept of "total allowable error", investigated by Westgard and co-workers over a quarter of a century for use in laboratory medicine, comprises bias as well as random elements. Yet, to minimize diagnostic misclassifications, it is necessary to have spatio-temporal comparability of results. This requires trueness obtained through metrological traceability based on a calibration hierarchy. Hereby, the result is associated with a final uncertainty of measurement purged of known biases of procedure and laboratory. The sources of bias are discussed and the importance of commutability of calibrators and analytical specificity of the measurement procedure is stressed. The practicability of traceability to various levels and the advantages of the GUM approach for estimating uncertainty are shown.