Comparison of different approaches to estimate the uncertainty of a liquid chromatographic assay

A measurement result cannot be properly interpreted without knowledge about its uncertainty. Several concepts to estimate the uncertainty of a measurement result have been developed. Here, four different approaches for uncertainty estimation are compared on the example of the RP-high-performance liquid chromatography (HPLC) assay for tylosin for veterinary use: the guide to the expression of uncertainty in measurement (GUM) approach, which derives the uncertainty of a measurement result by combining the uncertainties related to the uncertainty sources of the measurement process; the top-down approach, which uses the reproducibility estimate from an inter-laboratory study as uncertainty estimate; an approach recently presented by Barwick and Ellison, which combines precision, trueness and robustness data to obtain an uncertainty estimate of the measurement result and finally a further approach, which directly estimates the measurement uncertainty from a robustness test. The comparison shows that the different approaches lead to comparable uncertainty estimates.     

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.     

Gas chromatography analysis: method validation and measurement uncertainty evaluation for volume fraction measurements of gases in simulated reformate gas stream

It is important to understand each analytical system and its limitations when performing any chromatographic measurements. In the present paper, a methodology for method validation and measurement uncertainty evaluation for the measurement of volume fractions of selected gases (CO₂, CO, CH₄, H₂) in simulated reformate gas streams by using gas chromatography was developed. A detailed procedure for in-house method validation based on a simple experimental design and consistent statistics is presented. The analytical protocol allowed us to quantify gases in volume fractions from 2.00 to 100.0 mL/(100 mL) with satisfactory recoveries. We proved that the method was selective for the measurement of gases in simulated reformate gas stream. In addition, a step-by-step illustration of modelling approach for measurement uncertainty evaluation of each component is also provided. Uncertainty arising from repeatability and trueness is relatively low, while the contribution from reproducibility is of higher level for all the analytes tested. The main reason for this is changes in atmospheric pressure that affect gas chromatographic measurements. Solution of this problem could be more frequent calibration of apparatus, yielding to higher costs and more time-consuming process, or by measuring the atmospheric pressure and using it to correct the response of the gas chromatograph for resulting variations in sample size. The obtained results confirm that it is imperative to fully characterize the analytical system before proceeding with an analysis.     

基于自适应蒙特卡洛法评定测量不确定度程序开发与应用

进行了基于自适应蒙特卡洛法评定测量不确定度的程序开发与应用。基于Python语言,设计开发自适应蒙特卡洛法评定测量不确定度程序,包含评定过程框架、自定义变量名称模块、过程参数关联计算模块以及蒙特卡洛法采样计算模块。程序界面简洁,操作简单,计算准确,适用于任意多个独立变量、任意多个过程参数及单一被测量的数学模型,为利用自适应蒙特卡洛法评定测量不确定度提供了方便。     

Highly accurate gas chromatographic analysis of reference gases for use in vehicle emission measurements

 A highly accurate gas chromatographic analytical method has been developed for the determination of the composition of gas mixtures. It was tested using a reference gas as an example consisting of 3.5% of CO, 14% of CO₂, 0.2% of propane and residual N₂ intended for the use in vehicle emission measurements. The method is based on comparison measurements with samples of a calibration gas, whose composition is iteratively adapted to that of the sample investigated using a gravimetric gas mixing method. For the gas chromatographic measurement, a molecular sieve column and a polymer column are used in parallel and in isothermal operation. All gas components can be determined by a single gas chromatographic measurement, and the relative uncertainty of measurement achievable is ?0.4%. Received: 6 March 1996/Revised: 30 May 1996/Accepted: 6 June 1996     

Highly accurate gas chromatographic analysis of reference gases for use in vehicle emission measurements

A highly accurate gas chromatographic analytical method has been developed for the determination of the composition of gas mixtures. It was tested using a reference gas as an example consisting of 3.5% of CO, 14% of CO₂, 0.2% of propane and residual N₂ intended for the use in vehicle emission measurements. The method is based on comparison measurements with samples of a calibration gas, whose composition is iteratively adapted to that of the sample investigated using a gravimetric gas mixing method. For the gas chromatographic measurement, a molecular sieve column and a polymer column are used in parallel and in isothermal operation. All gas components can be determined by a single gas chromatographic measurement, and the relative uncertainty of measurement achievable is ?0.4%.     

Estimating uncertainty in matrix solid phase extraction methodology for the determination of chlorinated compounds in fish

In this study, a detailed process of uncertainty estimation associated with matrix solid phase dispersion method (MSPD) for the extraction of organochlorinated compounds from fish coupled with gas chromatography is given. The evaluation of uncertainty arising from trueness using fully nested experimental designs is presented by estimating proportional bias, in terms of recovery. The uncertainty of measurements associated with the actual chromatographic process was also estimated. The methodology was applied for the determination of selected organochlorine pesticides (OCPs) and polychlorinated biphenyls (PCBs) in real fish samples, from selected study locations in Western Greece (Mesologgi lagoon and Trichonida lake), based on chromatographic techniques. The sources of uncertainty are presented along with the calculated combined uncertainty terms derived from the analytical procedure and chromatographic process. The estimation of uncertainty was included in the presentation of final determinations.     

蒙特卡洛法评定橡胶门尼黏度测量的不确定度

根据高聚物流变学原理,建立了门尼黏度测量模型,分析了模型中各个变量的概率分布,利用Mathcad软件进行了测量模型的门尼黏度模拟,给出了模拟最佳值、不确定度及其包含区间,实现了门尼黏度测量不确定度的蒙特卡洛法评定。与GUM法相比,蒙特卡洛法评定门尼黏度测量不确定度具有编程模式化,过程简单等优点,适合多变量测量模型的不确定度评价。     

Measurement uncertainty of isotopologue fractions in fluxomics determined via mass spectrometry

Metabolic flux analysis implies mass isotopomer distribution analysis and determination of mass isotopologue fractions (IFs) of proteinogenic amino acids of cell cultures. In this work, for the first time, this type of analysis is comprehensively investigated in terms of measurement uncertainty by calculating and comparing budgets for different mass spectrometric techniques. The calculations addressed amino acids of Pichia pastoris grown on 10 % uniformly ¹³C labeled glucose. Typically, such experiments revealed an enrichment of ¹³C by at least one order of magnitude in all proteinogenic amino acids. Liquid chromatography–time-of-flight mass spectrometry (LC-TOFMS), liquid chromatography–tandem mass spectrometry (LC-MS/MS) and gas chromatography–mass spectrometry (GC-MS) analyses were performed. The samples were diluted to fit the linear dynamic range of the mass spectrometers used (10 μM amino acid concentration). The total combined uncertainties of IFs as well as the major uncertainty contributions affecting the IFs were determined for phenylalanine, which was selected as exemplary model compound. A bottom-up uncertainty propagation was performed according to Quantifying Uncertainty in Analytical Measurement and using the Monte Carlo method by considering all factors leading to an IF, i.e., the process of measurement and the addition of ¹³C-glucose. Excellent relative expanded uncertainties (k?=?1) of 0.32, 0.75, and 0.96 % were obtained for an IF value of 0.7 by LC-MS/MS, GC-MS, and LC-TOFMS, respectively. The major source of uncertainty, with a relative contribution of 20–80 % of the total uncertainty, was attributed to the signal intensity (absolute counts) uncertainty calculated according to Poisson counting statistics, regardless which of the mass spectrometry platforms was used. Uncertainty due to measurement repeatability was of importance in LC-MS/MS, showing a relative contribution up to 47 % of the total uncertainty, whereas for GC-MS and LC-TOFMS the average contribution was lower (30 and 15 %, respectively). Moreover, the IF actually present also depends on the isotopic purity of the carbon sources. Therefore, in the uncertainty calculation a carbon source purity factor was introduced and a minor contribution to the total uncertainty was observed. The results obtained by uncertainty calculation performed according to the Monte Carlo method were in agreement with the uncertainty value of the Kragten approach and showed a Gaussian distribution.     

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.     

Approach to accuracy assessment of the glass-electrode potentiometric determination of acid-base properties

Thermodynamic data are suitable subject for investigating strategies and concepts for the evaluation of complete measurement uncertainty budgets in situations where the measurand cannot be expressed in a mathematical formula. Some suitable approaches are the various forms of Monte Carlo simulations in combination with computer-intensive statistical methods that are directed to an evaluation of empirical distribution curves for the uncertainty budget. Basis of the analysis is a cause-and-effect diagram. Some experience is available with cause-and-effect analysis of thermodynamic data derived from spectrophotometric data. Another important technique for the evaluation of thermodynamic data is glass-electrode potentiometry. On basis of a newly derived cause-and-effect diagram, a complete measurement uncertainty budget for the determination of the acidity constants of phosphoric acid by glass-electrode potentiometry is derived. A combination of Monte Carlo and bootstrap methods is applied in conjunction with the commercially available code SUPERQUAD. The results suggest that glass-electrode potentiometry may achieve a high within-laboratory precision because major uncertainty contributions become evident via interlaboratory comparisons. This finding is further underscored by analysing available literature data.     

Calibration and uncertainty assessment for certified reference gas mixtures

The weighted least squares method to build an analysis function described in ISO 6143, Gas analysis—Comparison methods for determining and checking the composition of calibration gas mixtures, is modified to take into account the typically small number of instrumental readings that are obtained for each primary standard gas mixture used in calibration. The theoretical basis for this modification is explained, and its superior performance is illustrated in a simulation study built around a concrete example, using real data. The corresponding uncertainty assessment is obtained by application of a Monte Carlo method consistent with the guidance in the Supplement 1 to the Guide to the expression of uncertainty in measurement, which avoids the need for two successive applications of the linearizing approximation of the conventional method for uncertainty propagation. The three main steps that NIST currently uses to certify a reference gas mixture (homogeneity study, calibration, and assignment of value and uncertainty assessment), are described and illustrated using data pertaining to an actual standard reference material.     

A Monte Carlo approach for estimating measurement uncertainty using standard spreadsheet software

Despite the importance of stating the measurement uncertainty in chemical analysis, concepts are still not widely applied by the broader scientific community. The Guide to the expression of uncertainty in measurement approves the use of both the partial derivative approach and the Monte Carlo approach. There are two limitations to the partial derivative approach. Firstly, it involves the computation of first-order derivatives of each component of the output quantity. This requires some mathematical skills and can be tedious if the mathematical model is complex. Secondly, it is not able to predict the probability distribution of the output quantity accurately if the input quantities are not normally distributed. Knowledge of the probability distribution is essential to determine the coverage interval. The Monte Carlo approach performs random sampling from probability distributions of the input quantities; hence, there is no need to compute first-order derivatives. In addition, it gives the probability density function of the output quantity as the end result, from which the coverage interval can be determined. Here we demonstrate how the Monte Carlo approach can be easily implemented to estimate measurement uncertainty using a standard spreadsheet software program such as Microsoft Excel. It is our aim to provide the analytical community with a tool to estimate measurement uncertainty using software that is already widely available and that is so simple to apply that it can even be used by students with basic computer skills and minimal mathematical knowledge.     

The uncertainty in the calculation of the amount of blocked and reactive lysine in milk products,as determined by the furosine method

The measurement uncertainty in the calculation of the amount of blocked and reactive lysine (as determined by the furosine method) was evaluated according to the procedure described in the Eurachem/CITAC guide. The analytical method involves the chromatographic determination of lysine and furosine after acid hydrolysis. The calculation of blocked and reactive lysine in the initial protein is based on known conversion factors. The estimation of the uncertainty was performed in two steps: (1) determination of the uncertainty in the chromatographic determination of lysine and furosine, and (2) determination of the uncertainty in the calculation of blocked and reactive lysine. The individual contributions to the final uncertainty were identified, quantified, and combined in uncertainty budgets. The largest contribution to the calculation of blocked lysine came from estimating the conversion factor of blocked lysine into furosine during acid hydrolysis. For the calculation of reactive lysine, the main contribution came from the chromatographic determination of lysine. The uncertainty estimates were compared to available validation data (in-house and collaborative standard deviations of reproducibility).     

Effects of overlying soft tissue on X-ray fluorescence bone lead measurement uncertainty

The effects of overlying soft tissue on the measurement uncertainty of the in vivo ¹⁰⁹Cd K-shell X-ray fluorescence (XRF) technique were investigated, as applied to the tibia bone site. Experimental measurements were performed on a set of nine leg phantoms of different soft tissue thickness, intended to model the lower leg at mid-tibia. A standard bone phantom made from plaster-of-Paris and having a nominal lead concentration of 25.6 μg Pb per gram was used in all trials. Monte Carlo simulations of the experimental set-up were also performed. Results indicate a strong relation between measurement uncertainty and overlying tissue thickness (OTT) for the XRF bone lead method. In increasing the OTT from 3.2 to 14.6 mm, an increase in average measurement uncertainty by a factor of 2.40 was observed experimentally. Monte Carlo simulations indicated an increase in minimum detectable limit (MDL) by a factor of 2.46 over the same interval. Experimental and Monte Carlo results were generally in strong agreement. For subject screening purposes, direct measurement of soft tissue overlying the tibia is recommended whenever practical.     

Monte-carlo simulation of gas chromatographic separation for the prediction of retention times and peak half widths

Summary A new method for prediction of gas chromatographic retention times and peak half widths is based on the renewal theory. The only requirements are the heats of vaporization of the compounds to be separated and one calibration measurement. With this data, retention times and peak half widths can be predicted for isothermal as well as temperature-programmed gas chromatography. For the separation of non-polar substances on non-polar stationary phases the prediction error for retention times is approx. 1–2%. First simulations of polar molecules and polar stationary phases indicate that this method is also applicable in these cases but some extension will be required.     

Problems and risks occurred during uncertainty evaluation of a quantity calculated from correlated parameters: a case study of pH measurement

Parameters of a model describing a measurement process obtained during a calibration experiment allow one to calculate a measurement result, but a simple estimation of measurement uncertainties of the parameters is not sufficient to assess the uncertainty of the result. In this paper, an example of a pH measurement conducted using an ion-selective electrode is presented, in which the uncertainty is evaluated taking into consideration the existing correlation between the parameters of the electrode. The calculations apply either covariances or correlation coefficients that have to be computed additionally. The example presented in this paper illustrates that there are some problems with rounding of variables which, because of the existing very strong correlations, significantly changes the sought uncertainty. This approach is compared with other approaches, that is, usage of uncorrelated variables and Monte Carlo simulations that are described in an earlier work. It is concluded that the approach of uncertainty evaluation, in which covariances or correlation coefficients are explicitly calculated, is work-consuming and may cause significant discrepancies between correct and obtained assessments if some roundings or approximations are done, or if the correlation coefficient is obtained experimentally based on data including random errors.     

Application of capillary gas chromatography to studies on solvation thermodynamics

The potentiality of capillary gas chromatography (GC) as a means for research on solubility phenomena is focused. Basic thermodynamic information can be obtained in a simple and direct way from this technique relying on few parameters with their associated errors tightly controlled. An unexplored field of solvation phenomenology inaccessible to other techniques is revealed by the accuracy of capillary GC, provided that relevant chromatographic variables are utilized and an adequate treatment of the experimental information performed. The present article reviews different approaches for the attainment of basic thermodynamic information through capillary GC. Some traditional concepts on the treatment of chromatographic data for physicochemical measurement are questioned. Applications of the technique to research on solubility phenomena are depicted.     

化学计量中不确定度评定研究进展

测量不确定度是表征合理地赋予被测量之值的分散性的参数。本文针对化学计量不确定度评定基础模型仅适用于线性模型、概率分布为正态分布或缩放位移t分布等局限,介绍了近年来不确定度评定的研究热点:蒙特卡罗方法(Monte Carlo Method,MCM),不确定度评定的来源、评定概念、评估方法及其发展过程,扩大了测量不确定度评定与表示的适用范围。     

Capillary chromatographic analysis of volatile organic compounds in the indoor environment

The wide variety of volatile organic compounds found in the indoor environment can present a difficult chromatographic problem for the analyst. Capillary-column gas chromatography is required to achieve adequate separation. Thermal desorption/capillary gas chromatography/mass spectrometry is usually the preferred method for broad spectrum analysis of the volatile organics detected indoors. Cryogenic trapping is required for capillary gas chromatographic analysis of the more volatile components.