Multivariate control charts: Control charts for calibration curves

The proposed multivariate control charts for p-dimensional vectors are an extension of the conventional control charts for one variable. The controlling quantity is the Mahalanobis distance of vector x from the central value vector x..: D=(x-x..)^T-1.(x-x..), where is the covariance matrix estimate. The quantity D has Hotelling's T² distribution. A PC program was set up for the automatic graphical construction of such charts. The program draws the sequential chart of the quantity D as well as the position of the vectors x in the p dimensional control ellipsoid in the axes of the principal components. In this way a control chart was developed for the calibration curve in the photometric determination of Fe³⁺ with sulfosalicylic acid. Vector x was formed by absorbance values for the calibration curve points (p=5). The chart can assist in detection of even small disturbances of the calibration curve.     

Univariate analytical calibration methods and procedures. A review

An original focus on univariate calibration as an experimental process of quantitative analysis is presented. A novel classification system is introduced against the background of the present situation concerning nomenclature of calibration methods. Namely, it has been revealed that four methods well-known in analytical chemistry: the conventional method, the internal standard method, the indirect method and the dilution method, can be split into those carried out in both the interpolative and the extrapolative mode. It is then shown that the basic procedures of all these methods can be modified including different approaches, such as matrix-matched technique, spiking the sample with a reactant, bracketing calibration, and others. For the first time (as compared to monographies dealing with univariate calibration) it is reviewed how the methods are mixed and integrated with one another thereby creating new calibration strategies of extended capabilities in terms of enhanced resistance to the interference and non-linear effects – as the main sources of systematic calibration errors. As additional novelty, rationally possible combinations of the calibration methods – not met hitherto in the literature – have been predicted. Finally, some general rules relating to calibration are formulated and the main calibration problems that still need to be solved are displayed.     

Relation between prediction errors of inverse and classical calibration

The formulae for prediction errors of inverse and classical calibration derived by Centner, Massart and de Jong in the Fresenius’ Journal of Analytical Chemistry (1998) 361 : 2–9 are reconsidered. All calculations assume univariate calibration by ordinary least squares regression applied to an infinite number of data pairs. Inverse calibration gives rise to an error variance which is smaller by a certain factor than that of classical calibration. This factor amounts to unity plus the ratio of the variances of the measurement errors and of the responses used for the calibration. The root mean squared error of prediction is also smaller for inverse than for classical calibration, namely by the square root of this factor. A prediction error calculated in that way agrees well with a result obtained by Monte Carlo simulations. Received: 23 December 1999 / Revised: 14 February 2000 / Accepted: 15 February 2000     

External calibration strategy for trace element quantification in botanical samples by LA-ICP-MS using filter paper

The use of reference solutions dispersed on filter paper discs is proposed for the first time as an external calibration strategy for matrix matching and determination of As, Cd, Co, Cr, Cu, Mn, Ni, Pb, Sr, V and Zn in plants by laser ablation-inductively coupled plasma mass spectrometry (LA-ICP-MS). The procedure is based on the use of filter paper discs as support for aqueous reference solutions, which are further evaporated, resulting in solid standards with concentrations up to 250 μg g⁻¹ of each element. The use of filter paper for calibration is proposed as matrix matched standards due to the similarities of this material with botanical samples, regarding to carbon concentration and its distribution through both matrices. These characteristics allowed the use of ¹³C as internal standard (IS) during the analysis by LA-ICP-MS. In this way, parameters as analyte signal normalization with ¹³C, carrier gas flow rate, laser energy, spot size, and calibration range were monitored. The calibration procedure using solution deposition on filter paper discs resulted in precision improvement when ¹³C was used as IS. The method precision was calculated by the analysis of a certified reference material (CRM) of botanical matrix, considering the RSD obtained for 5 line scans and was lower than 20%. Accuracy of LA-ICP-MS determinations were evaluated by analysis of four CRM pellets of botanical composition, as well as by comparison with results obtained by ICP-MS using solution nebulization after microwave assisted digestion. Plant samples of unknown elemental composition were analyzed by the proposed LA method and good agreement were obtained with results of solution analysis. Limits of detection (LOD) established for LA-ICP-MS were obtained by the ablation of 10 lines on the filter paper disc containing 40 μL of 5% HNO₃ (v v⁻¹) as calibration blank. Values ranged from 0.05 to 0.81  μg g⁻¹. Overall, the use of filter paper as support for dried aqueous standards showed to be a useful strategy for calibration and plant analysis by LA-ICP-MS.     

Assessment of uncertainty in calibration of a gas mass flowmeter

 A primary calibration system was set up in Rafael some years ago, based on volumetric flow rate. The primary standard measures volumetric flow by means of the volume change of a dual piston over a specific time interval. This system serves to calibrate secondary standards of the thermal mass flowmeter type. Calibration procedures were prepared and validated. The paper describes the tests and calibration procedure conducted for the uncertainty assessment, the different components contributing to the measurement uncertainties, and the formulas involved with volumetric flow fates and with thermal mass flowmeters. Received: 10 July 1999 / Accepted: 15 February 2000     

Traceability in routine chemical measurements: an example of application in the determination of CO2 at atmospheric concentration

In routine chemical measurements traceability can be achieved by using analytical instruments calibrated against primary reference materials. In the present work the calibration of a CO₂ non-dispersive infrared (NDIR) analyzer with measuring range 0–2000 μmol/mol of CO₂ and a resolution of 5 μmol/mol is reported. A procedure with working reference gas mixtures (WRMs) has been adopted, which requires seven calibration points. Primary reference gas mixtures (PRMs) are used to validate WRMs in a narrower range around the average atmospheric CO₂ concentration value. In this range the relative uncertainty reached is of the order of some parts in 10³ and the corrections are between 1 μmol/mol and 5 μmol/mol. Received: 16 March 2000 Accepted: 27 November 2000     

Efficient use of pure component and interferent spectra in multivariate calibration

Partial Least Squares (PLS) is by far the most popular regression method for building multivariate calibration models for spectroscopic data. However, the success of the conventional PLS approach depends on the availability of a ‘representative data set’ as the model needs to be trained for all expected variation at the prediction stage. When the concentration of the known interferents and their correlation with the analyte of interest change in a fashion which is not covered in the calibration set, the predictive performance of inverse calibration approaches such as conventional PLS can deteriorate. This underscores the need for calibration methods that are capable of building multivariate calibration models which can be robustified against the unexpected variation in the concentrations and the correlations of the known interferents in the test set. Several methods incorporating ‘a priori’ information such as pure component spectra of the analyte of interest and/or the known interferents have been proposed to build more robust calibration models. In the present study, four such calibration techniques have been benchmarked on two data sets with respect to their predictive ability and robustness: Net Analyte Preprocessing (NAP), Improved Direct Calibration (IDC), Science Based Calibration (SBC) and Augmented Classical Least Squares (ACLS) Calibration. For both data sets, the alternative calibration techniques were found to give good prediction performance even when the interferent structure in the test set was different from the one in the calibration set. The best results were obtained by the ACLS model incorporating both the pure component spectra of the analyte of interest and the interferents, resulting in a reduction of the RMSEP by a factor 3 compared to conventional PLS for the situation when the test set had a different interferent structure than the one in the calibration set.     

Application of direct calibration in multivariate image analysis of heterogeneous materials

Many scientific instruments produce multivariate images characterized by three-way tables, an element of which represents the intensity value at a spatial location for a given spectral channel. A problem frequently encountered is to attempt estimating the contributions of some compounds at each location of these images. Usual regression methods of calibration, such as PLS, require having a matrix of calibration X (n × p) and the corresponding vector y of the dependent variable (n × 1). X can be built up by sampling pixel-vectors in the images, but y is sometimes difficult to obtain, if the surface of the samples is formed by chemically heterogeneous regions. In this case, the quantitative analyses related to y may be difficult, if the pixels represent very small areas (for example on microscopic images) or very large ones (satellite images). This is for example the case when dealing with biological solid samples representing different tissues. Direct Calibration (DC), sometimes referred to as “spectral unmixing”, do not require having such a calibration set. However, it is indeed needed to have both a matrix of “perturbing” pixel-vectors (noted K) and a vector of the “pure” component spectrum to be analyzed (p), which are more easily obtainable. For estimating the contribution, the unknown pixel vector x and the pure spectrum p are first projected orthogonally onto K giving the vectors x_⊥ onto p_⊥, respectively. The contribution is then estimated by a second projection of x_⊥ onto p_⊥. A method, based on principal component analysis, for determining the optimal dimensions of K is proposed. DC was applied on a collection of multivariate images of kernel of wheat to estimate the proportion of three tissues, namely out-layers, “waxy” endosperm and normal endosperm. The eventual results are presented as images of wheat kernels in false colors associated to the estimated proportions of the tissues. It is shown that DC is appropriate for estimating contributions in situations in which the more usual methods of calibration cannot be applied.     

Design and model of calibration for chemical measurements

The international standard ISO 11843 specifies basic methods to design experiments for estimation of critical values referring to the capability of detection. The detection capability depends on the experimental design, the calibration model used, and the errors of the measurement process. This study reports how the specification of the calibration points within the calibration range can be used as a-priori information for evaluation of calibration uncertainty without any consideration of the response variables of the calibration. As result of investigation of the experimental designs, calibration points within the calibration range can be omitted without significant changes of calibration uncertainty. The approach is demonstrated at a practical example, the determination of arsenic in surface water samples taken from a river in Germany.     

The measurement assurance concept in calibration and traceability at NBS/NIST

During the last quarter of the twentieth century, The United States National Bureau of Standards (NBS), later the National Institute of Standards and Technology (NIST), introduced a measurement quality control concept called ”measurement assurance,” and developed measurement assurance programs, or MAPs, for high-level calibration processes. The measurement assurance approach has, over time, become increasingly popular in the metrology community, and in recent years has become well accepted both inside and, to some extent, outside the United States as a rigorous way to ensure the quality of calibrations. The concept has also found application in defining traceability to national standards. This paper traces the history of the measurement assurance concept. Received: 23 October 2000 Accepted: 2 November 2000     

Continuous dynamic-gravimetric preparation of calibration gas mixtures for air pollution measurements

 The preparation of calibration gas mixtures for air pollution measurements by the dynamic-gravimetric method was investigated using sulphur dioxide in nitrogen as a model. The target mole fraction was 200×10^–9 mol/mol, with the option of also getting smaller mole fractions. Thermal mass flow meters calibrated with reference mass flows were used to measure the dilution gas flow (nitrogen). The relative standard uncertainty of the dilution gas flows between 10 mg/s (approx. 500 ml/min) and 40 mg/s (approx. 2000 ml/min) was 0.15%. The mass flow of the target component measured as the permeation rate was determined via the quasi-continuous observation of the loss in the permeation tube mass during the measuring time. A magnetic coupling system and an adapted microbalance were used for this purpose. The results presented show permeation rates measured over the lifetime of a tubular permeation source. The measurement cycles took between 3 days and 7 h at least. The relative standard uncertainty of the mixture composition did not exceed 2%. First comparisons with gas mixtures prepared by the static-gravimetric method show compatibility. The applicability of the system is not restricted to the SO₂/N₂ mixture. It can also be used for preparing other gas mixtures in this field of application. Received: 26 April 2000 / Accepted: 12 September 2000     

Bayesian calibration

Two methods of analyzing calibration data are compared: the familiar least-squares method and an empirical Bayes method. The least-squares method uses the information obtained from the current calibration run but ignores all information obtained from previous runs. The empirical Bayes method uses the current information plus summarized information from past calibration runs, e.g., estimates of the means and variances of the parameters. Both methods are applied to simulated and real data. The empirical Bayes method is defined for any number of calibration standards, while the definition of the least-squares method requires modification when the number of standards is less than the number of parameters. The absolute error in predicting unknown analyte concentrations is used as a measure of goodness of calibration. When only one or two standards are used, the Bayes method results in better calibration than the least-squares method; e.g., with one standard, the average error with the Bayes method is at least 30% less than that with the least-squares method. The Bayes method is most useful when (i) run-to-run variation in calibration parameters is small (yet large enough to warrant regular calibration), (ii) residual error is significant, and (iii) it is desirable to use few calibration standards.     

Efficiency calibration curves in gamma-ray spectrometry for samples with different matrices

A new method for determination of efficiency calibration curves in -spectrometry is described. From the calibration curves for a bulky sample and a point source, the curves for sources with the same geometrical parameters as the bulky sample but different attenuation coefficients may be calculated. The method was tested at an energy of 88 keV on a set of samples with attenuation coefficients between 0.2 and 2 cm² g^–1. The conditions where the new method is useful are discussed.     

New methods applicable for calibration of indicator electrodes

The new methods applicable for calibration of indicator electrodes, based on standard addition and standard subtraction methods, are suggested. Some of the methods enable the slope of an indicator electrode and equivalence volume V_eq to be determined simultaneously from a single set of potentiometric titration data. Some other methods known hitherto were also taken into account. A new model, based on a standard addition method, applicable also in nonlinear range for the ISE slope (S) is suggested, and its applicability was confirmed experimentally in calibration of calcium ISE.     

The maintenance of the second-order advantage: second-order calibration of excitation-emission matrix fluorescence for quantitative analysis of herbicide napropamide in various environmental samples

A rapid non-separative spectrofluorometric method based on the second-order calibration of excitation-emission matrix (EEM) fluorescence was proposed for the determination of napropamide (NAP) in soil, river sediment, and wastewater as well as river water samples. With 0.10 mol L⁻¹ sodium citrate-hydrochloric acid (HCl) buffer solution of pH 2.2, the system of NAP has a large increase in fluorescence intensity. To handle the intrinsic fluorescence interferences of environmental samples, the alternating penalty trilinear decomposition (APTLD) algorithm as an efficient second-order calibration method was employed. Satisfactory results have been achieved for NAP in complex environmental samples. The limit of detection obtained for NAP in soil, river sediment, wastewater and river water samples were 0.80, 0.24, 0.12, 0.071 ng mL⁻¹, respectively. Furthermore, in order to fully investigate the performance of second-order calibration method, we test the second-order calibration method using different calibration approaches including the single matrix model, the intra-day various matrices model and the global model based on the APTLD algorithm with nature environmental datasets. The results showed the second-order calibration methods also enable one or more analyte(s) of interest to be determined simultaneously in the samples with various types of matrices. The maintenance of second-order advantage has been demonstrated in simultaneous determinations of the analyte of interests in the environmental samples of various matrices.     

Flow-Injection Spectrophotometric Determination of Diclofenac or Mefenamic Acid in Pharmaceuticals

 Two flow injection (FI) spectrophotometric methods are proposed for the determination of diclofenac (DCF) or mefenamic acid (MF) in bulk samples and pharmaceuticals. Both methods are based on the reaction of DCF or MF with potassium ferricyanide in a sodium hydroxide medium. The absorbance of the orange products obtained is measured at 455 nm for DCF and 465 nm for MF. The corresponding calibration graphs are linear over the range 0.20–20.0 mg L⁻¹ for DCF and 1.00–100 mg L⁻¹ for MF, while the limits of detection were 0.05 and 0.18 mg L⁻¹, respectively. Received March 27, 2000. Revision November 15, 2000.     

Direct determination of reserpine in urine using excitation-emission fluorescence combined with three-way chemometric calibration methodologies

The concentration of reserpine in urine was directly and quantitatively measured by using the excitation-emission fluorescence (EEM) combined with three-way calibration methodologies. Two calibration methods are based on the alternating trilinear decomposition (ATLD) and the self-weighted alternating trilinear decomposition (SWATLD) algorithms, respectively. These chemometric methodologies have the second-order advantage, which is the ability to get accurate concentration estimates of interested analyte(s) even in the presence of uncalibrated interferences. The satisfactory results on spiked urine samples are obtained, when the component number was chosen to 3 (N = 3) for both the methods. This experiment is easily carried out without time-consuming and complicated pretreatment. It has proved that the three-way calibration methodologies based on ATLD and SWATLD can be feasible to directly quantify the medical content of reserpine in urine. __________ Translated from Chemical Journal of Chinese Universities, 2007, 28 (5): 827–830 [译自: 高等学校化学学报]     

Linearity of calibration curves: use and misuse of the correlation coefficient

The correlation coefficient is commonly used to evaluate the degree of linear association between two variables. However, it can be shown that a correlation coefficient very close to one might also be obtained for a clear curved relationship. Other statistical tests, like the Lack-of-fit and Mandel’s fitting test thus appear more suitable for the validation of the linear calibration model. A number of cadmium calibration curves from atomic absorption spectroscopy were assessed for their linearity. All the investigated calibration curves were characterized by a high correlation coefficient (r >0.997) and low quality coefficient (QC <5%), but the straight-line model was systematically rejected at the 95% confidence level on the basis of the Lack-of-fit and Mandel’s fitting test. Furthermore, significantly different results were achieved between a linear regression model (LRM) and a quadratic regression (QRM) model in forecasting values for mid-scale calibration standards. The results obtained with the QRM did not differ significantly from the theoretically expected value, while those obtained with the LRM were systematically biased. It was concluded that a straight-line model with a high correlation coefficient, but with a lack-of-fit, yields significantly less accurate results than its curvilinear alternative. Received: 15 January 2002 Accepted: 18 April 2002     

Effect of variation in argon content of calibration gases on determination of atmospheric carbon dioxide

Carbon dioxide (CO₂) is a greenhouse gas that makes by far the largest contribution to the global warming of the Earth's atmosphere. For the measurements of atmospheric CO₂ a non-dispersive infrared analyzer (NDIR) and gas chromatography are conventionally being used. We explored whether and to what degree argon content can influence the determination of atmospheric CO₂ using the comparison of CO₂ concentrations between the sample gas mixtures with varying Ar amounts at 0 and 18.6 mmol mol⁻¹ and the calibration gas mixtures with Ar at 8.4, 9.1, and 9.3 mmol mol⁻¹. We newly discovered that variation of Ar content in calibration gas mixtures could undermine accuracy for precise and accurate determination of atmospheric CO₂ in background air. The differences in CO₂ concentration due to the variation of Ar content in the calibration gas mixtures were negligible (<±0.03 μmol mol⁻¹) for NDIR systems whereas they noticeably increased (<±1.09 μmol mol⁻¹) especially for the modified GC systems to enhance instrumental sensitivity. We found that the thermal mass flow controller is the main source of the differences although such differences appeared only in the presence of a flow restrictor in GC systems. For reliable monitoring of real atmospheric CO₂ samples, one should use calibration gas mixtures that contain Ar content close to the level (9.332 mmol mol⁻¹) in the ambient air as possible. Practical guidelines were highlighted relating to selection of appropriate analytical approaches for the accurate and precise measurements of atmospheric CO₂. In addition, theoretical implications from the findings were addressed.     

Investigation of chromium(III) acetylacetonate as a calibration reference material for atomic absorption spectroscopy

It is shown that chromium(III) acetylacetonate meets the requirements of calibration reference materials: it is homogeneous, soluble in water and organic solvents, and its solutions are stable. Chromium(III) acetylacetonate has been successfully used as a calibration reference material in the analysis of chromium in iron-base alloys. The procedure involves chromium acetylacetonate extraction using methyl isobutyl ketone followed by direct atomic absorption analysis of the organic extract. The detection limit of the procedure is 10 ng g⁻¹. Correspondence: Konstantin Belikov, State Scientific Institution “Institute for Single Crystals” of the National Academy of Sciences of Ukraine, Kharkov, Ukraine