Test for non-linearity concerning linear calibrated chemical measurements

Linear calibrated chemical measurements are based on the assumption of linearity. Due to influences of matrices at real samples the condition of linearity can be violated. Therefore, a profound examination has to attach much importance on the linearity of calibration. However, different procedures have been applied for this purpose in literature. In order to examine linear calibration for non-linearity, a recently presented procedure is compared with conventional techniques. The associated statistical models and estimations are represented. The applicability of the different procedures is demonstrated in some practical examples, the determination of arsenic in surface water samples taken from different sites in abandoned mining areas in central Germany. Recommendations for using the indicators and tests of non-linearity are given.     

Validation steps for traceability of linear calibrated chemical measurements

Based on the new draft of the EURACHEM/CITAC Guide Traceability in Chemical Measurement, this publication describes how traceability can be achieved for chemical measurements using a linear calibration function. Traceability can be accomplished without larger expenditure, if the measurement is calibrated on the basis of appropriate reference standards and the linear regression employed is selected and validated statistically in a suitable form. The determination of nickel in aqua regia eluates of sediment samples, employed for an ICP-OES measurement, is used as a practical illustration of this approach.     

Comparison of regression techniques for linear calibration

A recently presented regression technique for linear calibration, which is based on a variance component model for univariate quantitative measurement data, is compared with the conventional and far spread regression techniques ordinary least squares regression and weighted least squares regression. The associated statistical models and estimations are represented. Its application is demonstrated at some practical examples. With consideration of special variation causes, like matrix influence or the influence of several operating conditions on the measurement response, it can be shown that the application of the variance component model is an advantage.     

Heteroscedastic controlled calibration model applied to analytical chemistry

In chemical analyses performed by laboratories, one faces the problem of determining the concentration of a chemical element in a sample. In practice, one deals with the problem using the so‐called linear calibration model, which considers that the errors associated with the independent variables are negligible compared with the former variable. In this work, a new linear calibration model is proposed assuming that the independent variables are subject to heteroscedastic measurement errors. A simulation study is carried out in order to verify some properties of the estimators derived for the new model and it is also considered the usual calibration model to compare it with the new approach. Three applications are considered to verify the performance of the new approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Evaluation of measurement uncertainty for analytical procedures using a linear calibration function

In the EURACHEM/CITAC draft ”Quantifying uncertainty in analytical measurement” estimations of measurement uncertainty in analytical results for linear calibration are given. In this work these estimations are compared, i.e. the uncertainty deduced from repeated observations of the sample vs. the uncertainty deduced from the standard residual deviation of the regression. As a result of this study it is shown that an uncertainty estimation based on repeated observations can give more realistic values if the condition of variance homogeneity is not correctly fulfilled in the calibration range. The complete calculation of measurement uncertainty including assessment of trueness is represented by an example concerning the determination of zinc in sediment samples using ICP-atomic emission spectrometry. Received: 9 February 2002 Accepted: 17 April 2002     

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.     

Generalization of a new calibration method based on linear correlation

This paper communicates modifications to our new calibration method based on linear correlation, described in detail in a former paper [Spectrochim. Acta 56B, 1159], which extend its applicability. The presented, generalized linear correlation method (GLCM) can be applied in any spectroscopic method for quantitation, and also when multielemental, trace solutions are analyzed and the analysis is not complete. Applications of the method to UV-Vis spectrophotometry and inductively coupled plasma mass spectrometry (ICP-MS) are also presented. The method showed a good, typically 1-5%, accuracy in all applications.     

A new formulation to estimate the variance of model prediction. Application to near infrared spectroscopy calibration

Evaluation of uncertainty affecting predictions is a major trend in analytical chemistry and chemometrics. Several approximate expressions and resampling methods have been proposed for the estimation of prediction uncertainty when using multivariate calibration. This article proposes a new expression for the variance of prediction, adapted to near infrared spectroscopy specificities and particularly to the spectral error structure, induced by the high colinearity of the variables. The proposed analytical expression enables a detailed evaluation of the different contributions and components of uncertainty affecting the model. An application to real data of feedstuff near infrared spectra related to protein content has shown its advantages.     

Using the L1 norm to select basis set vectors for multivariate calibration and calibration updating

With projection based calibration approaches, such as partial least squares (PLS) and principal component regression (PCR), the calibration space is spanned by respective basis vectors (latent vectors). Up to rank k basis vectors are formed where k ≤ min(m,n) with m and n denoting the number of calibration samples and measured variables. The user needs to decide how many and which respective basis vectors (tuning parameters). To avoid the second issue, basis vectors are selected top‐down starting with the first and sequentially adding until model criteria are satisfied. Ridge regression (RR) avoids the issues by using the full set of basis vectors. Another approach is to select a subset from the total available. The presented work develops a process based on the L₁ vector norm to select basis vectors. Specifically, the L₁ norm is used to select singular value decomposition (SVD) basis set vectors for PCR (LPCR). Because PCR, PLS, RR, and others can be expressed as linear combination of the SVD basis vectors, the focus is on selection and comparison using the SVD basis set. Results based on respective tuning parameter selections and weights applied to the SVD basis vectors for LPCR, top‐down PCR, correlation PCR (CPCR), PLS, and RR are compared for calibration and calibration updating using spectroscopic data sets. The methods are found to predict equivalently. In particular, the L₁ norm produces similar results to those obtained by the well‐studied CPCR process. Thus, the new method provides a different theoretical framework than CPCR for selecting basis vectors. Copyright © 2016 John Wiley & Sons, Ltd.     

Limits of determination and calibration in HPLC

Summary In HPLC calibration the expressions lowest calibration limit and determination limit are defined in statistical terms. The lowest calibration limit is the minimum mass in the measured series of calibration points. It is calculated from the confidence interval of the inverse of the calibration function as the lowest mass limit that may be differentiated from zero mass with a preset probability of error. If the calculated lowest calibration limit is lower than the actual data, points at lower concentration may be measured. The determination limit is the smallest concentration of an analysis that is differentiated from the concentration zero or an apparent blind value in the calibration curve with a given probability of error.Using two different UV-detectors (variable wavelength and photodiode-array) the lowest calibration limit is experimentally evaluated and compared with specific data for the detectors.Dedicated to Prof. Dr. E. Bayer, Tübingen on ocassion of his 60th birthday.     

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.     

Multivariate calibration of spectral data using dual-domain regression analysis

To date, few efforts have been made to take simultaneous advantage of the local nature of spectral data in both the time and frequency domains in a single regression model. We describe here the use of a novel chemometrics algorithm using the wavelet transform. We call the algorithm dual-domain regression, as the regression step defines a weighted model in the time-domain based on the contributions of parallel, frequency-domain models made from wavelet coefficients reflecting different scales. In principle, any regression method can be used, and implementation of the algorithm using partial least squares regression and principal component regression are reported here. The performance of the models produced from the algorithm is generally superior to that of regular partial least squares (PLS) or principal component regression (PCR) models applied to data restricted to a single domain. Dual-domain PLS and PCR algorithms are applied to near infrared (NIR) spectral datasets of Cargill corn samples and sets of spectra collected on batch chemical reactions run in different reactors to illustrate the improved robustness of the modeling.     

Building robust calibration models for the analysis of estrogens by gas chromatography with mass spectrometry detection

Five hormonal growth promotants (diethylstilbestrol, hexestrol, dienestrol, 17-β-estradiol and 17-α-ethynylestradiol) have been analysed by gas chromatography with mass spectrometry detection (GC/MS, SIM mode) for four non-consecutive days. The aim is to build models with stable predictions. The strategies applied are internal standardization and global models carried out by gathering signals recorded on several days. Two models were examined: univariate models (with standardized peak area) and PARAFAC2 (the analyte scores were standardized by the scores of the internal standard). Internal standardization has been proved to be efficient for both models of dienestrol and ethynylestradiol. The mean relative error in absolute value when samples recorded on a different day to the calibration set are quantified by PARAFAC2 is 7.00% and 7.11% for dienestrol and ethynylestradiol, respectively. For diethylstilbestrol and estradiol, internal standardization was combined with global calibration models built with signals recorded under several sources of variability (different days). Thus predictions become steadier over time and in the estradiol example, errors decrease from 33.10% to 9.76%. The mean relative error in absolute value with PARAFAC2 updated models oscillates between 6.34% for ethynylestradiol and 10.74% for diethylstilbestrol. For univariate updated models errors range from 6.42% to 14.19% for ethynylestradiol and estradiol respectively. The combination of both strategies has been proved to be efficient independently of the analyte and of the signal order.     

A novel algorithm for linear multivariate calibration based on the mixed model of samples

We present a novel algorithm for linear multivariate calibration that can generate good prediction results. This is accomplished by the idea of that testing samples are mixed by the calibration samples in proper proportion. The algorithm is based on the mixed model of samples and is therefore called MMS algorithm. With both theoretical support and analysis of two data sets, it is demonstrated that MMS algorithm produces lower prediction errors than partial least squares (PLS2) model, has similar prediction performance to PLS1. In the anti-interference test of background, MMS algorithm performs better than PLS2. At the condition of the lack of some component information, MMS algorithm shows better robustness than PLS2.     

NIR analysis of pharmaceutical samples without reference data: improving the calibration

Using an appropriate set of samples to construct the calibration set is crucial with a view to ensuring accurate multivariate calibration of NIR spectroscopic data. In this work, we developed and optimized a new methodology for incorporating physical variability in pharmaceutical production based on the NIR spectrum for the process. Such a spectrum contains the spectral changes caused by each treatment applied to the component mixture during the production process. The proposed methodology involves adding a set of process spectra (viz. difference spectra between those for production tablets and a laboratory mixture of identical nominal composition) to the set of laboratory samples, which span the wanted concentration range, in order to construct a calibration set incorporating all physical changes undergone by the samples in each step of the production process. The best calibration model among those tested was selected by establishing the influence of spectral pretreatments used to obtain the process spectrum and construct the calibration models, and also by determining the multiplying factor m to be applied to the process spectra in order to ensure incorporation of all variability sources into the calibration model. The specific samples to be included in the calibration set were selected by principal component analysis (PCA). To this end, the new methodology for constructing calibration sets for determining the Active Principle Ingredients (API) and excipients was applied to Irbesartan tablets and validated by application to the API and excipients of paracetamol tablets. The proposed methodology provides simple, robust calibration models for determining the different components of a pharmaceutical formulation.     

Analysis of long-term distributions of calibration parameters and calibration intervals for an atomic absorption spectrophotometer

 A combination of "black box" and "calendar-time" methods for the determination of calibration intervals of an analytical measuring instrument is discussed. Since the methods require information on the distributions of the calibration parameters, such information is described for an atomic absorption spectrophotometer, as an example. The hypotheses on the normal distribution of the calibration parameters are tested using the ω²-criterion and accepted at 0.90–0.95 levels of confidence. Corresponding control charts are designed for indication of warning and action limits of the calibration parameters, and diagnoses of outliers in further calibrations. Control charts indicate also when the calibration should be done according to the full program of the equipment manufacturer. Received: 15 April 2000 / Accepted: 24 July 2000     

Estimation of the uncertainty of mass measurements from in-house calibrated analytical balances

The uncertainty evaluation of mass measurements when using “in-house” calibrated analytical balances is revisited according to the Guide to the expression of Uncertainty Measurement (GUM). The calibration of analytical balances is discussed according to the guidelines of several bodies such as ASTM, UKAS and DKD/PTB. The remainder components of uncertainty can be estimated from the balance data sheet specifications.     

Use of factorial experimental design for the rapid evaluation of main and interactive factors affecting linearity in calibration curves for sulfate analysis by ion chromatography

Analytical results of anion determination by suppressed ion chromatography are significantly affected by calibration curve calculation. In this paper, as expected, eluent pK_a is shown to influence calibration linearity in the range 1–20 mg/l sulfate, with A carbonate-hydrogencarbonate mixture producing a larger non-linearity than NaOH. Evidence is given for very large errors (about 30–40%) in estimating sample sulfate concentration when linear regression is used instead of a quadratic calibration curve. This study was performed following a 2⁴ run full factorial experimental design, including eluent pK_a, counterion type, solution composition and current level for background suppression as main variables.     

Investigation of the performance of the instrument by nonlinear calibration

In the search for high reliability in chemical analysis, a straightforward method of nonlinear calibration is presented. The corresponding standard deviation (SD) was calculated by the law-of-propagation of errors that allowed the determination of uncertainties as a function of concentration within the range of concentrations defined by the lower limit of analysis (LLA) and the upper limit of analysis (ULA). The advantage of using nonlinear calibrations was demonstrated by the correspondence of average values to the peak position of the normal distribution. The concentration of cobalt in two certified reference materials (CRMs) was determined by flame absorption spectrometry (FAAS) that is recognised as an experiment of very high precision. In order to ensure the determination of a reliable value for the SD, a high number of repetitions were required for the analysis. It was thus demonstrated that results that apparently deviated significantly from the certified values, actually belonged to the overall normal distribution of results. The data of experiments were grouped according to Scott’s method, and the distribution of experiments showed a particularly high frequency of results at the peak position that exceeded the expected value predicted by the normal distribution. It also deviated from the distribution peak of experiments, which demonstrates the importance of a full investigation of the distribution of results using more than 100 repetitions. The skewness of the distribution of results was eliminated by the nonlinear calibration. Correspondence: Jens E. T. Andersen, Department of Chemistry, Technical University of Denmark, Kemitorvet building 207, DK-2800 Kgs. Lyngby, Denmark     

Fusion strategies for selecting multiple tuning parameters for multivariate calibration and other penalty based processes: A model updating application for pharmaceutical analysis

New multivariate calibration methods and other processes are being developed that require selection of multiple tuning parameter (penalty) values to form the final model. With one or more tuning parameters, using only one measure of model quality to select final tuning parameter values is not sufficient. Optimization of several model quality measures is challenging. Thus, three fusion ranking methods are investigated for simultaneous assessment of multiple measures of model quality for selecting tuning parameter values. One is a supervised learning fusion rule named sum of ranking differences (SRD). The other two are non-supervised learning processes based on the sum and median operations. The effect of the number of models evaluated on the three fusion rules are also evaluated using three procedures. One procedure uses all models from all possible combinations of the tuning parameters. To reduce the number of models evaluated, an iterative process (only applicable to SRD) is applied and thresholding a model quality measure before applying the fusion rules is also used. A near infrared pharmaceutical data set requiring model updating is used to evaluate the three fusion rules. In this case, calibration of the primary conditions is for the active pharmaceutical ingredient (API) of tablets produced in a laboratory. The secondary conditions for calibration updating is for tablets produced in the full batch setting. Two model updating processes requiring selection of two unique tuning parameter values are studied. One is based on Tikhonov regularization (TR) and the other is a variation of partial least squares (PLS). The three fusion methods are shown to provide equivalent and acceptable results allowing automatic selection of the tuning parameter values. Best tuning parameter values are selected when model quality measures used with the fusion rules are for the small secondary sample set used to form the updated models. In this model updating situation, evaluation of all possible models, thresholding, and iterative SRD performed equivalently for the three fusion rules with TR and PLS performed worse. While the application is model updating, the fusion processes are applicable to other situations requiring selection of multiple tuning parameter values.