An expression of uncertainty in calibration using stepwise or separate dilution of a stock solution.

The traditional method for linear calibration can estimate the confidence intervals of calibration lines from a set of experimental data for a single calibration line. However, the following situations, often encountered in laboratories, are out of reach of the method, since the concentrations of the standard solutions are not independent of each other: (A) a standard solution is diluted from a more concentrated one in a stepwise way (stepwise dilution); (B) every standard solution for a calibration experiment is prepared from a stock solution, but the stock solution is newly prepared for each calibration (separate dilution with the variable concentration of the stock solution). This paper puts forward a theory to calculate the confidence intervals of calibration lines in the above situations. Analyses made up of sample weighing, dilution, HPLC measurement and calibration with the linear least-squares fitting are taken as examples. The proposed theory is numerically compared to the traditional method.     

Non-linear and non-constant variance calibration curves in analysis of volatile organic compounds for testing of water by the purge-and-trap method coupled with gas chromatography/mass spectrometry

Currently used operating conditions for analysing volatile organic compounds (VOCs) by purge-and-trap gas chromatography/mass spectrometry (GC/MS) produced non-linear calibration curves with non-uniform variance. Second-order polynomial models therefore had to be used in weighted regression analysis of measurements of replicates spiked at various concentrations. A transparent procedure based on a reported method of very low computational complexity allowed calculation of the parameters of second-order models, confidence bands of regression lines, prediction bands, and confidence intervals of discriminated analyte concentrations. Tolerance intervals were introduced for this last purpose. Critical, detection and quantification levels drawn from the calibration curves were compared with those calculated by the EPA method.     

Model based robustness analysis of an ion-exchange chromatography step

Process development, optimization and robustness analysis for chromatographic separation are often entirely based on experimental work and generic knowledge. This paper describes a model-based approach that can be used to gain process knowledge and assist in the robustness analysis of an ion-exchange chromatography step using a model-based approach. A kinetic dispersive model, where the steric mass action model accounts for the adsorption is used to describe column performance. Model calibration is based solely on gradient elution experiments at different gradients, flow rates, pH and column loads. The position and shape of the peaks provide enough information to calibrate the model and thus single-component experiments can be avoided. The model is calibrated to the experiments and the confidence intervals for the estimated parameters are used to account for the model error throughout the analysis. The model is used to predict the result of a robustness analysis conducted as a factorial experiment and to design a robust pooling approach. The confidence intervals are used in a "worst case" approach where the parameters for the components are set at the edge of their confidence intervals to create a worst case for the removal of impurities at each point in the factorial experiment. The pooling limit was changed to ensure product quality at every point in the factorial analysis. The predicted purities and yields were compared to the experimental results to ensure that the prediction intervals cover the experimental results.     

Spectroscopic flame temperature measurements and their physical significance—V. Experimental determination of relative transition probabilities for 43 Fe I lines in the 3450–3950 Å wavelength region

Relative transition probabilities, as measured by the atomic emission spectroscopic technique, have been redetermined for 43 lines of Fe I under experimental conditions that in principle should yield accurate values. The Fe free-atoms are formed in a high temperature environment that has been shown to be isothermal and optically thin under the experimental conditions employed. The temperature of the iron atoms was measured by the reversal method, and the emission measurements were made so that the values obtained were strictly proportional to the line radiances. The high degree of correlation of the values reported with those obtained by the hook method is indicative of the overall reliability of these sets of data.     

Interpolation in the standard additions method

The standard additions method (SAM) has traditionally been performed by using extrapolation. This practice is suboptimal because predictions are affected by even slight departures of calibration points from a straight line. Despite this, most textbooks and papers in analytical chemistry still refer exclusively to extrapolation. In contrast, the use of interpolation is recommended in this paper as a way to get predictions on the central part of the regression line and thus minimize the bias in the prediction and the variance associated with the analytical result. Several scenarios were studied, with concentration errors simulated in different calibration solutions. It was found that translational effects due to variations at the central part of the calibration caused the lowest disturbances on the predicted concentrations. The differences between the interpolated and extrapolated predictions can be as large as ±30%. The confidence interval associated with the extrapolation result is wider than that due to interpolation by as much as 100%. It is shown that commonly used equations underestimate the correct confidence intervals. Both, absence of bias and improved precision, are of relevance in quality assurance, method validation and error propagation.     

Confidence intervals for calibration with neural networks

Based on neural network calibration the confidence intervals of aromaticity determination from infrared reflectance spectra of raw brown coals were estimated by means of the bootstrap method, a simplified Monte Carlo Simulation. The standard deviations and the confidence intervals were estimated to characterise the analysis error.It is shown that confidence intervals of non-linear analysis methods like Back Propagation Neural Networks (BPNN) can be estimated by the bootstrap method. The estimated confidence intervals of the calibration confirm the analysis by BPNN.     

Worst case uncertainty estimates for routine instrumental analysis

A methodology for the worst case measurement uncertainty estimation for analytical methods which include an instrumental quantification step, adequate for routine determinations, is presented. Although the methodology presented should be based on a careful evaluation of the analytical method, the resulting daily calculations are very simple. The methodology is based on the estimation of the maximum value for the different sources of uncertainty and requires the definition of limiting values for certain analytical parameters. The simplification of the instrumental quantification uncertainty estimation involves the use of the standard deviation obtained from control charts relating to the concentrations estimated from the calibration curves for control standards at the highest calibration level. Three levels of simplification are suggested, as alternatives to the detailed approach, which can be selected according to the proximity of the sample results to decision limits. These approaches were applied to the determination of pesticide residues in apples (CEN, EN 12393), for which the most simplified approach showed a relative expanded uncertainty of 37.2% for a confidence level of approximately 95%.     

Attempts to include uncorrected bias in the measurement uncertainty

In ISO Guide it is strictly recommended to correct results for the recognised significant bias, but in special cases some analysts find out practical to omit the correction and to enlarge the expanded uncertainty for the uncorrected bias instead. In this paper, four alternatively used methods computing these modified expanded uncertainties are compared according to the levels of confidence, widths and layouts of the obtained uncertainty intervals. The method, which seems to be the best, because it provides the same uncertainty intervals as in the case of the bias correction, has not been applied very much, maybe since these modified uncertainty intervals are not symmetric about the results. The three remaining investigated methods maintain their intervals symmetric, but only two of them provide intervals of the kind, that their levels of confidence reach at least the required value (95%) or a larger one. The third method defines intervals with low levels of confidence (even for small biases). It is proposed a new method, which gives symmetric intervals just with the required level of confidence. These intervals are narrower than those symmetric intervals with the sufficient level of confidence obtained by the two mentioned methods. A mathematical background of the problem and an illustrative example of calculations applying all compared methods are attached.     

Direct determination of manganese in vitamin-mineral tablets using solid sampling electrothermal atomic absorption spectrometry

Manganese in vitamin-minerals tablets was determined by solid sampling electrothermal atomic absorption spectrometry (SS-ETAAS) using three different calibration methods, namely calibration against aqueous standards, standard addition with aqueous standards on solid samples and calibration against solid certified standards. Samples were only finely ground and introduced directly into the furnace by means of solid autosampler system without any dissolving process. Effects of different calibration techniques, temperatures and heating rates of atomization and pyrolysis steps on the accuracy and precision of the analyte elements were investigated. After optimization of the experimental parameters, there is good agreement (at 95% confidence level) between the results obtained by solid sampling and those obtained by acid digestion of samples.     

Estimation of monomer reactivity ratios in free‐radical solution copolymerization of lauryl methacrylate–isobutyl methacrylate

Copolymers of isobutyl methacrylate (i‐BMA) and lauryl methacrylate (LMA) were prepared by free‐radical solution copolymerizations at 70 °C with azobisisobutyronitrile (AIBN) as an initiator. The synthesis of these copolymers was investigated over a wide composition range both at low and high conversion levels. Copolymer compositions were determined from the %C, %H, and %O contents of copolymer by elemental analysis. Monomer reactivity ratios were estimated by analyzing composition data with nonlinear least‐squares (NLLS) models based on Marquardt optimization and van Herk methods. The point estimates, 95% individual confidence intervals and 95% joint confidence intervals are obtained from differential and integral approaches. Even though no explicit integral form for penultimate unit model (PUM) is available, a numerical approach is developed for integral estimation of reactivity ratios from PUM. A simulator program was developed which upon coupling of experimental data, NLLS analysis, and D‐optimal criteria calculates the best optimized values of monomer reactivity ratios and monomer feed compositions in a sequential and iterative order for terminal and penultimate unit models. Moreover, the simulator has the capibilities to calculate all features of van Herk method, maximum compositional drift in each monomer feed composition, and data reconciliation. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 112–129, 2004     

Critical study of the determination of tin by electrothermal atomic-absorption spectrometry using resonance and non-resonance lines

The 235.48 nm non-resonance line of tin gives a sensitivity for AAS determination equal to that given by the most sensitive resonance line. Vaporization from a platform improves some twofold the sensitivities of all lines investigated. The atomization temperature has only slight influence on the relative sensitivities given by the non-resonance tin lines studied. Phosphoric acid shows a rather peculiar interference pattern, low concentrations (0.05%) depressing the signal, but high concentrations (1%) enhancing it. Contrary to expectation, the resonance and non-resonance lines are affected to the same extent by phosphoric acid. Platform-vaporization does not change the interference pattern of phosphoric acid. Calibration can be done by injecting various volumes of a single standard solution onto a platform coated with tantalum carbide. This method yields calibration graphs that are just as linear as those obtained with identical volume of standards of various concentrations. Some suggestions are made for further improvement of the analytical potential of non-resonance lines.     

Evaluation of self-absorption of manganese emission lines in Laser Induced Breakdown Spectroscopy measurements

This paper is part of a more general study aimed to the determination of the best experimental procedures for reliable quantitative measurements of Fe–Mn alloys by LIBS. In this work, attention is pointed on the self-absorption processes, whose effect deeply influences the LIBS measurements, reflecting in non-linear calibration curves. The effect of self-absorption on the line intensity can be quantified by defining a self-absorption coefficient, that measures the deviation of the line intensity from the linear extrapolation of the curve of growth in the optically thin regime. The authors demonstrated in a previous paper that self-absorption coefficients could be calculated once the electron density of the plasma is known and the Stark coefficients of the lines are available. However, when the Stark coefficients of the lines of interest are not known, a different approach is needed. In this work a new method for evaluation of self-absorption coefficients in LIBS measurements is presented, which does not require the knowledge of Stark coefficients. In order to understand the basic principles and setting out the theoretical tools that will be used for the analysis of the alloys, a preliminary study was done on pure Mn; LIBS spectra were acquired in different experimental conditions, at different laser energies and different delays after the laser irradiation of the sample. Moreover, collinear double pulse measurements were also performed. Analytical relations were derived and experimental procedures devised for evaluation of the self-absorption coefficients of several Mn lines, which are important for characterization and control of the experimental conditions in which the analysis is performed.     

Preconcentration of some metals using metalfix chelamine resin prior to ICP-AES analysis

Preconcentration and separation of Ni, Pb and Cd have been investigated using the metalfix chelamine resin prior to inductively coupled plasma – atomic emission spectrometry (ICP-AES) under various experimental conditions. The recoveries of the analytes obtained with 95% confidence level were 92 ± 1% for Ni, 100 ± 4% for Pb and 93 ± 3% for Cd.     

Non-linear least-squares fitting with microsoft excel solver and related routines in HPLC modelling of retention

Summary Two problems related to non-linear regression, the evaluation of the best set of fitting parameters and the reliability of the methods used for the estimation of the standard errors of these parameters, are examined. It is shown that a non-linear curve fitting routine, like the Microsoft Excel Solver, may give more than one solution for the same data set and a simple Monte Carlo routine is described for the evaluation of the bestfit. For standard errors, the reliability of two procedures based on the conventional curvature matrix method, four Jackknife techniques and the bootstrap method are examined by comparing their results to those obtained from a Monte Carlo simulation of the experimental data. It is shown that a fitting parameter may follow a nonnormal distribution when the equation to be fitted is complicated, even if the errors on the data are normally distributed. In this case only Monte Carlo methods of data simulation can give accurate information about the standard errors and the confidence intervals of these parameters.     

The quality coefficient as performance assessment parameter of straight line calibration curves in relationship with the number of calibration points

Knowledge of the response function (y?=?f(x)) is essential in the validation of quantitative analysis methods as it describes the mathematical relationship between measurable responses and the concentrations or quantities of the analyte in the sample within a suitable range. The most common response function used is a straight line obtained by ordinary least squares (OLS) regression. Suitability of calibration lines obtained by OLS regression might be verified by calculation of a quality coefficient (QC_mean). Mathematical modelling performed previously showed that with respect to critical limit values for g, which controls the symmetry of the prediction interval of the abscissa value obtained from the confidence intervals around the OLS calibration curve, a corresponding quality coefficient value exists as a quality performance parameter which is related to the spread of the abscissa values around their mean. In this paper, new mathematical models are developed to demonstrate to which extend also the number n of calibration points (x _i ,y _i ) defines the required value for the quality coefficient (QC_mean) for different values of g. From these models, it could be established that the attribution of a critical limit value to QC_mean as a performance parameter for straight line calibration cannot be arbitrary chosen but has to rely on the mathematical model relating QC_mean, the g-value, the number n of calibration points and the spread of the x _i -values around their mean. Practical measures for analysts are provided which tend to lower the g-value of straight calibration lines beneath critical values and enable to improve the quality of the calibration line applied for analysis, as demonstrated in an elaborated example.     

High-resolution photoelectron spectroscopy through deconvolution

A new iterative deconvolution algorithm for the development of very high resolution Hc(I)-excited photoelectron spectra of gases is presented. The algorithm accepts as input a medium-resolution spectrum and an instrument function obtained by scanning intrinsically narrow line (e.g. the Ar²P_3/2 line) under conditions identical to those used to acquire the medium resolution spectrum. The deconvolved partial spectrum of an O₂H₂O mixture is presented as a test case prior to presentation of results for three nitrogen lines. For comparison purposes directly obtained high-resolution spectra of the nitrogen lines are included. The shapes of the nitrogen lines are discussed within the framework of a one-center expansion theory of photoionization. The conditions under which deconvolution can be profitably applied are briefly discussed and an attempt is made to establish the deconvolution (contrary to still popular belief) is not in any way equivalent to curve fitting.     

Assessment of quality performance parameters for straight line calibration curves related to the spread of the abscissa values around their mean

In validation of quantitative analysis methods, knowledge of the response function is essential as it describes, within the range of application, the existing relationship between the response (the measurement signal) and the concentration or quantity of the analyte in the sample. The most common response function used is obtained by simple linear regression, estimating the regression parameters slope and intercept by the least squares method as general fitting method. The assumption in this fitting is that the response variance is a constant, whatever the concentrations within the range examined.The straight calibration line may perform unacceptably due to the presence of outliers or unexpected curvature of the line. Checking the suitability of calibration lines might be performed by calculation of a well-defined quality coefficient based on a constant standard deviation.The concentration value for a test sample calculated by interpolation from the least squares line is of little value unless it is accompanied by an estimate of its random variation expressed by a confidence interval. This confidence interval results from the uncertainty in the measurement signal, combined with the confidence interval for the regression line at that measurement signal and is characterized by a standard deviation s_x0 calculated by an approximate equation. This approximate equation is only valid when the mathematical function, calculating a characteristic value g from specific regression line parameters as the slope, the standard error of the estimate and the spread of the abscissa values around their mean, is below a critical value as described in literature.It is mathematically demonstrated that with respect to this critical limit value for g, the proposed value for the quality coefficient applied as a suitability check for the linear regression line as calibration function, depends only on the number of calibration points and the spread of the abscissa values around their mean.     

Fractal analysis methods for solid alkane monolayer domains at SiO2/air interfaces

A systematic evaluation of various fractal analysis methods is essential for studying morphologies of finite and noisy experimental patterns such as domains of long chain alkanes at SiO(2)/air interfaces. The derivation of trustworthy fractal dimensions crucially relies on the definition of confidence intervals for the assumed scaling range. We demonstrate that the determination of the intervals can be improved largely by comparing the scaling behavior of different morphological measures (area, boundary, curvature). We show that the combination of area and boundary data from coarse-grained structures obtained with the box-counting method reveals clear confidence limits and thus credible morphological data. This also holds for the Minkowski density method. It also reveals the confidence range. Its main drawback, the larger swing-in period at the lower cutoff compared to the box-counting method, is compensated by more details on the scaling behavior of area, boundary, and curvature. The sandbox method is less recommendable. It essentially delivers the same data as box-counting, but it is more susceptible to finite size effects at the lower cutoff. It is found that the domain morphology depends on the surface coverage of alkanes. The individual domains at low surface coverage have a fractal dimension of approximately 1.7, whereas at coverages well above 50% the scaling dimension is 2 with a large margin of uncertainty at approximately 50% coverage. This change in morphology is attributed to a crossover from a growth regime dominated by diffusion-limited aggregation of individual domains to a regime where the growth is increasingly affected by annealing and the interaction of solid growth fronts which approach each other and thus compete for the alkane supply.     

Evaluating reproducibility and similarity of mass and intensity data in complex spectra—applications to tubulin

We present a data processing approach based on the spectral dot product for evaluating spectral similarity and reproducibility. The method introduces 95% confidence intervals on the spectral dot product to evaluate the strength of spectral correlation; it is the only calculation described to date that accounts for both the non-normal sampling distribution of the dot product and the number of peaks the spectra have in common. These measures of spectral similarity allow for the recursive generation of a consensus spectrum, which incorporates the most consistent features from statistically similar replicate spectra. Taking the spectral dot product and 95% confidence intervals between consensus spectra from different samples yields the similarity between these samples. Applying the data analysis scheme to replicates of brain tubulin CNBr peptides enables a robust comparison of tubulin isotype expression and post-translational modification patterns in rat and cow brains.     

The widths and shapes of about 350 prominent lines of 65 elements emitted by an inductively coupled plasma

This work describes the measurement of the widths and shapes of about 350 prominent lines of 65 elements emitted by an inductively coupled plasma (ICP). The experimental procedure is an improved version of an earlier described approach using a 1.5-m echelle monochromator with predisperser. For most measurements the practical spectral bandwidth was smaller than 1.5 times the physical line width. Results are reported for both simple lines with chiefly Doppler broadening and complex structures with unresolved or partly resolved hyperfine structure (HFS). An atlas with the spectral scans of about 90 interesting line profiles is included and wavelengths of HFS components are tabulated. The latter were accurately determined in the case of well resolved structures and roughly estimated for poorly resolved or unresolved structures.

The data were collected chiefly with a view to spectrochemical analysis, with a threefold aim: (a) to provide the basic data needed for comparing detection limits obtained with spectroscopic apparatus of different bandwidths; (b) to identify HFS components which, under high resolution conditions, may be useful as separate prominent lines in order to circumvent spectral interference and (c) to establish a basis for models that can be used in software for line selection such that the number of data needed in view of differences among line shapes and widths can be reduced to a minimum.

It appears from the results that these targets are closely approached. Further work is required, however, for the full implementation of the results.