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.     

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     

The multivariate coefficient of variation for comparing serum protein electrophoresis techniques in external quality assessment schemes

External quality assessment (EQA) schemes are national or transnational programmes designed to control the analytical performance of clinical laboratories and to maintain inter-laboratory variability within acceptable limits. In such EQA programmes, participants are usually grouped by the type of assay technique/equipment they use. The coefficient of variation (CV) is a simple tool for comparing the inter-laboratory reproducibility of such techniques: the lower the CV, the better the analytical performance. Serum protein electrophoresis, a laboratory test profile consisting of five fractions (albumin, α ₁, α ₂, β and γ globulins) summing up to 100% of total proteins, can also be assayed in different ways depending on the media or the analytical principle. We propose a multivariate CV for comparing the performance of electrophoretic techniques in EQA, thus extending the univariate CV concept. First, the compositional nature of electrophoretic data requires a one-to-one transformation from the five-dimensional to the four-dimensional space. Next, robust estimations of the mean and the covariance matrix are needed to avoid the effect of outliers. The new approach is illustrated on electrophoretic datasets from the French and Belgian national EQA programmes.     

Analytical performance of some methods for the determination of trace elements in lichens used as air quality assessment

In this study, the validity of the effective (virtual) point-detector concept for the CdZnTe (CZT) detector for disc-shaped radioisotope sources has been investigated in the photon energy range of 50–662 keV. It has been found that the virtual point-detector model is valid for conventional CZT detectors. The photon energy dependence of the virtual interaction depth (VID) in the CZT detector does not seem to fit to a good trend. However, in general, the measured VID values are increasing with increasing γ-ray energy in the range of 50–662 keV, but the VID values always remain within the physical thickness of the CZT detector. For instance, the VID value was measured to be about 4.4±0.5 mm in the 5 mm thick CZT detector at 662 keV γ-ray energy. In application, the VID values for photons can be used for efficiency calculation models by scaling the measured detection efficiency as a function of energy and source-to-detector distance.     

Calculation of non-linearities of the calibration curves in atomic absorption flame spectroscopy

The deviations from the linear relationship between absorbance and concentration in atomic absorption measurements are calculated. Besides the chemical reactions there are four groups of interferences: (a) profiles of emission- and absorption lines (b) hyperfine structure (c) optics around the flame (d) non resonant light detected by the amplifier due to the optical arrangement and/or resolution of the monochromator.

The calculations show that curvature of experimental curves as obtained by apparatus for atomic absorption spectroscopy is caused partly by optical compromises and partly by the emission profile of the source. With a narrow simple emission line profile of the source and a precision monochromator a linear relationship within a few percent is possible up to 95 per cent absorption in a flame at atmospheric pressure.     

Evaluation of the calibration matrix condition number as a criterion for optimal derivative-spectrophotometric multicomponent quantitation

A criterion is described that will help predict the optimal derivative order to be used for a given quantitative procedure. The procedure is tested with a two-component sample of acid-base indicators at pH 4 and pH 6. The method predicted that the second derivative should be used at pH 4 while the first derivative is optimal at pH 6.     

Detector response and intensity cross-contribution as contributing factors to the observed non-linear calibration curves in mass spectrometric analysis

It is a common knowledge that detector fatigue causes a calibration curve to deviate from the preferred linear relationship at the higher concentration end. With the adaptation of an isotopically labeled analog of the analyte as the internal standard (IS), cross-contribution (CC) of the intensities monitored for the ions designating the analyte and the IS can also result in a non-linear relationship at both ends. A novel approach developed to assess 'the extent and the effect of [CC]… in quantitative GC-MS analysis' can be extended (a) to examine whether a specific set of CC values is accurate; and (b) to differentiate whether the observed non-linear calibration curve is caused by detector fatigue or the CC phenomenon. Data derived from the exemplar secobarbital (SB)/SB-d(5) system (as di-butyl-derivatives) are used to illustrate this novel approach. Comparing the non-linear nature of calibration data that are empirically observed to that derived from theoretical calculation (with the incorporation of adjustment resulting from the ion CC phenomenon), supports the conclusions that (a) both CC and detector fatigue contribute significantly to the observed non-linear nature of the calibration curve based on ion-pair m/z 207/212; and (b) detector fatigue is the dominating contributor when the calibration curve is based on ion-pair m/z 263/268.     

原子序数拓扑指数与脂肪族醛酮沸点的定量关系研究

按自定的编号规则,给醛酮分子中的碳原子和氧原子编号,并在此基础上建立新的拓扑指数(m^M),其中1^M对醛酮分子具有良好的结构选择性,并与72种醛酮的沸点显著相关：lg(660-Tbp)=5.8581-0.0111^M,γ=0.9970。引入0^M及其碳原子数的三元线性方程为：Log(679-Tbp)=6.045+0.0940^M-0.0201^M-0.194N,R=0.9991,明显优于文献的研究结果。并为预测脂肪族醛酮的沸点提供了有效的方法。     

The error in absorption measurements caused by the use of non-monochromatic light-III Application to linear calibration curves

When in spectrophotometry the absorption is measured with non-monochromatic light, a curved calibration curve is often obtained. Some errors are introduced if a linear calibration curve is still used. The magnitude of these errors has been calculated.     

The dependence of the variances of the parameters in non-linear regression analysis on the number of data points

In designing an experimentin which non-linear regression analysis is used to evaluate the physicochemical parameters that chracterize the behavior of the system being studied, it would be useful to be able to predict how the number of data points affects th reliabilities of the values obtained. The relation between these depends on hos the points are distributed as well as on the nature of the equation to which thd data are fitted. Weighted non-linear regression analysis has been applied to six common equations involving a total of fourteen parameters, using a number of n of points that was varied from 4 to     

The acceptor number — A quantitative empirical parameter for the electrophilic properties of solvents

The electrophilic properties of 34 solvents have been characterized by the Acceptor Number (AN) which has been derived from³¹P-n.m.r. measurements of triethylphosphine oxide dissolved in the respective solvents. Relationships are found between the acceptor numbers and theZ-values,E _T-values andY-values, as well as the free energies of solvation of anions and the redox potentials of the hexacyanoferrate(III)—hexacyanoferrrate(II) system in different solvents. The new parameter provides—together with the donor number—a useful guide in choosing the most appropriate solvent for a given reaction.     

Maintenance of the slope of linearized calibration curves in Zeeman graphite furnace atomic absorption spectrometry

L'vov and co-workers developed a theoretical model and computational procedure (B.V. L'vov, L.K. Polzik and N.V. Kocharova, Spectrochim. Acta Part B, 47 (1992) 889 and B.V. L'vov, L.K. Polzik, N.V. Kocharova, Yu.A. Nemets and A.V. Novichikhin, Spectrochim. Acta Part B, 47 (1992) 1187) that linearized calibration curves in Zeeman graphite furnace atomic absorption spectrometry by taking into account the presence of stray light. The calculations of L'vov and co-workers were based on three parameters: the rollover absorbance A_r, Zeeman sensitivity ratio R, and the original background corrected peak absorbance values A_z. In order to simplify the calculations, R was assumed to be unity. In the studies reported here, this simplification is shown to be unsatisfactory because the slope obtained in the upper portion of the calibration curve, after linearization, is found to be different from the slope obtained in the normal linear region. Deviations between these slopes were found to be as high as 30%. The present work also shows that the theoretical model of L'vov and co-workers does not have a mathematical solution at high values of A_z. This failure of the model prevents its use at high A_z values. The physical nature of this failure is still unclear, which points to the necessity for further work to understand the inadequacies of the present theory. In the present studies, calculations based on the Newton method of successive approximations (A.I. Yuzefovsky, E.G. Su, R.G. Michel, W. Slavin and J.T. McCaffrey, Spectrochim. Acta Part B, 49 (1994) 1643), allow incorporation of the experimental value of R at the rollover point R′, which better linearizes the calibration curves. By use of this approach, a satisfactory result is obtained for lead (R′ = 0.67) up to the point of failure of the model at high values of A_z.     

The influence of hollow-cathode lamp line profiles upon analytical curves in atomic absorption spectroscopy

Analytical curves of Ca, Al, Mn, Ga and In in the acetylene-nitrous oxide flame are calculated using line profiles measured with a pressure-scanned interferometer. The effects of hyperflne structure, flame Une shift and finite hollow-cathode lamp line width are determined and found to decrease sensitivity by 10 to 20% and to bend the curves towards the concentration axis according to a quadratic function that causes deviations from a straight line between 2 and 7% at absorbance one. An increase of the lamp current from 5 to 25 mA d.c. enhances the effects mentioned above. It is shown that cooling the hollow cathode raises the sensitivity by only a few per cent and in some cases even reduces sensitivity. The use of a tunable laser as a primary source can improve sensitivity, especially when the atomic line profile consists of several widely spaced hyperfine components.     

The effective charge number and diffusion coefficient of cationic cytochrome c in aqueous solution.

The diffusion coefficient and the effective charge number of cytochrome c as a function of ionic strength, temperature and pH have been measured. The measurements were carried out using a method based on a convective diffusion process across a porous membrane. The effect of ionic strength was studied in an NaCl solution the concentration of which varied from 0.001 to 1.0 M. The temperature range studied was 10-50 degrees C, and the pH values studied were 4.0, 6.5 and 8.25. The diffusion coefficient is fairly constant as a function of ionic strength and pH, and Walden's rule is valid in the temperature range studied. The effective charge number is practically constant (ca. 2) in the concentration range studied, except in 0.001 M solution, where it is the same as the titrated value. The charge number decreases slightly in the temperature range 10-30 degrees C, but seems to drop suddenly to zero at ca. 40 degrees C. Measurements using heavy water (D2O) as a solvent instead of water did not give zero charge at 40 degrees C for cytochrome c.     

Breakthrough curves and elution profiles of single solutes in case of adsorption isotherms with two inflection points

The shape of breakthrough curves and elution profiles depends strongly on the course of the specific equilibrium functions characterizing the chromatographic system. For a highly efficient system the equilibrium theory provides a methodology how to predict the band profiles. The concept is frequently applied to analyze single component systems characterized by isotherms possessing simple shapes (Langmuir or anti-Langmuir behaviour). However, adsorption isotherms often possess more complicated shapes and have inflection points in their courses. This leads to the development of composite concentration waves and results in complex shapes of breakthrough curves and elution profiles. In this paper, the equilibrium theory is used to predict breakthrough curves for a chromatographic system characterized by an adsorption isotherm with two inflection points. The results obtained are validated by comparing with numerical solutions of the equilibrium dispersive model.     

An IUPAC-based approach to estimate the detection limit in co-extraction-based optical sensors for anions with sigmoidal response calibration curves

An approach based on IUPAC methodology to estimate the limit of detection of bulk optode-based analytical methods for anions has been developed. The traditional IUPAC methodology for calculating the detection limit was modified to be adapted to particular cases where the calibration curves have a sigmoidal profile. Starting from the different full theoretical models for every co-extraction mechanism of the analyte in the membrane in bulk optodes, several particular simplified models at low analyte concentration were obtained and validated. The slope of the calibration curve at low analyte concentration was calculated from the first derivative of the simplified equation and, subsequently, the detection limit was estimated. This fitted-for-purpose estimation strategy was applied to anion quantification for in-house bulk optode-based analytical methods, and the estimated limits of detection were compared with those obtained by applying classical geometrical methodology. This way of establishing the detection limit yields values that maintain their true statistical and probabilistic aspects. It can be easily applied to any analytical system which yields non-linear calibration curves at low analyte concentration.     

Adsorption interaction parameter of polyethers in ternary mobile phases: The critical adsorption line

It is shown that in LC of polymers, the interaction parameter in ternary mobile phases can be described by a plane, which is determined by the dependencies in binary mobile phases. Instead of a critical adsorption point, critical conditions are observed along a straight line of composition between the two critical points in binary mobile phases. Consequently, a separation of block copolymers under critical conditions for one block by an adsorption mechanism for the other block can be achieved in ternary mobile phases of different compositions, which allows an adjustment of the retention of the adsorbing block.     

The method of concentration-dependent distribution in the quantitative use of radioisotopes : Theoretical calibration curves and comparison to the “substoichiometric principle”

The method is based on the utilization of a calibration curve which shows the dependence of the distribution ratio in a two-phase system containing the substance to be determined or a substance reacting with it, on the total concentration of the substance to be determined. The relationship of the proposed method to radiometric titration and to the “substoichiometric principle” in isotopic dilution is discussed. Formulae for the theoretical calibration curves in 4 systems are derived: sorption conforming to the Langmuir isotherm or to the Freundlich isotherm, extraction with a constant quantity of an extracting agent, and sorption or extraction in the presence of a constant quantity of a chelating agent. Theoretical sensitivities are given.     

Ultra‐performance liquid chromatography of amino acids for the quality assessment of pearl powder

Pearls have been widely used as a traditional medicine, in cosmetics, and as a health food supplement in China since ancient times. However, the identification and quality assessment of pearl powder have been challenging tasks because of the similar morphological features and chemical composition of its common adulterants, especially conch powders. In this study, ultra‐performance liquid chromatography was combined with pre‐column derivatization to rapidly quantify 14 amino acids in pearl powder and its analogues. Based upon the quantification results, a quality criterion of a total amino acid content of not less than 1.10% was proposed for pearl powder. Principal component analysis indicated that leucine and phenylalanine were the amino acids characteristic for distinguishing between pearls and nacres. The area ratio of leucine to phenylalanine was demonstrated to be an effective diagnostic marker to discriminate freshwater cultured pearls, natural seawater pearls, and nacres. The proposed method, involving both the qualitative and quantitative aspects, was subsequently applied to quality assessment of pearl powders purchased commercially in various parts of China; eight out of 18 batches were deemed authentic and unadulterated. In the future, this analytical process should play a significant role in standardizing and providing quality control to the pearl powder market.