Using Polynomial Regression to Objectively Test the Fit of Calibration Curves in Analytical Chemistry (original) (raw)
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Calibration and Validation of Analytical Methods - A Sampling of Current Approaches, 2018
Calibration curve is a regression model used to predict the unknown concentrations of analytes of interest based on the response of the instrument to the known standards. Some statistical analyses are required to choose the best model fitting to the experimental data and also evaluate the linearity and homoscedasticity of the calibration curve. Using an internal standard corrects for the loss of analyte during sample preparation and analysis provided that it is selected appropriately. After the best regression model is selected, the analytical method needs to be validated using quality control (QC) samples prepared and stored in the same temperature as intended for the study samples. Most of the international guidelines require that the parameters, including linearity, specificity, selectivity, accuracy, precision, lower limit of quantification (LLOQ), matrix effect and stability, be assessed during validation. Despite the highly regulated area, some challenges still exist regarding the validation of some analytical methods including methods when no analyte-free matrix is available.
Usefulness of Information Criteria for the Selection of Calibration Curves
Analytical Chemistry, 2013
The reliability of analytical results obtained with quantitative analytical methods is highly dependent on the selection of the adequate model used as the calibration curve. To select the adequate response function or model the most used and known parameter is to determine the coefficient R 2. However, it is well-known that it suffers many inconveniences, such as leading to overfitting the data. A proposed solution is to use the adjusted determination coefficient R adj 2 that aims at reducing this problem. However, there is another family of criteria that exists to allow the selection of an adequate model: the information criteria AIC, AICc, and BIC. These criteria have rarely been used in analytical chemistry to select the adequate calibration curve. This works aims at assessing the performance of the statistical information criteria as well as R 2 and R adj 2 for the selection of an adequate calibration curve. They are applied to several analytical methods covering liquid chromatographic methods, as well as electrophoretic ones involved in the analysis of active substances in biological fluids or aimed at quantifying impurities in drug substances. In addition, Monte Carlo simulations are performed to assess the efficacy of these statistical criteria to select the adequate calibration curve.
On the misuse of the correlation coefficient to assess linearity of calibration curves
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 clearly curved relationship. Other statistical tests, like the Lack-of-fit and Mandel's fitting test appear therefore 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 characterised 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 tests. Furthermore, significantly different results were achieved between a linear (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 straightline model with a high correlation coefficient, but with a lack of fit, yields significantly less accurate results than its curvilinear alternative.
Linearity of calibration curves: use and misuse of the correlation coefficient
Accreditation and Quality Assurance, 2002
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 ba-sis 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.
Analytical Letters, 1993
A model to calculate the analytical sensitivity, limit of detection, llmit of determination and precision of a method of instrumental analysis through a data set obtained by calibration experiment using the statistical analysis of linear regression, is proposed. This model has been applied to spectrophotometric, spectrofluorimetric and chromatographic methods. The values obtained are independent of the instrument used and can be applied as a criterion of comparison between differen? methods proposed for the same analyte. Also, these characteristics have been calculated using the IUPAC suggested model and * both results have been compared.
Effects of experimental design on calibration curve precision in routine analysis
The Journal of Automatic Chemistry, 1998
A computational program which compares the effciencies of different experimental designs with those of maximum precision (D-optimized designs) is described. The program produces confidence interval plots for a calibration curve and provides information about the number of standard solutions, concentration levels and suitable concentration ranges to achieve an optimum calibration. Some examples of the application of this novel computational program are given, using both simulated and real data.
The effect of influential data, model and method on the precision of univariate calibration
Talanta, 2002
Building a calibration model with detection and quantification capabilities is identical to the task of building a regression model. Although commonly used by analysts, an application of the calibration model requires at first careful attention to the three components of the regression triplet (data, model, method), examining (a) the data quality of the proposed model; (b) the model quality; (c) the LS method to be used or a fulfillment of all least-squares assumptions. This paper summarizes these components, describes the effects of deviations from assumptions and considers the correction of such deviations: identifying influential points is the first step in least-squares model building, the calibration task depends on the regression model used, and finally the least squares LS method is based on assumptions of normality of errors, homoscedasticity, independence of errors, overly influential data points and independent variables being subject to error. When some assumptions are violated, the ordinary LS is inconvenient and robust M-estimates with the iterative method of reweighted least-squares must be used. The effects of influential points, heteroscedasticity and non-normality on the calibration precision limits are also elucidated. This paper also considers the proper construction of the statistical uncertainty expressed as confidence limits predicting an unknown concentration (or amount) value, and its dependence on the regression triplet. The authors' objectives were to provide a thorough treatment that includes pertinent references, consistent nomeclature, and related mathematical formulae to show by theory and illustrative examples those approaches best suited to typical problems in analytical chemistry. Two new algorithms, calibration and linear regression written in S-PLUS and enabling regression triplet analysis, the estimation of calibration precision limits, critical levels, detection limits and quantification limits with the statistical uncertainty of unknown concentrations, form the goal of this paper.
Journal of AOAC INTERNATIONAL, 2010
Linearity assessment as required in method validation has always been subject to different interpretations and definitions by various guidelines and protocols. However, there are very limited applicable implementation procedures that can be followed by a laboratory chemist in assessing linearity. Thus, this work proposes a simple method for linearity assessment in method validation by a regression analysis that covers experimental design, estimation of the parameters, outlier treatment, and evaluation of the assumptions according to the International Union of Pure and Applied Chemistry guidelines. The suitability of this procedure was demonstrated by its application to an in-house validation for the determination of plasticizers in plastic food packaging by GC.
Analytica Chimica Acta, 2007
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.