Multiple straight-line least-squares analysis with uncertainties in all variables

doi:10.1016/0097-8485(90)80045-4

Computers & Chemistry

Volume 14, Issue 3, 1990, Pages 189-192

https://doi.org/10.1016/0097-8485(90)80045-4 Get rights and content

Abstract

A simple iterative method is described for fitting experimental data with uncertainties in all variables to straight-line models. The optimization algorithm does not require starting parameter guesses. The solution provides the “true” least-squares estimates of parameters and a complete variance-covariance matrix.

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In general, the calibration of an instrument involves a set of calibration standards with known concentration to measure the response of the instrument for each standard, and to fit a model to establish the relationship between the instrumental response and the concentration of the analyte, regression model that is used to transform the measurements made in a test sample into estimates of the quantity of the analyte. However, fitting empirical models involves specific peculiarities related to the data and problem being tackled. Hence, a net chemometric approach is adopted here that emphasizes the utility for calibration, determination of the sensitivity of a method, linear range, detection of possible matrix and interferent effects, comparison of two methods, calculation of a detection limit, decision limit, capability of detection, etc.
Measurement methods comparison with errors-in-variables regressions. From horizontal to vertical OLS regression, review and new perspectives
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This paper summarizes and confronts the relationships between six well-known regressions applied in the context of measurement methods comparison with or without replicated data. When two measurement methods are equivalent, it can be expected that there is no analytical bias between them and they must provide the same results on average notwithstanding the measurement errors. Usually, there is no golden-standard (like a calibration problem) and each measurement method is compared to another one and vice-versa with a linear regression analysis. If there is no bias, the results should not be far from the identity line Y = X. Unfortunately, the measurement errors in each axis must be taken into account in the regression analysis (and ideally the switching of axes) by applying a linear errors-in-variables regression. Therefore, the OLS (Ordinary Least Square) regression lines provide biased estimated lines while the errors-in-variables regressions lines lie between the two extreme OLS lines. DR (Deming Regression) and BLS (Bivariate Least Square) regressions are the most general regressions and provide (asymptotically) unbiased estimated lines which are confounded under homoscedasticity. The biases and coverage probabilities of the six regressions are compared with simulations where the results are displayed in charts instead of large data sets. As the precision ratio λ of two measurement methods is a very important parameter in errors-in-variables regressions, it will be a leitmotiv in this paper and all the charts will be linked to it. An unbiased estimator of λ is also given.
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Achieving second order advantage with multi-way partial least squares and residual bi-linearization with total synchronous fluorescence data of monohydroxy-polycyclic aromatic hydrocarbons in urine samples
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Table 3 summarizes the results obtained with the two algorithms. The statistical comparison of the results in Table 3 was made via the bivariate least-squares (BLS) regression method [36–38] and the elliptic joint confidence region (EJCR) test [39]. The plots of predicted versus actual concentrations provided the following slopes (b) and an intercepts (a): U-PLS/RBL, b = 1.019 ± 0.040 and a = –0.17 ± 1.00; N-PLS/RBL, b = 1.018 ± 0.035 and a = –0.28 ± 0.88.
An attractive approach to handle matrix interference in samples of unknown composition is to generate second- or higher-order data formats and process them with appropriate chemometric algorithms. Several strategies exist to generate high-order data in fluorescence spectroscopy, including wavelength time matrices, excitation–emission matrices and time-resolved excitation–emission matrices. This article tackles a different aspect of generating high-order fluorescence data as it focuses on total synchronous fluorescence spectroscopy. This approach refers to recording synchronous fluorescence spectra at various wavelength offsets. Analogous to the concept of an excitation–emission data format, total synchronous data arrays fit into the category of second-order data. The main difference between them is the non-bilinear behavior of synchronous fluorescence data. Synchronous spectral profiles change with the wavelength offset used for sample excitation. The work presented here reports the first application of total synchronous fluorescence spectroscopy to the analysis of monohydroxy-polycyclic aromatic hydrocarbons in urine samples of unknown composition. Matrix interference is appropriately handled by processing the data either with unfolded-partial least squares and multi-way partial least squares, both followed by residual bi-linearization.
Theil-Sen nonparametric regression technique on univariate calibration, inverse regression and detection limits
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Citation Excerpt :
Indeed, the effect of the “making up solutions” factor is taken into account through the overall measurement uncertainty at fixed concentration level which is comprehensive of the contributes of the instrumental and of the concentration variances. If certified reference materials are used as calibration standards and their uncertainty is of the same order of magnitude of that of the response variable, the bivariate least-squares (BLS) regression technique is found suitable to calculate the regression coefficients [6], and to construct both their joint confidence region [7] and the prediction intervals for the response [8]. As about the point (iii) the widely used OLS-based procedure assumes the responses of the reference method as free of error (x-axis) while y-axis refers to the responses obtained by the new method.
This paper reports the combined use of the nonparametric Theil–Sen (TS) regression technique and of the statistics of Lancaster–Quade (LQ) concerning the linear regression parameters to solve typical analytical problems, like method comparison, calculation of the uncertainty in the inverse regression, determination of the detection limit. The results of this new approach are compared to those obtained with appropriate reference methods, using simulated and real data sets. The nonparametric Theil–Sen regression technique appears a new robust tool for the problems considered because it is free from restrictive statistical constraints, avoids searching for the error nature on x and y, which may require long analysis times, and it is easy to use. The only drawback is that the intrinsic nature of the method may lead to a possible enlargement of the uncertainty interval of the discriminated concentration and to the determination of larger detection limits than those obtainable with the commonly used, less robust, regression techniques.
Validation and Error
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The estimation of prediction uncertainties and figures of merit for first- and higher-order multivariate calibration methods are discussed. Starting from a classification of multivariate data, instruments, and algorithms, error propagation is qualitatively used to discuss the estimation of standard errors of prediction, as a natural extension of the well-known univariate estimates. Figures of merit such as sensitivity and selectivity are discussed on the basis of the so-called net analyte signal concept, which allows to generalize the definitions both in the first- and in the second-order domain. In the latter case, the expressions cover all possible calibration cases, whether the so-called second-order advantage is achieved or not. Other figures of merit (limit of detection, limit of discrimination, limit of quantification, etc.) are also discussed. Finally, method validation and estimation of analytical properties relevant for validation of multivariate methods are reviewed, with emphasis on accuracy and precision studies.
Quality of Analytical Measurements: Univariate Regression
2009, Comprehensive Chemometrics
From a statistical point of view, the regression analysis is an area of ongoing research, so an ever-growing collection of techniques makes it difficult the selection of the most adequate one for a given problem.
From a chemical or biochemical point of view, instrumental calibration is an essential stage in many procedures of measurement for the quantitative determination of an analyte in a sample and also to evaluate the quality of the procedure.
In general, the calibration of an instrument supposes to prepare a set of calibration standards with known quantity of the analyte of interest, to measure the response of the instrument for each standard, and to establish the relationship between the response of the instrument and the concentration of the analyte. After that, this relationship is used to transform the measurements made in a test sample into estimates of the quantity of the analyte.
Consequently, it is necessary that the analysts have a good understanding of how to carry out a calibration experiment and how to evaluate the results obtained. This is the basic objective of this chapter. Section 1.05.1 gives a brief introduction, and Section 1.05.2 introduces the notation and the univariate linear regression model related to the calibration, and shows the least squares solution and the effect of supposing a normal distribution for the errors. The assessment of the hypotheses of the model is the subject of Section 1.05.3. Under the normality assumption, Section 1.05.4 summarizes the confidence intervals and the inferential procedures needed for an adequate treatment of the analytical questions, such as the uncertainty of the analytical determinations, the comparison of calibration models, or the evaluation of the bias.
In 1.05.5 The Design of a Calibration, 1.05.6 The Capability of Detection, the Decision Limit, and the Capability of Discrimination Computed from a Regression Model, 1.05.7 Standard Addition Method, practical questions are tackled, such as the problem of the distribution of the calibration standards to improve the precision, the capability of detection, and the standard addition method.
Finally, the last three sections, of methodological character, are dedicated to introduce generalizations of the least squares method. In Section 1.05.8, the lack of homoscedasticity in the data is approached by introducing the question as a particular case inside the generalized additive models. The diagnosis of outlier data is dealt with in Section 1.05.9 by using the least median squares regression, showing at the same time the analytical utility of the property of the exact fit. Finally, in Section 1.05.10, the case in which both the predictor and the response variables are affected by errors is exposed and some techniques that are used to handle this problem are shown.
The chapter concludes with an attention call on the necessity of using the least squares regression reflectively because the frequency with which a deviation of the hypotheses leads to very incorrect results is amply well known.

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Multiple straight-line least-squares analysis with uncertainties in all variables

Abstract

J. Chem. Educ.

Comput. J.