Reference material regression

Conventional XRF analysis uses calibration by regression, which is quite feasible for known matrices. Both single and multi-element standards are in use, prepared for example by vacuum evaporation of elements or compounds on a thin Mylar film. Comparing the X-ray intensities of the sample with those of a standard, allows quantitative analysis. Depending on the degree of similarity between sample and standard, a small or large correction for matrix effects is required. Calibration standards and samples must be carefully prepared standards must be checked frequently because of polymer degradation from continued exposure to X-rays. For trace-element determination, a standard very close in composition to the sample is required. This may be a certified reference material or a sample analysed by a primary technique (e.g. NAA). Standard reference material for rubber samples is not commercially available. Use can also be made of an internal standard,... 

Generally speaking, alternative methods (including on-line, off-line or in situ methods) may be used provided it can be demonstrated that equivalent results with those of reference procedures can be obtained. The experiments are generally carried out with standard solutions and reference materials for the determination of the method characteristics. The equivalence between methods must be statistically verified by plotting the results (Fig. 5) and checking the coordinates of the experimental regression fine (comparison of the slope and intercept values, which must be not statistically different from respectively 1 and 0 values of the theoretical fine). 

Calibration Most process analyzers are designed to monitor concentration and/or composition. This requires a calibration of the analyzer with a set of prepared standards or from well-characterized reference materials. The simple approach must always be adopted first. For relatively simple systems the standard approach is to use a simple linear relationship between the instrument response and the analyte/ standard concentration [27]. In more complex chemical systems, it is necessary to adopt either a matrix approach to the calibration (still relying on the linearity of the Beer-Lambert law) using simple regression techniques, or to model the concentration and/or composition with one or more multivariate methods, an approach known as chemometrics [28-30]. 

Method validation is important to ensure that the analytical method is in statistical control. A method may be validated by the so-called method evaluation function (MEF) (Christensen et al., 1993), which is obtained by linear regression analysis of the measured concentrations versus the true concentrations. A true concentration in a solution can be obtained by use of a high purity standard obtained from another manufacturer or batch than the one used for calibration. Both the high purity standard and the solvent are weighed using a traceable calibrated balance. If certified reference material is available this is preferred. The method evaluation includes the most important characteristics of the method as the following elements (see Figure 2.7) ... 

For chemical measurements with a linear calibration function, traceability of results can be formally established without great expenditure if the calibration is based on suitable reference standards and the linear regression is performed as shown above and (statistically) validated. The use of reference materials as samples make it possible to establish the traceability of a new analysis protocol by using an existing analysis method. 

Figure Liner regression plots for all secondary reference materials (SRMs) for potassium.

Preparation of Standards and Curves In the absence of a true blank control matrix, standards can be prepared in a protein buffer at multiple (generally 9 11) concentrations for the initial test of assay range. For method validation and sample assay, 6 8 nonzero concentrations of standards plus anchor points should be used to define a curvilinear standard curve. If a commercial kit is to be used, it is preferred that the standards be prepared from a bulk reference material to assure the consistencies of the standard as well as adding enough standard points to properly define the regression model. Before replacing the kit standards with those prepared... 

If several reference materials are available with a range of levels, a linear regression technique can be used to evaluate the bias. Again it is assumed that both types of bias may be present. The customary. w pair terminology for regression is used, and now the measured values are designated by rather than. v as used above, and. v is used for the reference v alues (//, ),. The linear equation used to evaluate the estimates of the slope and intercept obtained from several vr pairs is... 

By plotting the instrumental signal on the y axis against the known concentration (traceable to either a national or international standard or to a certified reference material) on the x axis a calibration plot is obtained. Calibration of analytical procedures is a dynamic process because the analytical method as well as the analyst are involved. Therefore, the analytical method should be in statistical control as the purpose of a calibration is to diminish bias. If a reasonable number of data points equally distributed over the range of interest (at least five in duplicate) valuable statistical tests for linearity, estimated errors, and confidence limits for the slope and the intercept can be performed. The weighed regression analysis should be considered, when the uncertainty depends on the concentration, and although the correlation coefficient is simple to calculate it is only a... 

When the assumption of error-free x-values is not valid, either in method comparisons or, in a conventional calibration analysis, because the standards are unreliable (this problem sometimes arises with solid reference materials), an alternative comparison method is available. This technique is known as the functional relationship by maximum likelihood (FREML) method, and seeks to minimize and estimate both x- and y-dlrection errors. (The conventional least squares approach can be regarded as a special and simple case of FREML.) FREML involves an iterative numerical calculation, but a macro for Minitab now offers this facility (see Bibliography), and provides standard errors for the slope and intercept of the calculated line. The method is reversible (i.e. in a method comparison it does not matter which method is plotted on the x-axis and which on the y-axis), and can also be used in weighted regression calculations (see Section 5.10). 

This expression has been used as the model equation for the regression model in the past. As pointed out by Baxter et aJ.[36], however, this is equivalent to assuming fip =fk/i- The pitfall of this approach is that the resultant value of the regression intercept is not used, rather its value is assumed from Eq. (5.35). To avoid the errors of Eq. (5.39), Baxter et al. proposed a revised regression model that combines two regressions one for the reference material of the analyte and internal standard, and the other that combines the sample and the internal standard [36]. Reliance on the reference material for the measurand, however, is unnecessary, provided that the isotope amount ratio of the internal standard is known [Eq. (5.39)]. However, it is a viable option nevertheless. 

The SN-curve is a line in the Oamp-N-plane (N is a measure of number of cycles to failure, S is a general symbol referring to stress, strain, load, displacement etc. In this case, S denotes aamp). When constant-amplitude fatigue data are available for the material, an SN-curve can be found using regression analysis of the test data. Usually, it is appropriate to... 

To solve the subclass size problem, it has been suggested to estimate regression-based reference intervals. Instead of dividing, for example, the total material into several age classes, one may construct continuous age-dependent reference limits and their confidence regions. Simulation studies have shown that this method produces reliable estimates with small sample sizes. ... 

For the first set of materials, and with the aim of assessing the dispatch conditions, a short-term stability study was conducted at 40°C. The layout chosen for the stability study was the so-called isochronous scheme samples were taken from the bulk, placed at 40° C and then moved back to the reference temperature (4°C), after 1 and 2 weeks. Then, at the same time, the samples were analysed for major components and trace elements. The results, 3 time-points (0, 1, 2 weeks) and 2 units analysed per time-point, were evaluated by one-way analysis of variance ANOVA. As some parameters (especially As, Cd, Cu, and to a minor extent also Mn, pH) showed a statistically significant slope of the regression line, it was decided to assure the dispatch of the samples at 4°C (with cooling elements). 

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