Statistical methods matrix effects

Quantitative XRF analysis has developed from specific to universal methods. At the time of poor computational facilities, methods were limited to the determination of few elements in well-defined concentration ranges by statistical treatment of experimental data from reference material (linear or second order curves), or by compensation methods (dilution, internal standards, etc.). Later, semi-empirical influence coefficient methods were introduced. Universality came about by the development of fundamental parameter approaches for the correction of total matrix effects... 

In addition to statistical peculiarities, special features may also result from certain properties of samples and standards which make it necessary to apply special calibration techniques. In cases when matrix effects appear and matrix-matched calibration standards are not available, the standard addition method (SAM, see Sect. 6.2.6) can be used. 

Standard addition is used to quantify the concentration in unknown samples when matrix interferences are present. The use of standard addition has been extensively discussed by Rodriguez et al., Cardone, and Honorato et al. [23-26]. Exact amounts of the analyte in increasing concentrations are added to the sample. The response (Y) is plotted vs. the added concentration (X). A straight line is regressed through the data points. The concentration of the analyte in the sample is given by the intercept on the X-axis (Xsample = a/b). Standard addition is probably the best way to correct for matrix effects. Rodriguez et al. have described the statistical techniques that can be used for the validation of analytical methods with standard addition [23],... 

In a statistical model, fixed effects have an influence on the mean value or average of the method s response while random effects have an influence on the variability of the method. Fixed effects are assessed in the context of accuracy. Random effects are assessed in the context of precision and become the intermediate precision components. In designing the validation design matrix the validation assays need to be balanced over both the fixed effects and the random effects. A mixed effects model (or design) occurs when both fixed effects and random effects are present (6). 

From the various results of the homogeneity and stability studies, it was concluded that the part of the uncertainty that cannot be evaluated by the statistical analysis (e.g. related to matrix effects, method stability, blank variations etc.) was in the order of 10% (expressed as standard uncertainty). The uncertainty related to the certification exercise was equal to Ucerj = 0.9166/ 6 = 0.0374 (or 2.0%), whereas the uncertainty factor related to homogeneity is 4.6% (see section 8.9.5). The combined uncertainty was thus estimated as follows U. = [(2.0) + (4.6) + (10) )] = 11%... 

When using the static headspace injector the possible matrix effect should be studied and can be evaluated by a statistical method. ... 

The reason for the lack of RMs is the absence of reliable and accurate methods of analysis. Microanalysis is, hence, in need of at least one method that can be used as a reference tool for other techniques and to link RMs or round robin exercises to the international unit of mass. Micro-XRF can be used for this potentially, especially when used for analyzing microscopic samples, where matrix absorption effects are relatively unimportant. At present, XRF is considered to be a rather poor method for certification purposes due to intense matrix effects resulting from intense radiation absorption and enhancement by secondary fluorescence. In wavelength-dispersive XRF, reliable results can only be obtained through calibration with a set of reference samples of closely similar composition to the unknown sample. In the case of energy-dispersive XRF using monochromatic excitation, the correction for matrix effects is simpler but in this case the method suffers from a number of other drawbacks, such as spectral overlap and poor statistics in the spectra. 

When the nature and composition of the sample is not well known, it is necessary to use influence correction methods, of which there are three primary types fundamental, derived, and regression. In the fundamental approach, the intensity of fluorescence can be calculated for each element in a standard sample from variables such as the source spectrum, the fundamental eiiuations for absorption and fluorescence, matrix effects the crystal reflectivity (in a WDXRF instrument), instrument aperture, the detector efficiency, and so forth. The XRF spectrum of the standard is measured, and in an iterative process the instrument variables are refined and combined with the fundamental variables to obtain a calibration function for the analysis. Then the spectrum for an unknown sample is measured, and the iterative process is repeated using initial estimates of the concentrations of the analytes. Iteration continues until the calculated spectrum matches the unknown spectrum according to appropriate statistical criteria. This method gives good results with accuracies on the order of I %-4% but is generally considered to be less accurate than derived... 

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