Analytical methods Regression analysis

In a curve-fitting method the concentration of a reactant or product is monitored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. Eor example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentration of a product is followed as a function of time, can be determined by fitting a rearranged form of equation 13.12... 

If a standard method is available, the performance of a new method can be evaluated by comparing results with those obtained with an approved standard method. The comparison should be done at a minimum of three concentrations to evaluate the applicability of the new method for different amounts of analyte. Alternatively, we can plot the results obtained by the new method against those obtained by the approved standard method. A linear regression analysis should give a slope of 1 and ay-intercept of 0 if the results of the two methods are equivalent. 

A reading of Section 2.2 shows that all of the methods for determining reaction order can lead also to estimates of the rate constant, and very commonly the order and rate constant are determined concurrently. However, the integrated rate equations are the most widely used means for rate constant determination. These equations can be solved analytically, graphically, or by least-squares regression analysis. 

If an analytical solution is available, the method of nonlinear regression analysis can be applied this approach is described in Chapter 2 and is not treated further here. The remainder of the present section deals with the analysis of kinetic schemes for which explicit solutions are either unavailable or unhelpful. First, the technique of numerical integration is introduced. 

Fourier transform infrared (FTIR) spectroscopy of coal low-temperature ashes was applied to the determination of coal mineralogy and the prediction of ash properties during coal combustion. Analytical methods commonly applied to the mineralogy of coal are critically surveyed. Conventional least-squares analysis of spectra was used to determine coal mineralogy on the basis of forty-two reference mineral spectra. The method described showed several limitations. However, partial least-squares and principal component regression calibrations with the FTIR data permitted prediction of all eight ASTM ash fusion temperatures to within 50 to 78 F and four major elemental oxide concentrations to within 0.74 to 1.79 wt % of the ASTM ash (standard errors of prediction). Factor analysis based methods offer considerable potential in mineral-ogical and ash property applications. 

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) ... 

To establish chiral separation method for donepezil hydrochloride enantiomers by capillary electrophoresis (CE) and to determine the two enantiomers in plasma [39], alkalized plasma was extracted by isopropa-nol-n-hexane (3 97) and L-butefeina was used as the IS. Enantioresolution was achieved using 2.5% sulfated-beta-cyclodextrin as chiral selector in 25 mmol/1 triethylammonium phosphate solution (pH 2.5) on the uncoated fused-silica capillary column (70 cm x 50 fim i.d.). The feasibility of the method to be used as quantitation of donepezil HC1 enantiomers in rabbit plasma was also investigated. Donepezil HC1 enantiomers were separated at a baseline level under the above condition. The linearity of the response was evaluated in the concentration range from 0.1 to 5 mg/1. The linear regression analysis obtained by plotting the peak area ratio (A(s)/A(i)) of the analyte to the IS versus the concentration (C) showed excellent correlation coefficient The low limit of detection was 0.05 mg/1. The inter- and intra-day precisions (RSD) were all less than 20%. Compared with chiral stationary phase by HPLC, the CE method is simple, reliable, inexpensive, and suitable for studying the stereoseletive pharmacokinetics in rabbit. 

The online statistical calculations can be performed at http //members.aol.com/ johnp71/javastat.html. To carry out linear regression analysis as an example, select Regression, correlation, least squares curve-fitting, nonparametric correlation, and then select any one of the methods (e.g., Least squares regression line, Least squares straight line). Enter number of data points to be analyzed, then data, x and y . Click the Calculate Now button. The analytical results, a (intercept), b (slope), f (degrees of freedom), and r (correlation coefficient) are returned. 

There are several ways to study the effect of various experimental factors on an analytical method. The classical method of studying one variable at a time while holding others constant is extremely inefficient. Other approaches such as regression analysis and complete factorial designs involve a large number of experiments and are also inefficient. For example, a factorial design of seven factors at two levels requires 2 or 128 experiments. Therefore, alternate approaches which reduce the experimental work are very attractive. 

Definition of Linearity The linearity of an analytical method is its ability (within a given range) to elicit test results that are directly, or by a well-defined mathematical transformation, proportional to the concentration of analyte in samples within a given range. Linearity is usually expressed in terms of the variance around the slope of the regression line (correlation coefficient), calculated according to an established mathematical relationship from test results obtained by the analysis of samples with varying concentrations of analyte. 

Determination of Linearity and Range Determine the linearity of an analytical method by mathematically treating test results obtained from analysis of samples with analyte concentrations across the claimed range of the method. The treatment is normally a calculation of a regression line by the method of least squares of test results versus analyte concentrations. In some cases, to obtain proportionality between assays and sample concentrations, the test data may have to be subjected to a mathematical transformation before the regression analysis. The slope of the regression line and its variance (correlation coefficient) provide a mathematical measure of linearity the y-intercept is a measure of the potential assay bias. 

These special cases of multiple linear regression analysis have been developed for the determination of the impact of individual molecular substructures (independent variables) on one dependent variable. Both techniques are similar yet, the Free-Wilson method considers the retention of the unsubstituted analyte as base, while Fujita-Ban analysis uses the less substituted molecule as reference. These procedures have not been frequently employed in chromatography only their application in QSRR studies in RP TLC and HPLC have been reported. 

Inspired by the work of statisticians, chemists have also placed the use of the robust methods on the agenda. Phillips and Eyring [8] were among the first ones who applied an M-estimator, the biweight function of Tukey, in regression analysis of analytical data. They concluded that the efficiency of the robust regression was about the same or superior to the least squares... 

This relationship is established by measurement of samples with known amounts of analyte (calibrators). One may distinguish between solutions of pure chemical standards and samples with known amounts of analyte present in the typical matrix that is to be measured (e.g., human serum). The first situation applies typically to a reference measurement procedure, which is not influenced by matrix effects, and the second case corresponds typically to a field method that often is influenced by matrix components and so preferably is calibrated using the relevant matrix. Calibration functions may be linear or curved, and in the case of immunoassays often of a special form (e.g., modeled by the four-parameter logistic curve) This model (logistic in log x) has been used for both radioimmunoassay and enzyme immunoassay techniques and can be written in several forms as shown (Table 14-1). Nonlinear regression analysis is applied to estimate the relationship, or a logit transforma-... 

Regression analysis is commonly applied in comparing the results of analytical method comparisons. Typically an experiment is carried out in which a series of paired values is collected when comparing a new method with an established method. This series of paired observations (xl,-, x2,) is then used to establish the nature and strength of the rela-... 

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