Nonlinear least-squares regression analysis

FIGURE 19.4 Relationsliip behA een plasma concentrations of tocainide and suppression of ventricular premature beats (VPBs) for four representative patients. The relationship betwreen VPB frequency and tocainide concentrations shown by the solid curves was obtained from a nonlinear least-squares regression analysis of the data using Equation 19.10. The estimate of n for each patient can be compared with the shape of the tocainide concentration-antiarrhythmic response curve. (Reproduced with permission from Meffin PJ, Winkle RA, Blaschke TF, Fitzgerald J, Harrison DC. Clin Pliarmacol Ther 1977 22 42-57.)... 

The most common fitting analysis employed is to fit the experimental data to nonlinear least squares regression analysis (NLLS). This analysis is employed for a single decay curve obtained at set experimental parameters, such as fixed excitation and emission wavelengths and fixed concentrations of probe, supramolecular structure, and quencher. The experimental data are fitted to equations derived from assumed mechanistic models. The fitting procedure, which... 

The Nonlinear Least-Squares Regression Analysis of Kinetic Data... 

Toward these ends, the kinetics of a wider set of reaction schemes is presented in the text, to make the solutions available for convenient reference. The steady-state approach is covered more extensively, and the mathematics of other approximations ( improved steady-state and prior-equilibrium) is given and compared. Coverage of data analysis and curve fitting has been greatly expanded, with an emphasis on nonlinear least-squares regression. 

In a well-behaved calibration model, residuals will have a Normal (i.e., Gaussian) distribution. In fact, as we have previously discussed, least-squares regression analysis is also a Maximum Likelihood method, but only when the errors are Normally distributed. If the data does not follow the straight line model, then there will be an excessive number of residuals with too-large values, and the residuals will then not follow the Normal distribution. It follows, then, that a test for Normality of residuals will also detect nonlinearity. 

The following protocol was proposed and consisted of 4 measuring days. Each day, four (or six at day 1) standards and four samples are analyzed. The calibration curves are constructed by least squares regression analysis and statistically tested for nonlinearity by means of an F-test on the residuals. The amount of cortisol in the serum samples is obtained by linear interpolation on the daily calibration curve. Preliminary experiments were also set up to determine the influence of the use of peak height or peak area ratios. For the cortisol measurement, some separation takes place between syn and anti isomers, therefore the use of peak heights is less favorable. 

As discussed in Section 8.3.8, the confidence limits (CLs) obtained from nonlinear least-squares regression are not very reliable. If time, availability of standards and finances permit, a better feel for the true CLs is obtained by repeating the same experiment (calibration) several times. Similar comments apply to CLs of interpolated values of (Qa /Qsis ) from measured values of (Ra /Rsis) in actual analysis using the nonlinear regression parameters evaluated from the calibration experiments (compare Equation [8.32] for the linear regression case). Of course, as before, the calibrators can in principle be made up either as mixed solutions of pure analytical and internal standards or as extracts of spiked matrix blanks. 

The effect of PDMS molecular weight on the interfacial tension at constant temperature for a constant molecular weight of PBD (M = 980, MJM = 1.07) is illustrated in Fig. 4. The molecular weight dependence was obtained by performing nonlinear least-squares regression of the data to an expression of the form y = This analysis yielded z = 0.54 for the present PDMS/PBD... 

Fig. 18.3 Calibration plot relevant to the analysis of uric acid at a taurine-modified glassy carbon electrode. Experimental point can be fitted by nonlinear least-square regression. The two additional lines show the fitting of the curvilinear dynamic range by two subsequent linear ranges...

Since the nonlinear least-squares method requires initial guesses to start the procedure, three different initial trials were performed (1) (0,0), (2) (1,1), and (3) the values obtained from the Lineweaver-Burk plot in Example 4.2.4. All three initial trials give the same result (and thus the same relative error). Note the large differences in the values obtained from the nonlinear analysis versus those from the linear regression. If the solutions are plotted along with the experimental data as shown below, it is clear that the Lineweaver-Burk analysis does not provide a good fit to the data. 

If an independent measure of [ ]q is available from chemisorption, the constants k2, Ki, and K2 can be obtained from linear regression. It should be noted that many kineticists no longer use the linearized form of the rate equation to obtain rate constants. Inverting the rate expression places greater statistical emphasis on the lowest measured rates in a data set. Since the lowest rates are usually the least precise, a nonlinear least squares analysis of the entire data set using the normal rate expression is preferred. 

However, the regression theory requires that the errors be normally distributed around (—7 a). and not around f as in the linearized version just described. Hence use the values determined as initial estimates to obtain more accurate values of the constants by minimizing the sum of squares of the residuals of the rates directly from the raw rate equation by nonlinear least squares analysis. 

Changing the regression method by using either weighted, least-squares analysis, if the variance is not homoscedastic, or nonlinear, least-squares analysis to determine the parameter values. 

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