Parabolic regression

If the regression expression is a polynomial, then, by applying the method of least squares to identify the coefficients and compute the values of the coefficients, we obtain a simple linear system. If we particularize the case for a regression expression given by a polynomial of second order, the general relation (5.3) is reduced to  

The same procedure is used if we increase the polynomial degree given by the regression equation. In this case, the tests of the coefficient significance and model confidence are implemented as shown in the example developed in Section 5.4.1.1. It is important to note that we must use relation (5.59) for the calculation of the variances around the mean value of Pj. 

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A multivariate linear regression can successfully be employed for moderately wide range calibration. As an example parabolic regression (17) 31,72) should be mentioned. Higher order polynomials have disadvantages in terms of degree of freedom and error propagation, and the results are less precise. 

Note that the lipophilicity parameter log P is defined as a decimal logarithm. The parabolic equation is only non-linear in the variable log P, but is linear in the coefficients. Hence, it can be solved by multiple linear regression (see Section 10.8). The bilinear equation, however, is non-linear in both the variable P and the coefficients, and can only be solved by means of non-linear regression techniques (see Chapter 11). It is approximately linear with a positive slope (/ ,) for small values of log P, while it is also approximately linear with a negative slope b + b for large values of log P. The term bilinear is used in this context to indicate that the QSAR model can be resolved into two linear relations for small and for large values of P, respectively. This definition differs from the one which has been introduced in the context of principal components analysis in Chapter 17. 

A non-linear Hansch model has been applied to the bactericidal concentrations (O of 17 doubly substituted phenols (Table 37.2) which have been reported by Klarmann et al. [20]. By means of multiple linear regression we obtain the parabolic Hansch model of eq. (37.9) ... 

Partitioning into the CNS will be important for hallucinogens, as for any drug that acts centrally. Correlation between 1-octanol/water partition coefficients and human activity has been reported (13). Regression analysis of log human activity on log P yielded a parabolic fit with an optimum at log P 3.14. The derived equation accounted for only 62% of the variance but included compounds with a variety of substitution patterns and, presumably, qualitative differences in activity. 

More careful examination of this shape reveals two important facts, (a) Plots of ssq as a function of k at fixed Io are not parabolas, while plots of ssq vs. Io at fixed k are parabolas. This indicates that Io is a linear parameter and k is not. (b) Close to the minimum, the landscape becomes almost parabolic, see Figure 4-6. We will see later in Chapter 4.3, Non-Linear Regression, that the fitting of non-linear parameters involves linearisation. The almost parabolic landscape close to the minimum indicates that the linearisation is a good approximation. 

The a and n constants of substituents are often useful when correlated to biological activity in the statistical procedure known as multivariate regression analysis. As is well known from pharmacological testing of various drug series, such correlations can be either linear or parabolic. The linear relationship is described by the equation... 

In order to make the comparison between Ep and Ep/2 measurements summarized in Table 9, the two quantities were measured in separate experiments. A recent study by Eliason and Parker has shown that this is not necessary [57]. Analysis of theoretical LSV waves by second-order linear regression showed that data in the region of Ep are very nearly parabolic. The data in Fig. 9 are for the LSV wave for Nernstian charge transfer. The circles are theoretical data and the solid line is that described by a second-order polynomial equation. It was concluded that no detectable error will be invoked in the measurement of LSV Ep and Ip by the assumption that the data fit the equation for a parabola as long as the data is restricted to about 10 mV on either side of the maximum. This was verified by experimental measurements on both a Nernstian and a kinetic system. 

Figure 8. Comparison of the regression coefficients for parabolic and linear rate laws for plagioclase dissolution at 25°C and a pH range of 3-8...

The regression procedure is strongly influenced by stochastic errors or noise in the measurement. One effect is illustrated in comparison of Figure 19.1 to Figure 19.3, in which stochastic noise with a standard deviation equal to 1 percent of the modulus was added to the synthetic data. Solid lines have been drawn on the bottom contour map to indicate the values for which the function is inmimized. The presence of stochastic errors in the data does not introduce roughness in the parabolic... 

Data regressions based on the law of mass action are generally adequate for most situations. However, this model only retains validity in liquid-phase reactions at equilibrium without cooperativity. Reactions that involve solid-phase, multiple cooperative binding, and not reaching equilibrium, deviate from the model. Therefore, empirical equations that are not based on the law of mass action have been used for curve fitting also. Among these, polynomial (205) and spline functions are often used (206-209). Polynomial regression can be a second-order (parabolic) or third-order (cubic) function ... 

Hall and Kier reexamined this BCF data set using the response surface optimization technique as reported for a neurotoxicity data set.In this approach the nonlinear parabolic form is extended to a general two-variable parabolic form. The analysis can be performed using ordinary multiple linear regression programs or an extended form of the analysis can be performed using SAS. For the 20 compounds investigated by Sabljic, Hall and Stewart " used the sum and difference of the zero order chi indexes, "x "id defined as follows ... 

Table 3. Parabolic rate constant kp for oxidation of P e52AI4S from linear regression of (Am/A)2 versus t...

As the literature indicated that calibration graphs may be expressed as functions close to parabolic ones, the regression lines sought (within the second order) were in the form... 

Sigma-rho corrections were assigned equally to the substituents at positions 6 and 7. Revised STERIMOL (L, B1 and B5) parameters were calculated following established methods.f27.28l The data matrices consist of lipophilicity (F), molar refractivity (MR), STERIMOL (L, Bl, B5) parameters and de novo indicator variables for substituents in positions 1, 6 and 7. The contribution to the partition coefficient by the substituents at positions 6 and 7 were summed, 2F(6,7), as were the molar refractivity contributions, 2MR(6,7). These two sets of summed variables paralleled a similar approach used by Koga in his QSAR analysis of a set of quinoline derivatives. (231 At the same time, the limitations that apply to summed n values, particularly where there is a significant a-p interaction were kept in mind.f22.29.301 The latter is represented as API in the data tables and its significance was evaluated in some of the regression models. In order to check for parabolic relationships, squared terms for the partition coefficient, molar refraction, and STERIMOL parameters were evaluated. 

However, the parabolic model is still valuable for structure-activity analyses. It is the simpler model, easier to calculate, and most often a sufficient approximation of the true structure-activity relationship. The calculation of bilinear equations is relatively time-consuming, as compared to the parabolic model strange results may be obtained in ill-conditioned data sets. On the other hand, in many cases the, bilinear model gives a better description of the data, especially if additional physicochemical parameters are included in the regression equation. The lipophilicity optimum of symmetrical curves is precisely described by both, the parabolic model (optimum log P = — b/2a) and the bilinear model (optimum log P = — log P). In the case of unsymmetrical curves the site of the lipophilicity optimum is described much better by the bilinear model (optimum log P = log a — log p — log (b — a) eq. 93) than by the parabolic model. 

The discipline of quantitative structure-activity relationships (QSAR), as we define it nowadays, was initiated by the pioneering work of Corwin Hansch on growthregulating phenoxyacetic acids. In 1962—1964 he laid the foundations of QSAR by three important contributions the combination of several physicochemical parameters in one regression equation, the definition of the lipophilicity parameter jt, and the formulation of the parabolic model for nonlinear lipophilicity-activity relationships. 

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