Nonlinear regression described

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. 

Kinetic studies at several temperatures followed by application of the Arrhenius equation as described constitutes the usual procedure for the measurement of activation parameters, but other methods have been described. Bunce et al. eliminate the rate constant between the Arrhenius equation and the integrated rate equation, obtaining an equation relating concentration to time and temperature. This is analyzed by nonlinear regression to extract the activation energy. Another approach is to program temperature as a function of time and to analyze the concentration-time data for the activation energy. This nonisothermal method is attractive because it is efficient, but its use is not widespread. ... 

One shortcoming of Schild analysis is an overemphasized use of the control dose-response curve (i.e., the accuracy of every DR value depends on the accuracy of the control EC o value). An alternative method utilizes nonlinear regression of the Gaddum equation (with visualization of the data with a Clark plot [10], named for A. J. Clark). This method, unlike Schild analysis, does not emphasize control pECS0, thereby giving a more balanced estimate of antagonist affinity. This method, first described by Lew and Angus [11], is robust and theoretically more sound than Schild analysis. On the other hand, it is not as visual. Schild analysis is rapid and intuitive, and can be used to detect nonequilibrium steady states in the system that can corrupt... 

The partial differential equations describing the catalyst particle are discretized with central finite difference formulae with respect to the spatial coordinate [50]. Typically, around 10-20 discretization points are enough for the particle. The ordinary differential equations (ODEs) created are solved with respect to time together with the ODEs of the bulk phase. Since the system is stiff, the computer code of Hindmarsh [51] is used as the ODE solver. In general, the simulations progressed without numerical problems. The final values of the rate constants, along with their temperature dependencies, can be obtained with nonlinear regression analysis. The differential equations were solved in situ with the backward... 

Finally, we should refer to situations where both independent and response variables are subject to experimental error regardless of the structure of the model. In this case, the experimental data are described by the set (yf,x,), i=l,2,...N as opposed to (y,Xj), i=l,2,...,N. The deterministic part of the model is the same as before however, we now have to consider besides Equation 2.3, the error in Xj, i.e., x, = Xj + ex1. These situations in nonlinear regression can be handled very efficiently using an implicit formulation of the problem as shown later in Section 2.2.2... 

In addition to the three methods described above, nonlinear regression methods or other transform approaches may be used to determine the dispersion parameter. For a more complete treatment of the use of transform methods, consult the articles by Hopkins et al. (15) and Ostergaard and Michelsen (14). 

This observation is expected from theory, as the observed thickness distributions are exactly the functions by which one-dimensional short-range order is theoretically described in early literature models (Zernike and Prins [116] J. J. Hermans [128]). From the transformed experimental data we can determine, whether the principal thickness distributions are symmetrical or asymmetrical, whether they should be modeled by Gaussians, gamma distributions, truncated exponentials, or other analytical functions. Finally only a model that describes the arrangement of domains is missing - i.e., how the higher thickness distributions are computed from two principal thickness distributions (cf. Sect. 8.7). Experimental data are fitted by means of such models. Unsuitable models are sorted out by insufficient quality of the fit. Fit quality is assessed by means of the tools of nonlinear regression (Chap. 11). 

If the structural entities are lamellae, Eq. (8.80) describes an ensemble of perfectly oriented but uncorrelated layers. Inversion of the Lorentz correction yields the scattering curve of the isotropic material I (5) = I (s) / (2ns2). On the other hand, a scattering pattern of highly oriented lamellae or cylinders is readily converted into the ID scattering intensity /, (53) by ID projection onto the fiber direction (p. 136, Eq. (8.56)). The model for the ID intensity, Eq. (8.80), has three parameters Ap, dc, and <7C. For the nonlinear regression it is important to transform to a parameter set with little parameter-parameter correlation Ap, dc, and oc/dc. When applied to raw scattering data, additionally the deviation of the real from the ideal two-phase system must be considered in an extended model function (cf. p. 124). 

The authors describe the use of a Taylor expansion to negate the second and the higher order terms under specific mathematical conditions in order to make any function (i.e., our regression model) first-order (or linear). They introduce the use of the Jacobian matrix for solving nonlinear regression problems and describe the matrix mathematics in some detail (pp. 178-181). 

The concepts described so far are usually not applied in the original data space but in a transformed and enlarged space using basis expansions (see Section 4.8.3 about nonlinear regression). Thus each observation jc, is expressed by a set of basis functions (object vectors jc, with m dimensions are replaced by vectors h(x,) with r dimensions)... 

To verify such a steric effect a quantitative structure-property relationship study (QSPR) on a series of distinct solute-selector pairs, namely various DNB-amino acid/quinine carbamate CSPpairs with different carbamate residues (Rso) and distinct amino acid residues (Rsa), has been set up [59], To provide a quantitative measure of the effect of the steric bulkiness on the separation factors within this solute-selector series, a-values were correlated by multiple linear and nonlinear regression analysis with the Taft s steric parameter Es that represents a quantitative estimation of the steric bulkiness of a substituent (Note s,sa indicates the independent variable describing the bulkiness of the amino acid residue and i s.so that of the carbamate residue). For example, the steric bulkiness increases in the order methyl < ethyl < n-propyl < n-butyl < i-propyl < cyclohexyl < -butyl < iec.-butyl < t-butyl < 1-adamantyl < phenyl < trityl and simultaneously, the s drops from -1.24 to -6.03. In other words, the smaller the Es, the more bulky is the substituent. The obtained QSPR equation reads as follows ... 

Thereafter, a reference text such as Enzyme Kinetics (Segel, 1993) should be consulted to determine whether or not the proposed mechanism has been described and characterized previously. For the example given, it would be found that the proposed mechanism corresponds to a system referred to as partial competitive inhibition, and an equation is provided which can be applied to the experimental data. If the data can be fitted successfully by applying the equation through nonlinear regression, the proposed mechanism would be supported further secondary graphing approaches to confirm the mechanism are also provided in texts such as Enzyme Kinetics, and values could be obtained for the various associated constants. If the data cannot be fitted successfully, the proposed reaction scheme should be revisited and altered appropriately, and the whole process repeated. 

It is possible, of course, that a devised reaction scheme may be novel and that no existing equation can be identified. In such a case, it will be necessary to derive an equation to describe the proposed mechanism, and thereafter to attempt once again to fit the experimental data by nonlinear regression. If this is deemed necessary, there are two approaches that might be taken. 

Representative data for [ H]acetylcholine binding to the membrane-bound Torpedo nAChR. Bindng was measured either by equilibrium dialysis (closed circles) as described in Protocol 4.1 or by centrifugation (open squares, see Protocol 4.2). Estimated Kd values from nonlinear regression curve fitting were 12 nM and 10 nM, respectively with corresponding Rq values of 0.14 ulM and 0.135 ulM... 

Fig. 7.7 Representative binding curve obtained by nonlinear regression from a competitive MS binding assay for dopamine Di receptors, in which (+)-butaclamol competes with SCH 23390 as marker. The points describe nonbound SCH 23390 quantified by LC-ESI-MS/MS. Data reflect means (+s) from binding samples, each performed in quadruplicate.

Nonlinear regression analysis, taking into account all the equilibria, seems to be a reasonable way to get a true picture of the processes taking place and finally to get the relevant data. In practice, however, it is not possible to decide whether the mathematical model describes the reality... 

The first work on pKa determination by zone electrophoresis using paper strips was described by Waldron-Edward in 1965 (15). Also, Kiso et al. in 1968 showed the relationship between pH, mobility, and p/C, using a hyperbolic tangent function (16). Unfortunately, these methods had not been widely accepted because of the manual operation and lower reproducibility of the paper electrophoresis format. The automated capillary electrophoresis (CE) instrument allows rapid and accurate pKa determination. Beckers et al. showed that thermodynamic pATt, (pATf) and absolute ionic mobility values of several monovalent weak acids were determined accurately using effective mobility and activity at two pH points (17). Cai et al. reported pKa values of two monovalent weak bases and p-aminobenzoic acid (18). Cleveland et al. established the thermodynamic pKa determination method using nonlinear regression analysis for monovalent compounds (19). We derived the general equation and applied it to multivalent compounds (20). Until then, there were many reports on pKa determination by CE for cephalosporins (21), sulfonated azo-dyes (22), ropinirole and its impurities (23), cyto-kinins (24), and so on. 

In this chapter, three chemometric methods of increasing importance to SEC are examined nonlinear regression, graphics and error propagation analysis. These three methods are briefly described with emphasis on SEC applications and on critical concerns in their correct implementation. In addition to the specific references cited, further information on these methods and others may be found in a recent book vdiich examines chemometrics in both SEC and HPIiC together (J ) as well as in periodic reviews (2) ... 

From an examination of Eq. (6) for a two-compartment model it is evident that Vss is dependent on the quantification of K12 and K21. For this model K12 and K21 can be determined by nonlinear regression analysis of plasma concentration-time data, either by deriving them from the fitted values of the coefficients and exponentials of the bi-exponential expression describing the concentration-time data, or by coding them directly into the modeling program. For the case where tissue elimination exists, it is possible to code into the model the existence of a K20, but the convergence process will not be able to resolve the appropriate micro rate constant. 

This process is partially overlapped with the next process, the j3 relaxation. To analyze the loss permittivity in the subglass zone in a more detailed way, the fitting of the loss factor permittivity by means of usual equations is a good way to get confidence about this process [69], Following procedures described above Fig. 2.42 represent the lost factor data and deconvolution in two Fuoss Kirwood [69] as function of temperature at 10.3 Hz for P4THPMA. In Fig. 2.43 show the y and relaxations that result from the application of the multiple nonlinear regression analysis to the loss factor against temperature. The sum of the two calculated relaxations is very close to that in the experimental curve. 

Curvilinear regression should not be confused with the nonlinear regression methods used to estimate model parameters expressed in a nonlinear form. For example, the model parameters a and b in y = axb cannot be estimated by a linear least-squares algorithm. Information in Chapter 7 describes nonlinear approaches to use in this case. Alternatively, a transformation to a linear model can sometimes be used. Implementing a logarithmic transformation on yt = ax/ produces the model log y = log a + blog x which can now be utilized with a linear least-squares algorithm. The literature [4, 5] should be consulted for additional information on linear transformations. 

In actual practice, nonlinear regression is used to fit a suitable pharmacokinetic model described by the function c (t) to time—concentration data. Then, the estimated parameters are used as constants in the pharmacodynamic model to estimate the pharmacodynamic parameters. Alternatively, simultaneous fitting of the model to the concentration-effect—time data can be performed. This is recommended as c (t) and E (t) time courses are simultaneously observed. 

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