Regression analysis methods

Try a graphical method and a regression analysis method for solving... 

The large deviation between the estimated rate constants for a given temperature extrapolated from classical (Fig. 35) and modified Arrhenius regression analysis (Fig. 37) and the experimental rate constants determined at the actual temperature is due to the linear regression analysis method used. That is, the values of the logarithms of the rate constants do not reflect directly the errors in the experimental data. As Bentley pointed out,317 when the error term e is added to the logarithm of k in accordance with Eq. (2.76), in the linear... 

The regression analysis method can be considered as a generalization of the other two types of methods. It can estimate simultaneously some additional transport properties it is analyzed in detail in Section 4.7. 

With the application of the Unear regression analysis method to polyester of NPG-IPA, residual mass can be... 

The corresponding three-dimensional grid plots for the S-values [derived according to Eq. (5)] of these two polypeptides as the temperature and j/ values were systematically varied are shown in Fig. 26a-d. In each case, the S values for polypeptides 1 and 2 were derived by regression analysis methods from the gradient of the experimental plots of log k, vemus i/> at the specified j/ and T values with the regression eoefficients >0.9985. In turn, the S value of a peptide or protein in the presence of an RPC sorbent can be related [16,20,211,212] to extrathermodynamic parameters, such as the accessible molecular surface area, A/l, ui, through the expression... 

The simplified, regular regime, and regression analysis methods are particularly relevant for drying processes. In them, the samples are placed in a dryer and moisture diffusivity is estimated from drying data. All the drying methods are based on Pick s... 

Various regression analysis methods for fitting the above equations to experimental data have been discussed in the literature. The direct nonlinear regression exhibits several advantages over indirect... 

In agreement with previous studies [3, 4], motor power is shown to be linear in both Q and N. Table 1 shows the steady state model parameters Ci, C2 and C3 for this and subsequent set of data obtained using linear data regression analysis method. A different way to describe this model is that the differential specific energies, 9P/9Q and 9P/5N remain constant for all Q/N. 

Another problem is to determine the optimal number of descriptors for the objects (patterns), such as for the structure of the molecule. A widespread observation is that one has to keep the number of descriptors as low as 20 % of the number of the objects in the dataset. However, this is correct only in case of ordinary Multilinear Regression Analysis. Some more advanced methods, such as Projection of Latent Structures (or. Partial Least Squares, PLS), use so-called latent variables to achieve both modeling and predictions. 

To gain insight into chemometric methods such as correlation analysis, Multiple Linear Regression Analysis, Principal Component Analysis, Principal Component Regression, and Partial Least Squares regression/Projection to Latent Structures... 

Multiple linear regression analysis is a widely used method, in this case assuming that a linear relationship exists between solubility and the 18 input variables. The multilinear regression analy.si.s was performed by the SPSS program [30]. The training set was used to build a model, and the test set was used for the prediction of solubility. The MLRA model provided, for the training set, a correlation coefficient r = 0.92 and a standard deviation of, s = 0,78, and for the test set, r = 0.94 and s = 0.68. 

Two additional methods for determining the composition of a mixture deserve mention. In multiwavelength linear regression analysis (MLRA) the absorbance of a mixture is compared with that of standard solutions at several wavelengths. If Asx and Asy are the absorbances of standard solutions of components X and Y at any wavelength, then... 

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. 

W. Mendenhall, Introduction to EinearMode/s and the Design andAna/ysis of Experiments, Duxbury Press, Belmont, Calif., 1968. This book provides an introduction to basic concepts and the most popular experimental designs without going into extensive detail. In contrast to most other books, the emphasis in the development of many of the underlying models and analysis methods is on a regression, rather than an analysis-of-variance, viewpoint. 

These were developed using constrained regression analysis or other suitable methods such that the following objective function is minimized. 

There have been also found the quantitative characteristics of the methods. They are as follows for HPLC method the linearity is 0.1 ng to 2 ng the detecting limit is 0.1 ng the limit of the quantitative estimation makes up 0.0004 mg/kg a coefficient of variation is 2.74% for the chromatodensitometry method the linearity is 2 ng to 10 ng the detecting limit is 0.6 ng the coefficient of variation is 2.37%. The data obtained have been treated using a regressive analysis. 

Multiple regression analysis can be executed by various programs. The one shown in the Appendix is from Mathcad 6 Plus, the regress method. Taking the log of the rates first and averaging later gives somewhat different result. 

If the rate law depends on the concentration of more than one component, and it is not possible to use the method of one component being in excess, a linearized least squares method can be used. The purpose of regression analysis is to determine a functional relationship between the dependent variable (e.g., the reaction rate) and the various independent variables (e.g., the concentrations). 

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. 

Miller first used Eq. (7-41) to correlate multiple variations, and this approach has more recently been subjected to considerable development. Many cross-interaction constants have been evaluated multiple regression analysis is one technique, but Miller and Dubois et ah discuss other methods. Lee et al. consider Pxy to be a measure of the distance between groups x and y in the transition state... 

We now consider a type of analysis in which the data (which may consist of solvent properties or of solvent effects on rates, equilibria, and spectra) again are expressed as a linear combination of products as in Eq. (8-81), but now the statistical treatment yields estimates of both a, and jc,. This method is called principal component analysis or factor analysis. A key difference between multiple linear regression analysis and principal component analysis (in the chemical setting) is that regression analysis adopts chemical models a priori, whereas in factor analysis the chemical significance of the factors emerges (if desired) as a result of the analysis. We will not explore the statistical procedure, but will cite some results. We have already encountered examples in Section 8.2 on the classification of solvents and in the present section in the form of the Swain et al. treatment leading to Eq. (8-74). 

FIGURE 3.3 The enthalpy change, ATT, for a reaction can be determined from the slope of a plot of R In versus l/T. To illns-trate the method, the values of the data points on either side of the 327.5 K (54.5 C) data point have been nsed to calculate ATT at 54.5 C. Regression analysis would normally be preferable. (Adapted from Brandts, ]. F., 1964. Tim thermo-dynamics of protein denatnration. I. The denatnration of ehy-motrypsinogen. om Q.7A of the American Chemical Society m 429 -430L)... 

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