Regression analysis, parameters from equations

As a result, interest began to develop in empirical relationships between characteristic x-ray wavelength emission and chemical composition. Sherman himself had discussed this [23] in 1954, the year before the publication of his theoretical equations, and his scheme involved the use of regression analysis to solve equations, the parameters of which were to be calculated with data from samples of known composition. The following equations are typical of this approach, which is generally attributed to Beattie and Brissey [24]. 

The reflectivity R = 0.5[ r + / p ], can be measured. R is independent of both A and 4 and thus provides a third variable. In order to obtain nf, kf and L, values of these parameters are estimated. R, A and T are then calculated from equations (2.84) to (2.92) and compared to the experimentally observed values. nt, kt and Lare altered and the calculations repeated. Regression analysis eventually yields values of the thickness and refractive index of the film that would give rise to the observed R, 4 and A. 

The solution of problems in chemical reactor design and kinetics often requires the use of computer software. In chemical kinetics, a typical objective is to determine kinetics rate parameters from a set of experimental data. In such a case, software capable of parameter estimation by regression analysis is extremely usefiil. In chemical reactor design, or in the analysis of reactor performance, solution of sets of algebraic or differential equations may be required. In some cases, these equations can be solved an-... 

The separation of synthetic red pigments has been optimized for HPTLC separation. The structures of the pigments are listed in Table 3.1. Separations were carried out on silica HPTLC plates in presaturated chambers. Three initial mobile-phase systems were applied for the optimization A = n-butanol-formic acid (100+1) B = ethyl acetate C = THF-water (9+1). The optimal ratios of mobile phases were 5.0 A, 5.0 B and 9.0 for the prisma model and 5.0 A, 7.2 B and 10.3 C for the simplex model. The parameters of equations describing the linear and nonlinear dependence of the retention on the composition of the mobile phase are compiled in Table 3.2. It was concluded from the results that both the prisma model and the simplex method are suitable for the optimization of the separation of these red pigments. Multivariate regression analysis indicated that the components of the mobile phase interact with each other [79],... 

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 ... 

Algebraic expressions for terms M and C were derived using Dewar s PMO method (for C in a version similar to the co-technique [57] in order to calculate carbocation stabilization energies). The size factor S is simply a cubic function of the number of carbon atoms [97], The three independent variables of the model were assumed to be linearly related to the experimental Iball indices (vide supra). By multilinear regression analysis (sample size = 26) an equation was derived for calculating Iball indices from the three theoretical parameters. The correlation coefficient for the linear relation between calculated and experimental Iball indices is r = 0.961. 

Hansch analysis Hansch analysis is a common quantitative structure-activity relationship approach in which a Hansch equation predicting biological activity is constructed. The equation arises from a multiple linear regression analysis of both observed biological activities and various molecular property parameters (Hammett, Hansch, and Taft parameters). 

From an inspection of the RSQUARE output, the five-variable equation with the highest correlation was selected for a more complete regression analysis. The five-variable equation (Equation 3) represents the best balance between high correlation and economy in the number of variable parameters. A serious disadvantage of having numerous independent variables in an empirical equation is the increased risk of a chance correlation (12). Consequently, the number of experimental observations required to establish statistical significance increases rapidly with the number of independent variables. In this study, l i experimental determinations were required to obtain statistical significance at the 95% confidence level. 

We chose 60 compounds with pl50 values ranging from 7.1 to 4.9 and subjected them to regression analysis using several physicochemical parameters (Table I). The 60 compounds contained variations in six positions of the basic structure. Quantitative structure-activity correlations with as many individual uncouplers in one equation have not yet been published. As far as we know, the Hansch approach has been applied to uncouplers of oxidative phosphorylation only twice first in 1965 by Hansch and co-workers to phenols and recently by Muraoka and Terada to N-phenylanthranilic acids. From Muraoka s data we recalculated the correlation with w and o- and obtained an equation which gave the best fit (last equation, Figure 3). 

Equation 8 is the preferred integrated form of the EE model. It can be readily used with linear regression analysis to fit the three parameters (a, E, or r) to experimental X vs. r data from an isothermal reactor (see for example, Table I and Figure 1). 

Some simple reaction kinetics are amenable to analytical solutions and graphical linearized analysis to calculate the kinetic parameters from rate data. More complex systems require numerical solution of nonlinear systems of differential and algebraic equations coupled with nonlinear parameter estimation or regression methods. 

From the theoretical point of view, on the basis of the complex physico-chemical analysis of these melts, considerations of their structure, i.e. the ionic composition was made. Using the multiple linear regression analysis, the equations describing molar volume and the surface tension on composition were obtained. From the individual interaction parameters, the formation of different structural entities in the melts was proposed. 

The model parameters are determined in the following manner the functional relations =/( — 8) and Rincorporated into B.C. of Equation (5.21), are obtained by the polynomial regression analysis of the electrode potential curves and the Kceii versus E curves determined from the Jin, versus AE plots, respectively. It should be borne in mind that (1 — 8) does not represent the average lithium content in the electrode, but the lithium content at the surface of the electrode. In other words, the electrode potential (t) in Equation (5.21) is the potential at the electrode surface. As the relationship —/(I — 8) includes information about the phase transition, we can consider the effect of phase transition on the theoretical CT with the functional relationship =fil — 8), without taking any of the intercalation isotherm. 

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