Linear regression analysis, multiple

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... 

Step S Building a Multiple Linear Regression Analysis (MLRA) Model... 

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. 

In the above paragraphs we saw that multiple linear regression analysis on equations of the form... 

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). 

Walters [24] examined the effect of chloride on the use of bromide and iodide solid state membrane electrodes, and he calculated selectivity constants. Multiple linear regression analysis was used to determine the concentrations of bromide, fluorine, and iodide in geothermal brines, and indicated high interferences at high salt concentrations. The standard curve method was preferred to the multiple standard addition method because of ... 

Once suitable parameters are available the values of g can be correlated with them by means of either simple linear regression analysis if the model requires only a single variable, or multiple linear regression analysis if it requires two or more variables. Such a correlation results in a SPQR. In this work we consider only those parameters that are defined directly or indirectly from suitable reference sets or, in the case of steric parameters, calculated from molecular geometries. 

Simple and valence indices up to sixth order were computed for all the PAHs used in the present study database. The program MOLCONN2 [133, 152,154, 156] performed these calculations using the chemical structural formula as input. SAS [425] was used on a mainframe computer to perform statistical analyses. First, indices were selected which explained the greatest amount of variance in the data (i.e., R2 procedure). These indices were then used in a multiple linear regression analysis (REG procedure). 

Pihlaja and Rossi [83ACSA(B)289] prepared l,3-dioxan-2-one and all of its methyl derivatives, recorded their C NMR spectra, and derived the methyl substituent shift parameters by a multiple linear regression analysis of the anancomeric and two equivalent chair conformers (Table X). With these values, the authors estimated the conformational equilibria for two unequally populated chair conformations (Nos. 2, 3, 9, 11, and 14 in Table X). A consistent picture of the predominance of the chair conformation and the corresponding chair chair equilibria in l,3-dioxan-2-ones was obtained in complete agreement with earlier H NMR results. 

Regression equation The equation obtained by the correlation of a data set with a correlation equation by means of simple or multiple linear regression analysis. 

Another QSAR study utilizing 14 flavonoid derivatives in the training set and 5 flavonoid derivatives in the test set was performed by Moon et al. (211) using both multiple linear regression analysis and neural networks. Both statistical methods identified that the Hammett constant a, the HOMO energy, the non-overlap steric volume, the partial charge of C3 carbon atom, and the HOMO -coefficient of C3, C3, and C4 carbon atoms of flavonoids play an important role in inhibitory activity (Eqs. 3-5, Table 5). 

The effects of both alkyl and aryl substituents can be observed in the two-component tautomeric equilibria of 3-alkyl-l-aryl-2,3-dihydro-177-naphth[l,2-r ][l,3]oxazines containing C-3-epimeric naphthoxazines 52B-58B and 52G-58C (Scheme 7). The influence of the Meyer parameters (V ) of the alkyl substituents on the epimerization constants (K d ( r= [B]/[G]) can be characterized by Equation (3). Multiple linear regression analysis of log A)r according to Equation (4) leads to the conclusion that these equilibria are also influenced significantly by the inductive effect of substituent Y 0.48) <2004JOC3645>. 

There are a number of different types of source apportionment models, including the chemical mass balance method, factor analysis, multiple linear regression analysis, and Lagrangian modeling. The chemical... 

Our objective in this work is to present surveys of the methods now available for the quantitative treatment of steric effects in the design of bioactive molecules. Commonly, this consists in the modification of a lead compound by structural changes which result in a set of bioactive substances. The bioactivity is determined and then related to structure. This is generally carried out by means of multiple linear regression analysis using a correlation equation of the type... 

Table V. Multiple linear regression analysis of foam capacity...

Figure 6. Experimentally observed and mathematically simulated regression lines of foam capacity of different percentages of glandless cottonseed flour in suspensions at various pH values. Experimental 4%, 10%, and 16% suspensions were run at pH 3.5, 6.5, and 9.5 to test the reliability of the multiple linear regression analysis. Quantitative data used in this analysis are in Figures 2 and 4.

Table 3 (73) compares the retention coefficients for synthetic peptides from various sources. To ensure comparability, the data has been standardized with respect to lysine and assigned a value of 100. The table shows that there are discrepancies between the results obtained using different chromatographic systems. Predictions of retention times should therefore be made using chromatographic systems similar to those used to calculate the retention coefficients for the amino acids. Casal et al. (75a) have made a comparative study of the prediction of the retention behavior of small peptides in several columns by using partial least squares and multiple linear regression analysis. 

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). 

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