Multiple linear regression coefficients

Table 9 Multiple Linear Regression Coefficients for Standard Molar Gibbs Energies of Transfer for Alkali Metal Cations Based on Solvent Properties t, DN, 1/e, and...

Table 10 summarizes the multiple linear regression coefficients obtained from the analysis by reference to Eqs. (30) and (31) [138. The signs... 

The partition coefficients of the probe compounds were determined in solutions with different concentrations of sodium lauryl sulfate (SLS) surfactant. For example, the partition coefficients of the probe compoimds in PDMS/10% SLS (P/LIO) system are given in Table 5.1. The log /Tp/uo values of the probe compounds and their solute descriptors formed a new LFER equation matrix. The system coefficients of the P/LIO system were obtained by multiple linear regression analysis of the LFER equation matrix. The system coefficients of the P/LIO system were [0.82, 0.25, -1.16, -1.93, -1.61, 0.86]p/Lio with a multiple linear regression coefficient of 0.89. When the P/W system was used as the reference system, the changes in system... 

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. 

Experimental polymer rheology data obtained in a capillary rheometer at different temperatures is used to determine the unknown coefficients in Equations 11 - 12. Multiple linear regression is used for parameter estimation. The values of these coefficients for three different polymers is shown in Table I. The polymer rheology is shown in Figures 2 - 4. 

The study is based on four iinear hydrocarbons (in Ci, Ce to Ca) and the model uses Antoine and Clapeyron s equations. The flashpoints used by the author do not take into account all experimental values that are currently available the correlation coefficients obtained during multiple linear regression adjustments between experimental and estimated values are very bad (0.90 to 0.98 see the huge errors obtained from a correlation study concerning flashpoints for which the present writer still has a coefficient of 0.9966). The modei can be used if differences between pure cmpounds are still low regarding boiling and flashpoints. 

Canonical Correlation Analysis (CCA) is perhaps the oldest truly multivariate method for studying the relation between two measurement tables X and Y [5]. It generalizes the concept of squared multiple correlation or coefficient of determination, R. In Chapter 10 on multiple linear regression we found that is a measure for the linear association between a univeiriate y and a multivariate X. This R tells how much of the variance of y is explained by X = y y/yV = IlylP/llylP. Now, we extend this notion to a set of response variables collected in the multivariate data set Y. 

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. 

Crystallinity aside, the two physical attributes of a drug that most control its skin permeability are its physical size and its lipophilicity [44,45]. When all extant human permeability coefficients (at the time over 90 compounds) were subjected to multiple linear regression by Potts and Guy using the following semi-empirical equation [45] ... 

FIGURE 4.24 PLS as a multiple linear regression method for prediction of a property y from variables xi,..., xm, applying regression coefficients b1,...,bm (mean-centered data). From a calibration set, the PLS model is created and applied to the calibration data and to test data. 

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). 

With the aid of multiple linear regression, model coefficients were calculated, which describe the effect of the variables pn the physical stability of the tablets. Since two levels of each variable were studied it was possible to calculate the linear contribution of the variables. The general form of the model which describes the effect of the variables is given by the following formula ... 

In practice, the values of coefficients A, B and C can be determined by conducting several experiments for the same solute at different flow rates. The method of multiple linear regression is then used to find the hyperbolic function that best matches the experimental values. 

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. 

The usefulness of the LSER approach hinges on the similarity of the partitioning coefficients obtained from the sensing experiments (Ks) and the gas chromatographic experiments (Kqc)- In other words, it is assumed that the relationship Ks Kgc holds. This is how LSER is used for evaluation of a new sensing material. First, the coefficient Kqc is obtained from the tabulated database or experimentally. Second, the multiple linear regression technique (see Chapter 10) is used to obtain the best fit for the sensor test data, and the individual coefficients in (2.3) are evaluated. This approach has been used successfully in evaluation of multiple materials for gas sensors (Abraham et al., 1995 Grate et al., 1996). 

The multiple linear regression (MLR) method was historically the first and, until now, the most popular method used for building QSPR models. In MLR, a property is represented as a weighted linear combination of descriptor values F=ATX, where F is a column vector of property to be predicted, X is a matrix of descriptor values, and A is a column vector of adjustable coefficients calculated as A = (XTX) XTY. The latter equation can be applied only if the matrix XTX can be inverted, which requires linear independence of the descriptors ( multicollinearity problem ). If this is not the case, special techniques (e.g., singular value decomposition (SVD)26) should be applied. 

The simplest form of regression is multiple linear regression (MLR), Y = XB + E. Here, X contains the descriptors, [di... dn] B contains the regression coefficients, Y contains the figures of merit and E contains the residuals. One well known example of MLR is the relationship shown in Equation (6.9). This model requires a few well-characterized parameters d. .. dn, which are usually derived from experimental measurements or from QM calculations. There are several applications of MLR in catalysis, eg., the quantitative analysis of ligand effects (QALE) model developed by Fernandez et al. [90]. 

---