Regression definition

Regression analyses revealed systematic differences between experimental log P and log P calculations based on the summation of fragment values. These differences could be attributed to chemical characteristics of the molecules, which in turn allowed the definition of correction rules such as chain conjugation, electronegativity facing bulk or the proximity effect, which describes the presence of electronegative centers in a molecule separated by one or two carbons. Correction values needed for log P calculation were shown to represent multiples of a constant value of 0.289, which is known as the magic constant (CM). 

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. 

The method of PCA can be used in QSAR as a preliminary step to Hansch analysis in order to determine the relevant parameters that must be entered into the equation. Principal components are by definition uncorrelated and, hence, do not pose the problem of multicollinearity. Instead of defining a Hansch model in terms of the original physicochemical parameters, it is often more appropriate to use principal components regression (PCR) which has been discussed in Section 35.6. An alternative approach is by means of partial least squares (PLS) regression, which will be more amply covered below (Section 37.4). 

EWLS regression is by statistical definition the optimum method, and has the added benefits that (a) uncertainties of the slope and intercept of the regression are easily estimated, and (b) the probability that the data support assumption (1) above can be readily calculated. 

It should be noted that the above definition of Xj is different from the one often found in linear regression books. There X is defined for the simple or multiple linear regression model and it contains all the measurements. In our case, index i explicitly denotes the i"1 measurement and we do not group our measurements. Matrix X, represents the values of the independent variables from the i,h experiment. 

PCR is based on a PCA input data transformation that by definition is independent of the Y-data set. The approach to defining the X-Y relationship is therefore accomplished in two steps. The first is to perform PCA on the. Y-data, yielding a set of scores for each measurement vector. That is, if xk is the fcth vector of d measurements at a time k, then zk is the corresponding kth vector of scores. The score matrix Z is then regressed onto the Y data, generating the predictive model... 

This definition is convenient because it allows us to then jump directly to what is arguably the simplest Chemometric technique in use, and consider that as the prototype for all chemometric methods that technique is multiple regression analysis. Written out in matrix notation, multiple regression analysis takes the form of a relatively simple matrix equation ... 

Tom the appropriate regression equations in Charton s review109. cBy definition, see text. 

All regression methods aim at the minimization of residuals, for instance minimization of the sum of the squared residuals. It is essential to focus on minimal prediction errors for new cases—the test set—but not (only) for the calibration set from which the model has been created. It is relatively easy to create a model— especially with many variables and eventually nonlinear features—that very well fits the calibration data however, it may be useless for new cases. This effect of overfitting is a crucial topic in model creation. Definition of appropriate criteria for the performance of regression models is not trivial. About a dozen different criteria— sometimes under different names—are used in chemometrics, and some others are waiting in the statistical literature for being detected by chemometricians a basic treatment of the criteria and the methods how to estimate them is given in Section 4.2. 

Multilinear Regression Analysis. As an entry to the problem we have selected simple gas phase reactions involving proton or hydride ion transfer which are influenced by only a few effects and for which reactivity data of high accuracy are available. In these situations where a larger set of numerial data are available multilinear regression analysis (MLRA) was applied. Thus, the simplest mathematical form, a linear equation is chosen to describe the relationship between reactivity data and physicochemical factor. The number of parameters (factors) simultaneously applied was always kept to a minimum, and a particular parameter was only included in a MLRA study if a definite indication of its relevance existed. 

Competitive, 249, 123, 146, 190 [partial, 249, 124 progress curve equations for, 249, 176, 180 for three-substrate systems, 249, 133, 136] competitive-uncompetitive, 249, 138 concave-up hyperbolic, 249, 143 dead-end, 249, 124 [for bireactant kinetic mechanism determination, 249, 130-133 definition of kinetic constants, 249, 220-221 effects on enzyme progress curves, nonlinear regression analysis, 249, 71-72 inhibition constant evaluation, 249, 134-135 kinetic analysis with, 249, 123-143 one-substrate systems, 249, 124-126 unireactant systems, theory,... 

MeasurementCs), redundant Regression, see Calibration Residuals concentration definition of, 7 interpretatiun of, 198-200 definition of, 7, 90... 

This pH definition for non-aqueous and mixed solvent systems is practically the same as that for aqueous solutions (Section 6.2.1). Thus, if a pH standard is available for the solvent or mixed solvent under study, the glass electrode is calibrated with it and then the pH of the sample solution is measured. The pHRVs values for 0.05 mol kg-1 KHPh have been assigned to aqueous mixtures of eight organic solvents (see 5 for pHRVs at 25 °C). Although they are for discrete solvent compositions, the pHRVs in between those compositions can be obtained by use of a multilinear regression equation [14b],... 

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