Hansch regression equation

Quantitative stmcture—activity relationships have been estabUshed using the Hansch multiparameter approach (14). For rat antigoiter activities (AG), the following (eq. 1) was found, where, as in statistical regression equations, n = number of compounds, r = regression coefficient, and s = standard deviation... 

A difficulty with Hansch analysis is to decide which parameters and functions of parameters to include in the regression equation. This problem of selection of predictor variables has been discussed in Section 10.3.3. Another problem is due to the high correlations between groups of physicochemical parameters. This is the multicollinearity problem which leads to large variances in the coefficients of the regression equations and, hence, to unreliable predictions (see Section 10.5). It can be remedied by means of multivariate techniques such as principal components regression and partial least squares regression, applications of which are discussed below. 

A recent re-examination of the 19F NMR data for me/a-subsl.ihil.cd fluorobenzenes in hydrocarbon solvents by Hansch, Leo and Taft92 led to a slightly different regression equation. When this was applied to the vinyl group, o/ was found to be 0.07, in fair agreement with the reactivity-based value of 0.10. 

As the equations show, linear correlations with the variables tr and a gave satisfactory results. This is certainly a simplification resulting from limited variance in the substituents. One would assume that square terms of the hydrophobic parameter are necessary in every correlation with biological activity not only to account for the random walk penetration process as in the original derivation of his equation by Hansch, but also, or even predominantly, as a description of the fact that numerous indifferent hydrophobic sites within the biological system compete with the site of action for the active molecule. In a first attempt we calculated regression equations for our hydrazones with the molecular parameter... 

An example of a Hansch analysis (see section III. B. 2.a) using MLR is a study on substituted tetrahydroisoquinolines with affinity for both phenylethanolamine iV-methyltrans-ferase (PNMT) and the a2-adreno-receptor " (see Figure 23.10). The multiple regression equations obtained were ... 

Quantitative structure-activity relationships QSAR. The QSAR approach pioneered by Hansch and co-workers relates biological data of congeneric structures to physical properties such as hydrophobicity, electronic, and steric effects using linear regression techniques to estimate the relative importance of each of those effects contributing to the biological effect. The molecular descriptors used can be 1-D or 3-D (3D-QSAR). A statistically sound QSAR regression equation can be used for lead optimization. 

The most comprehensive set of solvent regression equations is that given by Leo and Hansch [27] and repeated in the subsequent publication by Leo, Hansch, and Elkins (28). Their original equations were written with log KSw as the dependent variable but are restated here in the following form ... 

Examining 18 substituted 3-phenylthio-1,1,1-trlfluoro-2-propanones, regression equations were obtained between the inhibitory activities and the Hammett (ct), Taft (E ) steric and Hansch (ir) hydrophobicity constants (H). In the fiope of increasing the significance of these equations and to better distinguish between the Importance of various substituent positions, several new compounds of the related structure were synthesized, a much larger set of substituent parameters was applied, and instead of the arbitrary choice of these values, the variables were selected into the equations by a more sophisticated tool, linear stepwise regression analysis. 

The discipline of quantitative structure-activity relationships (QSAR), as we define it nowadays, was initiated by the pioneering work of Corwin Hansch on growthregulating phenoxyacetic acids. In 1962—1964 he laid the foundations of QSAR by three important contributions the combination of several physicochemical parameters in one regression equation, the definition of the lipophilicity parameter jt, and the formulation of the parabolic model for nonlinear lipophilicity-activity relationships. 

Yes, and the predictions are quite good. Two different methods emerge from a host of others and are most commonly used to predict octanol-water partition coefficients for the many organic compounds that exist. One approach is to calculate from a knowledge of structural constants, whereas the second approach requires that a chemical s partition coefficient be measured between a solvent other than octanol and water, Kf, . Kf, can then be calculated from linear regression equations that relate log (for a particular solvent) and log Kf, . Two forms of the structural constant approach are most popular (1) the Hansch n hydrophobic character of... 

Recalculating the published data on bio-depressants, Hansch and his colleagues found that the following regression equation made a good fit (the squared term ensured a parabolic relationship) ... 

Several Taft values can be used in one regression equation, as was done by Kutter and Hansch (1969) who studied the preventive effect of substituted benzoic acids on the combination of ovalbumin antigen with its antibody, each benzoic acid being incorporated as a hapten in the antigen. They obtained this multiple regression equation ... 

Another use of rho is to convert sigma values, derived from benzoic acid, for use in other nuclei. This was first achieved for naphthalene the results were then extended to quinoline, and then to other heterocycles [Perrin, 1965c Perrin, Dempsey and Serjeant, 1981 (their Section 7.2)]. Although rho continues to appear in formal statements of Hansch s multiple regression equation, it is usually only indirectly present, namely as a nucleus-modified sigma value. 

One may first consider the relation between log K and log S (aqueous solubility) since S is the major parameter that influences the partitioning of slightly soluble organic compounds. Aqueous solubility may also be thought of as a special form of partition coefficient in which the compound distributes between an ideal solvent (itself) and water. Hansch et aL (44) were the first to correlate octanol-water partition coefficients with aqueous solubilities for various types of low-molecular-weight organic liquids. Their results are shown in Table II. The regression equation between log and log S for a total of 156 compounds from various classes is... 

Let us consider a pair of regression equations to illustrate the qualitative nature of them. The selected equation, which represents the Hansch-type QSAR, is... 

Quantitative structure-activity relationships (QSAR), a concept introduced by Hansch and Fujita (1964) is a kind of formal system based on a kinetic model, which in turn is expressed in term of a first-order linear differential equation. Solution of the differential equation leads to a linear equation ( Hansch-Fujita equation ), the coefficients of which are determined by regression analysis resulting in a QSAR equation of a particular compound series. For a prediction, the dependent variable of the QSAR equation is calculated by algebraic operations. 

In (eco)toxicological QSAR studies, the molecular descriptor of choice is the n-octanol/water partition coefficient (log P), generally used in a simple regression equation. However, sometimes a simple linear regression model is inadequate to model properly the dependence of biological activity (BA) on logF. For example, fish exposed to very hydrophobic chemicals for a limited test duration have insufficient time to achieve a pseudo-steady state partitioning equilibrium between the toxicant concentration in aqueous circumambient phase and the hydrophobic site of action within the fish. Hansch initiated the use of a parabolic model in log P (equation 1) to overcome... 

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). 

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