Statistical models multilinear regression

More than just a few parameters have to be considered when modelling chemical reactivity in a broader perspective than for the well-defined but restricted reaction sets of the preceding section. Here, however, not enough statistically well-balanced, quantitative, experimental data are available to allow multilinear regression analysis (MLRA). An additional complicating factor derives from comparison of various reactions, where data of quite different types are encountered. For example, how can product distributions for electrophilic aromatic substitutions be compared with acidity constants of aliphatic carboxylic acids And on the side of the parameters how can the influence on chemical reactivity of both bond dissociation energies and bond polarities be simultaneously handled when only limited data are available ... 

The surface tensions themselves in the GB/SA and MST-ST models were developed by taking collections of experimental data for the free energy of solvation in a specific solvent, removing the electrostatic component as calculated by the GB or MST model, and fitting the surface tensions to best reproduce the residual free energy given the known SASA of the solute atoms. Such a multilinear regression procedure requires a reasonably sized collection of data to be statistically robust, and limitations in data have thus restricted these models to water, carbon tetrachloride, chloroform, and octanol as solvents. 

It is clear that for an unsymmetrical data matrix that contains more variables (the field descriptors at each point of the grid for each probe used for calculation) than observables (the biological activity values), classical correlation analysis as multilinear regression analysis would fail. All 3D QSAR methods benefit from the development of PLS analysis, a statistical technique that aims to find the multidimensional direction in the X space that explains the maximum multidimensional variance direction in the F space. PLS is related to principal component analysis (PCA)." ° However, instead of finding the hyperplanes of maximum variance, it finds a linear model describing some predicted variables in terms of other observable variables and therefore can be used directly for prediction. Complexity reduction and data... 

In determining a mathematical model, whether by linear combinations or by multilinear regression, we have assumed the standard deviation of random experimental error to be (approximately) constant (homoscedastic) over the experimental region. Mathematical models were fitted to the data and their statistical significance or that of their coefficients was calculated on the basis of this constant experimental variance. Now the standard deviation is often approximately constant. All experiments may then be assumed equally reliable and so their usefulness depends solely on their positions within the domain. 

Direct information on structural regions dominated by different Q" species in Alkaline/alkaline earth silicate glasses have been obtained from linear and multilinear regressions. The statistical models achieved an accuracy in prediction of about 2 ppm for the Si 5iso [87], 10 ppm for the Na 5iso [91], of 2°-4 for the mean value of the Si-O-Si bond angle distribution, 2°-4° [87], and of less than 10 ppm for the chemical shift anisotropy Si Acs of the species [91]. 

The field of nonlinear regression, of which multilinear models are one part, is an area of statistics under rapid development, the results of which are likely to improve the utility of multilinear methods in the future. 

---