Statistical methods multiple regression analysis

For statistical reasons, multiple regression analysis cannot be used for 3D-QSAR methods that consider many more 3D descriptors than compounds or for which the descriptors are mutually correlated. The alternative strategies described next can be used to find a quantitative model in such situations. As will be seen, cross-validation is an important technique for assessing the robustness of a proposed model. 

Bruntz et al. applied multiple regression analysis and found that the method of least squares yielded a set of coefficients that produced a 0.84 correlation of ozone concentration with the data. Adding mixing height to the correlation yielded no statistically significant improvement in agreement with the assertions of Hanna. ... 

Optimization techniques may be classified as parametric statistical methods and nonparametric search methods. Parametric statistical methods, usually employed for optimization, are full factorial designs, half factorial designs, simplex designs, and Lagrangian multiple regression analysis [21]. Parametric methods are best suited for formula optimization in the early stages of product development. Constraint analysis, described previously, is used to simplify the testing protocol and the analysis of experimental results. 

Because the British group applied extensively their statistical method to determine causation of large interindividual pharmacokinetic variations without describing its strengths and weaknesses, others have attempted to assess critically the application of multiple regression analysis for this particular purpose (31,32) While this statistical method has great potential, it requires considerable modification beyond its initial applications in this area (27-29), if that potential is to be realized (H,32). Thus far, its applications in pharmacokinetics (27-29) have been disappointing because those who have employed it neither formulated nor addressed, much less demonstrated fulfillment of, several fundamental assumptions inherent in its use ( 1, 32). 

The most important statistical parameters r, s, and F and the 95% confidence intervals of the regression coefficients are calculated by Eqs. (20) to (23) (for details on Eqs. (20) to (23), see Refs. 39 to 42). For more details on linear (multiple) regression analysis and the calculation of different statistical parameters, as well as other validation techniques (e.g., the jackknife method and bootstrapping), see Refs. 33,39 12 ... 

One of the simplest methods of idraitifying outlie from a multiple regression analysis is the use of the standard residual. This provides a measure of how well or poorly a compound is predicted by that model. Again, it should be stressed that this is a simple statistic and is intonal to the model that is, the prediction is made for the compound already in the model. More severe methods for the identification of outliers are available (Eriksson et al., 2(XB Tropsha et al., 2003). The sign on the standard residual also indicates if a compound is over- or underpredicted by a model. Examination of the standard residuals (which are provided automatically by most... 

Part V will cover several techniques for working on prevention that apply multiple factor models. Multiple factor models may use quantitative or qualitative analysis. Statistical techniques, such as factor analysis, multiple regression analysis and other multivariate methods may be useful. Fault tree analysis, failure mode and effects analysis and other approaches help identify characteristics that together can lead to undesired events. 

Under circiim.stances in which the molecular descriptors are highly intercorrelated (e.g., molecular connectivity indices), there are statistical limitations with respect to the use of a classical multiple regression analysis. Such data sets can be satisfactorily treated by the application of principal components regression (PCR) and partial least squares (PLS) methods.Numerous environmental QSAR model.s use... 

Linear or nonlinear multiple regression analysis is used as a statistical tool to derive quantitative models, to check the significance of these models and of each individual term in the regression equation. Other statistical methods, such as discriminant analysis, principal component analysis (PCA), or partial least squares (PLS) analysis (see Partial Least Squares Projections to Latent Structures (PLS) in Chemistry) are alternatives to regression analysis (see Che mo me tries Multivariate View on Chemical Problems)Newer approaches compare the similarity of molecules with respect to different physicochemical or other properties with their biological activities. 

Multiple linear regression is strictly a parametric supervised learning technique. A parametric technique is one which assumes that the variables conform to some distribution (often the Gaussian distribution) the properties of the distribution are assumed in the underlying statistical method. A non-parametric technique does not rely upon the assumption of any particular distribution. A supervised learning method is one which uses information about the dependent variable to derive the model. An unsupervised learning method does not. Thus cluster analysis, principal components analysis and factor analysis are all examples of unsupervised learning techniques. 

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). 

L. Elden, Partial least-squares vs. Lanczos Bidiagonalization - I Analysis of a projection method for multiple regression. Computational Statistics Data Analysis, 46, 11-31, (2004). 

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). 

Statistical methods. Certainly one of the most important considerations in QSAR is the statistical analysis of the correlation of the observed biological activity with structural parameters - either the extrathermodynamic (Hansch) or the indicator variables (Free-Wilson). The coefficients of the structural parameters that establish the correlation with the biological activity can be obtained by a regression analysis. Since the models are constructed in terms of multiple additive contributions the method of solution is also called multiple linear regression analysis. This method is based on three requirements (223) i) the independent variables (structural parameters) are fixed variates and the dependent variable (biological activity) is randomly produced, ii) the dependent variable is normally and independently distributed for any set of independent variables, and iii) the variance of the dependent variable must be the same for any set of independent variables. 

The ultimate development in the field of sample preparation is to eliminate it completely, that is, to make a chemical measurement directly without any sample pretreatment. This has been achieved with the application of chemometric near-infrared methods to direct analysis of pharmaceutical tablets and other pharmaceutical solids (74-77). Chemometrics is the use of mathematical and statistical correlation techniques to process instrumental data. Using these techniques, relatively raw analytical data can be converted to specific quantitative information. These methods have been most often used to treat near-infrared (NIR) data, but they can be applied to any instrumental measurement. Multiple linear regression or principal-component analysis is applied to direct absorbance spectra or to the mathematical derivatives of the spectra to define a calibration curve. These methods are considered secondary methods and must be calibrated using data from a primary method such as HPLC, and the calibration material must be manufactured using an equivalent process to the subject test material. However, once the calibration is done, it does not need to be repeated before each analysis. 

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