Multiple regression techniques, application

At this point. It seems useful to examine an example of the application of multiple regression techniques to analysis of experimental data for which results have already been obtained and published. 

To this point our discussions have largely focused on the application of matrices to linear problems associated with simultaneous equations, applications that commonly arise in least-square, multiple regression techniques. One further important function that occurs in multivariate analysis and the analysis of variance is the quadratic form. 

To test the applicability of statistical techniques for determination of the species contributions to the scattering coefficient, a one-year study was conducted in 1979 at China Lake, California. Filter samples of aerosol particles smaller than 2 ym aerodynamic diameter were analyzed for total fine mass, major chemical species, and the time average particle absorption coefficient, bg. At the same time and location, bgp was measured with a sensitive nephelometer. A total of 61 samples were analyzed. Multiple regression analysis was applied to the average particle scattering coefficient and mass concentrations for each filter sample to estimate aj and each species contribution to light scattering, bgn-j. Supplementary measurements of the chemical-size distribution were used for theoretical estimates of each b pj as a test of the effectiveness of the statistical approach. 

The main application for computers in statistical calculations has been in the field of correlation studies. The technique of multiple-regression analysis is now widely used to examine the effects of one or more independent variables on a dependent variable. This important statistical... 

It must be emphasized that the potential of multiple regression analysis to resolve sources of pharmacokinetic variations is much greater than has been realized by the particular canned1 model used previously. The technique itself is both sensitive and powerful. However, for multiple regression analysis to be used appropriately, a model must be developed that encompasses non-linear as well as linear relationships (JJ4). Error terms especially need to be appropriately modelled, rather than treated in a simply additive manner as in previous applications of this method. 

Application of equation 1 using the volumes in Table I to the polymer database (Appendix) results in a matrix of 87 equations and 35 unknowns. The solution was obtained by a least squares fit of the data using multiple linear regression techniques. 

Although the term theoretical techniques in relation to electronic effects may commonly be taken to refer to quantum-mechanical methods, it is appropriate also to mention the application of chemometric procedures to the analysis of large data matrices. This is in a way complementary to analysis through substituent constants based on taking certain systems as standards and applying simple or multiple linear regression. Chemometrics involves the analysis of suitable data matrices through elaborate statistical procedures,... 

Some of the earliest applications of chemometrics in PAC involved the use of an empirical variable selection technique commonly known as stepwise multiple linear regression (SMLR).8,26,27 As the name suggests, this is a technique in which the relevant variables are selected sequentially. This method works as follows ... 

These special cases of multiple linear regression analysis have been developed for the determination of the impact of individual molecular substructures (independent variables) on one dependent variable. Both techniques are similar yet, the Free-Wilson method considers the retention of the unsubstituted analyte as base, while Fujita-Ban analysis uses the less substituted molecule as reference. These procedures have not been frequently employed in chromatography only their application in QSRR studies in RP TLC and HPLC have been reported. 

NIR spectroscopy became much more useful when the principle of multiple-wavelength spectroscopy was combined with the deconvolution methods of factor and principal component analysis. In typical applications, partial least squares regression is used to model the relation between composition and the NIR spectra of an appropriately chosen series of calibration samples, and an optimal model is ultimately chosen by a procedure of cross-testing. The performance of the optimal model is then evaluated using the normal analytical performance parameters of accuracy, precision, and linearity. Since its inception, NIR spectroscopy has been viewed primarily as a technique of quantitative analysis and has found major use in the determination of water in many pharmaceutical materials. 

These techniques span the entire field from multiple linear regression (MLR)-type methods and various forms of neural network architectures to rule-based techniques of different kinds. These approaches also span from single models to multiple models, that is, consensus or ensemble modeling. Terms like machine learning and data or information fusion are also frequently encountered in this area of research, as well as the concepts of applicability domain and validation. 

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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