Multiple linear regression applications

Most of the 2D QSAR methods are based on graph theoretic indices, which have been extensively studied by Randic [29] and Kier and Hall [30,31]. Although these structural indices represent different aspects of molecular structures, their physicochemical meaning is unclear. Successful applications of these topological indices combined with multiple linear regression (MLR) analysis are summarized in Ref. 31. On the other hand, parameters derived from various experiments through chemometric methods have also been used in the study of peptide QSAR, where partial least square (PLS) [32] analysis has been employed [33]. 

Classical least-squares (CLS), sometimes known as K-matrix calibration, is so called because, originally, it involved the application of multiple linear regression (MLR) to the classical expression of the Beer-Lambert Law of spectroscopy ... 

Aqueous solubility is selected to demonstrate the E-state application in QSPR studies. Huuskonen et al. modeled the aqueous solubihty of 734 diverse organic compounds with multiple linear regression (MLR) and artificial neural network (ANN) approaches [27]. The set of structural descriptors comprised 31 E-state atomic indices, and three indicator variables for pyridine, ahphatic hydrocarbons and aromatic hydrocarbons, respectively. The dataset of734 chemicals was divided into a training set ( =675), a vahdation set (n=38) and a test set (n=21). A comparison of the MLR results (training, r =0.94, s=0.58 vahdation r =0.84, s=0.67 test, r =0.80, s=0.87) and the ANN results (training, r =0.96, s=0.51 vahdation r =0.85, s=0.62 tesL r =0.84, s=0.75) indicates a smah improvement for the neural network model with five hidden neurons. These QSPR models may be used for a fast and rehable computahon of the aqueous solubihty for diverse orgarhc compounds. 

A possible relationship between DFR and the application rate, as well as the crop volume estimate (CrV), was investigated using a multiple linear regression model (ADFR = a + b AR + c CrV). No significant contribution of crop volume to the variation of ADFR was observed (p = 0.19 and p = 0.87 for high-volume applications and all applications, respectively). 

Since most quantitative applications are on mixtures of materials, complex mathematical treatments have been developed. The most common programs are Multiple Linear Regression (MLR), Partial Least Squares (PLS), and Principal Component Analyses (PCA). While these are described in detail in another chapter, they will be described briefly here. 

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,... 

A.J. O Neil, R.D. lee and A.C. Moffat, The application of multiple linear regression to the measurement of the median particle size of drugs and pharmaceutical excipients by near-infrared spectroscopy. Analyst, 123, 2297-2302 (1998). 

Four types of emissions related to distance from a sampling site are compiled using standard emissions factors where applicable. Pace investigated many other variables in multiple linear regressions with TSP but found these to be the significant ones. 

In the chapter, we report a successful application of the Free-Wilson (26-30) methodology to model structure-activity/selectivity relationships. The Fujita-Ban (31-34) modification of Free-Wilson coupled with multiple linear regression... 

Only a small number of applications combining multiple linear regression and atomic spectrometry have been published. Most of them have been covered in two reviews [23,24], which readers are strongly encouraged to consult. Since most applications of MLR models have been studied together with the use of PLS models, the practical examples will be referred to in Chapter 4, in order to avoid unnecessary repetitions here. 

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 ... 

The simplest form of regression is multiple linear regression (MLR), Y = XB + E. Here, X contains the descriptors, [di... dn] B contains the regression coefficients, Y contains the figures of merit and E contains the residuals. One well known example of MLR is the relationship shown in Equation (6.9). This model requires a few well-characterized parameters d. .. dn, which are usually derived from experimental measurements or from QM calculations. There are several applications of MLR in catalysis, eg., the quantitative analysis of ligand effects (QALE) model developed by Fernandez et al. [90]. 

There is a fairly confusing literature on the use of multiple linear regression for calibration in chemometrics, primarily because many workers present their arguments in a very formalised manner. However, the choice and applicability of method depends on three main factors ... 

For inttoductory purposes multiple linear regression (MLR) is used to relate the experimental response to the conditions, as is common to most texts in this area, but it is important to realise that odter regression methods such as partial least squares (PLS) are applicable in many cases, as discussed in Chapter 5. Certain designs, such as dtose of Section 2.3.4, have direct relevance to multivariate calibration. In some cases multivariate methods such as PLS can be modified by inclusion of squared and interaction terms as described below for MLR. It is important to remember, however, diat in many areas of chemistry a lot of information is available about a dataset, and conceptually simple approaches based on MLR are often adequate. 

Chemometrics is the discipline concerned with the application of statistical and mathematical methods to chemical data [2.18], Multiple linear regression, partial least squares regression and the analysis of the main components are the methods that can be used to design or select optimal measurement procedures and experiments, or to provide maximum relevant chemical information from chemical data analysis. Common areas addressed by chemometrics include multivariate calibration, visualisation of data and pattern recognition. Biometrics is concerned with the application of statistical and mathematical methods to biological or biochemical data. 

Numerous authors have devised multiple linear regression approaches to the correlation of solvent effects, the intent being to widen the applicability of the correlation and to develop insight into the molecular factors controlling the correlated process.For example, Zilian treated polarity as a combination of effects measured by molar refraction, AN, and DN. Koppel and Palm write... 

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. 

---