Multivariate regression analysis

The total residual sum of squares, taken over all elements of E, achieves its minimum when each column Cj separately has minimum sum of squares. The latter occurs if each (univariate) column of Y is fitted by X in the least-squares way. Consequently, the least-squares minimization of E is obtained if each separate dependent variable is fitted by multiple regression on X. In other words the multivariate regression analysis is essentially identical to a set of univariate regressions. Thus, from a methodological point of view nothing new is added and we may refer to Chapter 10 for a more thorough discussion of theory and application of multiple regression. 

Beilken et al. [ 12] have applied a number of instrumental measuring methods to assess the mechanical strength of 12 different meat patties. In all, 20 different physical/chemical properties were measured. The products were tasted twice by 12 panellists divided over 4 sessions in which 6 products were evaluated for 9 textural attributes (rubberiness, chewiness, juiciness, etc.). Beilken etal. [12] subjected the two sets of data, viz. the instrumental data and the sensory data, to separate principal component analyses. The relation between the two data sets, mechanical measurements versus sensory attributes, was studied by their intercorrelations. Although useful information can be derived from such bivariate indicators, a truly multivariate regression analysis may give a simpler overall picture of the relation. 

The above study was replicated later with 75 asymptomatic black children, 3-7 years old, of uniformly low socioeconomic status (Hawk et al. 1986 Schroeder and Hawk 1987). Backward stepwise multivariate regression analysis revealed a highly significant negative linear relationship between Stanford-Binet IQ scores and contemporary PbB levels over the entire range of 6-47 pg/dL (mean,... 

Multiple linear regression (MLR) is a classic mathematical multivariate regression analysis technique [39] that has been applied to quantitative structure-property relationship (QSPR) modeling. However, when using MLR there are some aspects, with respect to statistical issues, that the researcher must be aware of ... 

The separation of synthetic red pigments has been optimized for HPTLC separation. The structures of the pigments are listed in Table 3.1. Separations were carried out on silica HPTLC plates in presaturated chambers. Three initial mobile-phase systems were applied for the optimization A = n-butanol-formic acid (100+1) B = ethyl acetate C = THF-water (9+1). The optimal ratios of mobile phases were 5.0 A, 5.0 B and 9.0 for the prisma model and 5.0 A, 7.2 B and 10.3 C for the simplex model. The parameters of equations describing the linear and nonlinear dependence of the retention on the composition of the mobile phase are compiled in Table 3.2. It was concluded from the results that both the prisma model and the simplex method are suitable for the optimization of the separation of these red pigments. Multivariate regression analysis indicated that the components of the mobile phase interact with each other [79],... 

Experimental data with the independent variables L and I can be plotted, producing a 3-dimensional binding isotherm, and a multivariate regression analysis gives the association constants KSL and KSI as well as the mobility of the corresponding complexes. 

The a and n constants of substituents are often useful when correlated to biological activity in the statistical procedure known as multivariate regression analysis. As is well known from pharmacological testing of various drug series, such correlations can be either linear or parabolic. The linear relationship is described by the equation... 

The multivariate regression analysis let to build up a model capable of describing the relationship between the analytical responses of the... 

As it turns out, methods using quantum chemical parameters alone have been applied successfully only to very restricted structural types (Reddy, 1996). When applied to a larger variety of structures, non-quantum chemical parameters, such as an indicator for alkanes and a simple count of hetero atoms, are required (Bodor, 1992), and multivariate regression analysis of a database of measured values determines the contribution of each parameter. This certainly undermines any claim to a superior theoretical basis for quantum calculations. 

Potyrailo and May used the acoustic-wave sensor to quantify cresol and benzo-quinone [21]. They tested sensors for the quantification of cresol and benzoqui-none in mixtures with multivariate regression analysis tools. Only 2 mL of solution was used for analysis, which included cresol and/or benzoquinone on the order of pg. After the sensor had been calibrated with a standard library of pure cresol or benzoquinone solution, 19 model libraries, including mixtures of cresol and benzoquinone, were analyzed (Fig. 8.9). A linear correlation occurs between actual... 

Ulberth, F. (1995) Quantitation of foreign fat in foreign fat/milkfat mixtures by multivariate regression analysis of fatty acid data. J. Agr. Food Chem., 43, 1556-1560. 

Molina, E., Martfn-Alvarez R J., Ramos M. (1999). Analysis of cows , ewes and goats milk mixtures by capillary electrophoresis Quantification by multivariate regression analysis. Int. 

Kinetic parameters for the model with a single step first order reaction have also been calculated by a multivariate regression analysis of the DTG results for all heating rates. The best fit kinetic parameters are logic (Ws ") = 14.75 and = 196.7 kJ/mol demonstrating that the result for the kinetic parameters is not influenced by choosing either the TGA or the DTG curves for the multivariate regression. 

In general BC(DEF) parameters describe molecular properties related to nonspecific intermolecular interactions in the liquid state and could therefore be useful in predicting biological activity or physico-chemical properties depending on such nonspecific interactions 29 linear models were calculated by multivariate regression analysis that correlates BC(DEF) parameters to 29 different physico-chemical properties. 

Due to the large number of descriptors (commonly 15,000 - 20,000 for each field), the multivariate regression analysis is usually performed by partial least squares regression (PLS), with or without - variable selection. Moreover, a similarity matrix can be calculated from distance functions based on interaction fields between pairs of molecules. 

A set of 120 atom types was proposed. The corresponding hydrophobic constants were evaluated by multivariate regression analysis using a training set of 893 compounds, = 0.856 and s = 0.496. This approach is actually called the ALOGP method [Viswanadhan et al, 1993], and a similar approach was also proposed for - molar refractivity. As in the Broto-Moreau-Vandycke approach, correction factors are avoided, while hydrogen atoms are considered instead. 

Not all relationships can be adequately described using the simple linear model, however, and more complex functions, such as quadratic and higher-order polynomial equations, may be required to fit the experimental data. Finally, more than one variable may be measured. For example, multiwavelength calibration procedures are finding increasing applications in analytical spectrometry and multivariate regression analysis forms the basis for many chemometric methods reported in the literature. 

Multivariate regression analysis plays an important role in modem process control analysis, particularly for quantitative UV-visible absorption spectrometry and near-IR reflectance analysis. It is conunon practice with these techniques to monitor absorbance, or reflectance, at several wavelengths and relate these individual measures to the concentration of some analyte. The results from a simple two-wavelength experiment serve to illustrate the details of multivariate regression and its application to multivariate calibration procedures. 

Of course, the reason for the improvement in the calibration model when the second term is included is that A21 serves to compensate for the absorbance due to the tyrosine since X21 is in the spectral region of a tyrosine absorption band with little interference from tryptophan. Figure 6. In general, the selection of variables for multivariate regression analysis may not be so obvious. 

Multiple linear regression is a classic mathematical multivariate regression analysis... 

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