Multiple linear regression prediction

Figure 10.16 Multiple linear regression prediction of olive oil variety from NMR data containing five varieties (at best, all but four predictions are correct with selection by...

Multiple linear regression analysis is a widely used method, in this case assuming that a linear relationship exists between solubility and the 18 input variables. The multilinear regression analy.si.s was performed by the SPSS program [30]. The training set was used to build a model, and the test set was used for the prediction of solubility. The MLRA model provided, for the training set, a correlation coefficient r = 0.92 and a standard deviation of, s = 0,78, and for the test set, r = 0.94 and s = 0.68. 

Using a multiple linear regression computer program, a set of substituent parameters was calculated for a number of the most commonly occurring groups. The calculated substituent effects allow a prediction of the chemical shifts of the exterior and central carbon atoms of the allene with standard deviations of l.Sand 2.3 ppm, respectively Although most compounds were measured as neat liquids, for a number of compounds duplicatel measurements were obtained in various solvents. 

We will explore the two major families of chemometric quantitative calibration techniques that are most commonly employed the Multiple Linear Regression (MLR) techniques, and the Factor-Based Techniques. Within each family, we will review the various methods commonly employed, learn how to develop and test calibrations, and how to use the calibrations to estimate, or predict, the properties of unknown samples. We will consider the advantages and limitations of each method as well as some of the tricks and pitfalls associated with their use. While our emphasis will be on quantitative analysis, we will also touch on how these techniques are used for qualitative analysis, classification, and discriminative analysis. 

Two models of practical interest using quantum chemical parameters were developed by Clark et al. [26, 27]. Both studies were based on 1085 molecules and 36 descriptors calculated with the AMI method following structure optimization and electron density calculation. An initial set of descriptors was selected with a multiple linear regression model and further optimized by trial-and-error variation. The second study calculated a standard error of 0.56 for 1085 compounds and it also estimated the reliability of neural network prediction by analysis of the standard deviation error for an ensemble of 11 networks trained on different randomly selected subsets of the initial training set [27]. 

L. Breiman and J.H. Friedman, Predicting multivariate responses in multiple linear regression. J. Roy. Stat. Soc. B59 (1997) 3-37. 

Beside mid-IR, near-IR spectroscopy has been used to quantitate polymorphs at the bulk and dosage product level. For SC-25469 [34], two polymorphic forms were discovered (a and /3), and the /3-form was selected for use in the solid dosage form. Since the /3-form can be transformed to the a-form under pressure by enantiotropy, quantitation of the /3-form in the solid dosage formulation was necessary. Standard mixtures of both forms in the formulation matrix were prepared, and spectra were measured in the near-IR via diffuse reflectance. Utilizing a standard, near-IR multiple linear regression, statistical approach, the a- and /3-forms could be predicted to within 1% of theoretical. This extension of the diffuse reflectance IR technique shows that quantitation of polymorphic forms at the bulk and/or dosage product level can be performed. 

Wajima and coauthors offer an alternative approach to utilize animal VD data to predict human VD [13]. Several compound descriptors that included both chemical structural elements as well as animal VDSS values were subject to multiple linear regression and partial least squares statistical analyses, with human VDSS as the independent parameter to be predicted using a dataset of 64 drugs. Methods derived in this manner were compared to simple allometry for overall accuracy. Their analyses yielded the following regressions ... 

FIGURE 4.24 PLS as a multiple linear regression method for prediction of a property y from variables xi,..., xm, applying regression coefficients b1,...,bm (mean-centered data). From a calibration set, the PLS model is created and applied to the calibration data and to test data. 

General regression neural network methodology is potentially more useful for predicting the QSPR relationships as compared to multiple linear regressions. 

In a paper that addresses both these topics, Gordon et al.11 explain how they followed a com mixture fermented by Fusarium moniliforme spores. They followed the concentrations of starch, lipids, and protein throughout the reaction. The amounts of Fusarium and even com were also measured. A multiple linear regression (MLR) method was satisfactory, with standard errors of prediction (SEP) for the constituents being 0.37% for starch, 4.57% for lipid, 4.62% for protein, 2.38% for Fusarium, and 0.16% for com. It may be inferred from the data that PLS or PCA (principal components analysis) may have given more accurate results. 

Neural networks are a relatively new tool in data modelling in the field of pharmacokinetics [54—56]. Using this approach, non-linear relationships to predicted properties are better taken into account than by multiple linear regression [45]. Human hepatic drug clearance was best predicted from human hepatocyte data, followed by rat hepatocyte data, while in the studied data set animal in vivo data did not significantly contribute to the predictions [56]. 

Multiple linear regression is an extension of simple linear regression the difference being more than one independent variable (descriptor) is used in the prediction of the dependent variable ... 

In the previous section we saw how to study the dependence of an outcome variable on another variable measured at baseline. It could well be that there are several baseline variables which predict outcome and in this section we will see how to incorporate these variables simultaneously through a methodology termed multiple (linear) regression. 

Multiple linear regression (MLR), although a popular technique, does not meet the requirements of the experimental design describe above. MLR can only deal with one dependent variable at a time and assumes that all variables are orthogonal (uncorrelated), and tiiat they are all completely relevant to the experiment, en dealing with an experimental system for the first time, it is not always possible to predict which variables will be relevant to the experiment, and which will not. So a technique is needed that can reconcile such uncertainties. 

Table 3 (73) compares the retention coefficients for synthetic peptides from various sources. To ensure comparability, the data has been standardized with respect to lysine and assigned a value of 100. The table shows that there are discrepancies between the results obtained using different chromatographic systems. Predictions of retention times should therefore be made using chromatographic systems similar to those used to calculate the retention coefficients for the amino acids. Casal et al. (75a) have made a comparative study of the prediction of the retention behavior of small peptides in several columns by using partial least squares and multiple linear regression analysis. 

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