Linear regression case studies

Gonzalez, A. G., TWo Level Factorial Experimental Designs Based on Multiple Linear Regression Models A Tutorial Digest Illustrated by Case Studies, Analytica Chimica Acta 360, 1998, 227-241. 

Often, it is not quite feasible to control the calibration variables at will. When the process under study is complex, e.g. a sewage system, it is impossible to produce realistic samples that are representative of the process and at the same time optimally designed for calibration. Often, one may at best collect representative samples from the population of interest and measure both the dependent properties Y and the predictor variables X. In that case, both Y and X are random, and one may just as well model the concentrations X, given the observed Y. This case of natural calibration (also known as random calibration) is compatible with the linear regression model... 

The data were statistically analyzed using the SOLO Statistical System (BMDP Statistical Software, Inc., Los Angeles, CA) on a personal computer. Differences between groups were tested by the Mann-Whitney test or a paired t-test in cases where paired data sets were tested. Possible relationships were studied with (multiple) linear regression using least-square estimates. 

At this point, a considerable amount of theory on Hansch analysis has been presented with almost no examples of practice. The next three Case Studies will hopefully solidify ideas on Hansch analysis that have already been discussed. Each Case Study introduces a different idea. The first is an example of a very simple Hansch equation with a small data set. The second demonstrates the use of squared parameters in Hansch equations. The third and final Case Study shows how indicator variables are used in QSAR studies. If you are unfamiliar with performing linear regressions, be sure to read Appendix B on performing a regression analysis with the LINEST function in almost any common spreadsheet software. A section in the appendix describes in great detail how to derive Equations 12.20 through 12.22 in the first Case Study. 

Many conventional texts use summations rather than matrices for determination of regression equations, but both approaches are equivalent. In Fig. 1, the absorbance of the spectra of case study 1A at 336 nm is plotted against the concentration of pyrene (Table 1). The graph is approximately linear, and provides a best fit slope calculated by /... 

When the studied case concerns obtaining a relationship for the characterization of a process with multiple independent variables and only one dependent variable, we can use a multiple linear regression ... 

The kinds of calculations described above are done for all the molecules under investigation and then all the data (combinations of 3-point pharmacophores) are stored in an X-matrix of descriptors suitable to be submitted for statistical analysis. In theory, every kind of statistical analysis and regression tool could be applied, however in this study we decided to focus on the linear regression model using principal component analysis (PCA) and partial least squares (PLS) (Fig. 4.9). PCA and PLS actually work very well in all those cases in which there are data with strongly collinear, noisy and numerous X-variables (Fig. 4.9). 

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. 

The retention model developed by Eon and Guiochon [7,8] to describe the adsorption effects at both gas-liquid and liquid-solid interfaces, which was later modified by Mdckel et al. [6] to account for the retention at chemically bonded reversed-phase materials in HPLC, is not applicable to ion chromatography. But if the dependence of the capacity factors of various inorganic anions on the column temperature is studied, certain parallels with HPLC are observed. The linear dependences shown in Fig. 3-2 are obtained for the ions bromide and nitrate when the In k values are plotted versus the reciprocal temperature (van t Hoff plot). However, in the case of fluoride, chloride, nitrite, orthophosphate, and sulfate, the k values were found to be constant within experimental error limits in the temperature range investigated. Upon linear regression of the values in Table 3-1, the following relations are derived for bromide and nitrate ... 

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