Appendix 7.1 Linear Regression Analysis

Appendix 7.1 Linear Regression Analysis 275 For the assumed linear form of Equation 7.50,... 

Using the actual dimensions of commercial steel pipe from Appendix F, plot the pipe wall thickness versus the pipe diameter for both Schedule 40 and Schedule 80 pipe, and fit the plot with a straight line by linear regression analysis. Rearrange your equation for the line in a form consistent with the given equation for the schedule number as a function of wall thickness and diameter ... 

Comparisons made below refer to kinetic data obtained for processes proceeding under similar conditions. All available values of (log A, E) within each group of related reactions were included in the linear regression analysis (Appendix II) and the compensation line was calculated using these formulas. Unless otherwise stated, the units of A are always molecules m-2 sec-1 at 1 Torr pressure of reactants and those of E are kJ mole-1. The compilation of Arrhenius parameters referred to identical reaction conditions is not always easy (or, indeed, possible in some instances) and it may be necessary to recalculate data from literature sources using an extrapolation. Not all details of the necessary corrections are recorded below, but such estimations were always minimized to preserve the objectivity of the conclusions reached. 

Multiple linear regression analysis of Equation (1) can also be used and for this kxy is determined with as many different combinations of and self-interaction coefficients (p or p ) can be measured by fitting the Bronsted or Hammett data to a binomial expression (logA = a + bx + cx ) by regular statistical software packages or by a program based on the statistical equation in Appendix 1 (Section A 1.1.4.4). 

For the regression analysis, the experimental data are fixed and the model parameters are varied to minimize SS nsing any of the optimization techniques discussed in Chapter 6. An analytical solution to the minimization problem is possible when the model has a linear form such as odei = ka. The fitting process is then known as linear regression analysis and is summarized in Appendix 7.1. Unfortunately, the more complex rate expressions are nonlinear. It is sometimes possible to transform the model to a linear form, but the transformation causes a bias so that some portions of the operating space will be better fit than other portions. This book... 

Our previous MNR Arrhenius parameters have been revised based on weighted linear regression analysis. The present temperature corrections have been based on the new results, which have been included in Appendix A. 

Because a regression analysis of lethality data for squirrel monkeys and dogs showed an approximately linear response (n = 0.97 and 0.99, respectively, see Appendix B), the lethality threshold estimate (27.3 ppm) was linearly scaled (C1 xt=k) t° the AEGL time periods using the methods often Berge et al. (1986) (Appendix A). 

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. 

In organic compound analysis, the instrument response is expressed as a response factor (RF), which is the ratio of the concentration (or the mass) of the analyte in a standard to the area of the chromatographic peak. Conversely, a calibration factor (CF) is the ratio of the peak area to the concentration (or the mass) of the analyte. Equation 1, Appendix 22, shows the calculation of RF and CF. In trace element and inorganic compound analyses, the calibration curve is usually defined with a linear regression equation, and response (calibration) factors are not used for quantitation. 

Fig. 6. 15 A suite of five whole-rock samples of the Wyatt Formation collected by V.H. Minshew from outcrops between Mt. Wyatt and Mt. Gardiner in the Scott Glacier area define two straight lines labeled A and B derived by least-squares linear regression. Line A which includes all five samples plus one duplicate analysis yields a date of 524 13 Ma and a high initial Sr/ Sr ratio of 0.711919 0.000547 (la). Line B is defined by three selected samples and corresponds to a precise date of 802 2 Ma but it has an impossibly low initial Sr/ Sr ratio of 0.69775. We conclude that the igneous rocks of the Wyatt Formation are older than 524 13 Ma but younger that 802 2 Ma. These previously unpublished data by G. Faure are presented in Appendix 6.133. In addition. Appendices 6.7.3.1 and 6.7.3.2 contain modal analyses and chemical compositions of rocks of the Wyatt Formation from Minshew (1967)...

Appendix A. Mathematical appendix contains a survey of selected mathematical-physical methods that are often used for solution of practical problems. These include numerical processing and uncertainty calculation, dimensional analysis, linear regression, iterative root finding, etc. 

Quite the most popular of these approaches is multiple regression analysis introduced by Hansch in 1968. In this method, the most favoured variables are (i) partition coefficients (P), from the system octanol/water (Section 3.2) (ii) the sigma (a) and rho (p) values from Hammett s Linear Free Energy Equation (Appendix III) and (iii) Taft s steric factorst E ) which are... 

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