Regression analysis linear least squares method

If the rate law depends on the concentration of more than one component, and it is not possible to use the method of one component being in excess, a linearized least squares method can be used. The purpose of regression analysis is to determine a functional relationship between the dependent variable (e.g., the reaction rate) and the various independent variables (e.g., the concentrations). 

To gain insight into chemometric methods such as correlation analysis, Multiple Linear Regression Analysis, Principal Component Analysis, Principal Component Regression, and Partial Least Squares regression/Projection to Latent Structures... 

Lowdin, P. O. (1992) On linear algebra, the least square method, and the search for linear relations by regression analysis in quantum chemistry and other sciences. Adv. Quantum Chem. 23, 83-126. 

The rate expressions Rj — Rj(T,ck,6m x) typically contain functional dependencies on reaction conditions (temperature, gas-phase and surface concentrations of reactants and products) as well as on adaptive parameters x (i.e., selected pre-exponential factors k0j, activation energies Ej, inhibition constants K, effective storage capacities i//ec and adsorption capacities T03 1 and Q). Such rate parameters are estimated by multiresponse non-linear regression according to the integral method of kinetic analysis based on classical least-squares principles (Froment and Bischoff, 1979). The objective function to be minimized in the weighted least squares method is... 

Application of a least-squares method to the linearized plots (e.g., Scatchard and Hames) is not reasonable for analysis of drug-protein binding or other similar cases (e.g., adsorption) to obtain the parameters because the experimental errors are not parallel to the y-axis. In other words, because the original data have been transformed into the linear form, the experimental errors appear on both axes (i.e., independent and dependent variables). The errors are parallel to the y-axis at low levels of saturation and to the x-axis at high levels of saturation. The use of a double reciprocal plot to determine the binding parameters is recommended because the experimental errors are parallel to the y-axis. The best approach to this type of experimental data is to carry out nonlinear regression analysis on the original equation and untransformed data. 

Since the nonlinear least-squares method requires initial guesses to start the procedure, three different initial trials were performed (1) (0,0), (2) (1,1), and (3) the values obtained from the Lineweaver-Burk plot in Example 4.2.4. All three initial trials give the same result (and thus the same relative error). Note the large differences in the values obtained from the nonlinear analysis versus those from the linear regression. If the solutions are plotted along with the experimental data as shown below, it is clear that the Lineweaver-Burk analysis does not provide a good fit to the data. 

It is then necessary to draw the best possible line through the data points on the graph. This is done by a procedure known as linear regression analysis by the least squares method . This is quite a mouthful and can produce a glazed expression on any chemist who is not mathematically orientated. In fact, the principle is quite straightforward. 

The basic principle of experimental design is to vary all factors concomitantly according to a randomised and balanced design, and to evaluate the results by multivariate analysis techniques, such as multiple linear regression or partial least squares. It is essential to check by diagnostic methods that the applied statistical model appropriately describes the experimental data. Unacceptably poor fit indicates experimental errors or that another model should be applied. If a more complicated model is needed, it is often necessary to add further experimental runs to correctly resolve such a model. 

A recalculation of the data, using the method of non-linear least squares analysis, in the present review indicated that the data supported the formation of a sixth fluoride species, ZrF , and that regression analysis could be used to derive an uncertainty for the third sulphate species, Zr(S04)j . The stability constant values derived in the present review were for equilibria of the form... 

The adaptive least squares (ALS) method [396, 585 — 588] is a modification of discriminant analysis which separates several activity classes e.g. data ordered by a rating score) by a single discriminant function. The method has been compared with ordinary regression analysis, linear discriminant analysis, and other multivariate statistical approaches in most cases the ALS approach was found to be superior to categorize any numbers of classes of ordered data. ORMUCS (ordered multicate-gorial classification using simplex technique) [589] is an ALS-related approach which... 

A second method was used to determine if there was a significant difference between HPS, DPS, and the 50 50 mixture. Linear least-squares regression analysis was used to correlate the spectra of HPS and DPS with the 50 50 mixture. The point of maximmn correlation was calculated as 60 4% HPS and 40 7% DPS. The residuals showed no systematic series of peaks indicating that the deviations from 50 50 were due to random chance. Therefore, based on two data analysis methods, one can conclude that there is not a significant amount... 

Before the advent of computer technology and the computerized statistical methods for data analysis, a procedure that was employed extensively in the analysis of enzyme kinetic data was linear regression. It is important to point out that linear regression performed by the least squares method should not be used unless the values are weighted. If it is used without proper weighting one can get bad results. 

All calculations of Dl can be carried out simultaneously by a GW-BASIC personal computer program, written for non-linear least-squares regression analysis and based on the experimental pair values H and t in centimeters and seconds, respectively. The experimental Dl values are of the same order of magnitude and, in some cases, very close to those obtained by other techniques or calculated theoretically from empirical equations. The precision ( 13%) and the accuracy (8 7%) of the RF-GC method, as determined from the Dl values of Refs. compared to those computed from the more accurate empirical equation of Wilke-Qiang," are relatively satisfactory, considering the difficulties in obtaining experimental Dl values, and the large dispersion of the predicted diffusion coefficients. ... 

---