Stepwise regression analysis

To benchmark our learning methodology with alternative conventional approaches, we used the same 500 (x, y) data records and followed the usual regression analysis steps (including stepwise variable selection, examination of residuals, and variable transformations) to find an approximate empirical model, / (x), with a coefficient of determination = 0.79. This model is given by... 

A study of 398 male and 133 female civil servants in London, England, measured blood pressure, PbB, and serum creatinine concentration the study found no correlation between blood pressure and PbB after adjustment for significant covariates, including sex, age, cigarette smoking, alcohol intake, and body mass index in a stepwise multiple regression analysis (Staessen et al. 1990). 

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,... 

Principal component analysis (PCA) of the soil physico-chemical or the antibiotic resistance data set was performed with the SPSS software. Before PCA, the row MPN values were log-ratio transformed (ter Braak and Smilauer 1998) each MPN was logio -transformed, then, divided by sum of the 16 log-transformed values. Simple linear regression analysis between scores on PCs based on the antibiotic resistance profiles and the soil physico-chemical characteristics was also performed using the SPSS software. To find the PCs that significantly explain variation of SFI or SEF value, multiple regression analysis between SFI or SEF values and PC scores was also performed using the SPSS software. The stepwise method at the default criteria (p=0.05 for inclusion and 0.10 for removal) was chosen. 

Fu et al. [16] analyzed a set of 57 compounds previously used by Lombardo and other workers also. Their molecular geometries were optimized using the semiempirical self-consistent field molecular orbital calculation AMI method. Polar molecular surface areas and molecular volumes were calculated by the Monte Carlo method. The stepwise multiple regression analysis was used to obtain the correlation equations between the log BB values of the training set compounds and their structural parameters. The following model was generated after removing one outlier (Eq. 50) ... 

A training set of 57 compounds studied by Lombardo et al., Norinder et al., and Clark was used to develop the following model using stepwise multiple regression analysis, after removing one outlier (Eq. 65) ... 

Source Apportionment Models for the Cyclohexane-Soluble Fraction of Respirable Suspended Particulate Matter. Stepwise multiple regression analysis was used to determine the coefficients of the source tracers for the models proposed for CYC in equations (7)-(9). These models are shown in Table IV. As expected from the factor analyses, the coefficient for V, accounting for the greatest proportion of the variance of CYC, was fitted first into the equation. Equation (14) was the simplest and the F value was slightly higher than for equations (15) and (16). In addition, as will be discussed later in this paper, the coefficient for PB was in reasonable agreement with the ratio of CYC /PB for samples collected in the Allegheny Tunnel. 

The inverse relationship between limonin content and taste preference was confirmed in another study (41) using a stepwise multiple regression analysis of data from 60 samples of commercial frozen-concentrated orange juice (FCOJ) packed during two seasons. This and a latter report (42) concluded that limonin content was highly correlated with the flavor quality of the juice. 

For the biotransformation assay results, concentration-normalized maximum induction values were modeled. Stepwise multiple linear regression analysis was used to select the most suitable parameters from log Kow, EHOMO, ELUMO, and the difference in EHOMO and ELUM0 (ELUM0 - EH0M0). 

Figure 5.29 illustrates the pH-rate constant profile for the hydrolysis of L-phenylalanine methyl ester (weak base, pKa = 7.11) at 25°C. When attempts are made to simulate the experimental data with Equation (5.167) over a wide range of pH values, the model seldom fits well, because the values of kobs differ by several orders of magnitude and nonlinear regression analysis does not converge. Therefore, it is recommended that the kinetic values be within less than a few orders of magnitude. A localized and stepwise simulation process is recommended. At very low or high pH, Equation (5.167) simplifies to... 

Based on the earlier work of Meyer and Overton, who showed that the narcotic effect of anesthetics was related to their oil/water partition coefficients, Hansch and his co-workers have demonstrated unequivocally the importance of hydrophobic parameters such as log P (where P is, usually, the octanol/water partition coefficient) in QSAR analysis.28 The so-called classical QSAR approach, pioneered by Hansch, involves stepwise multiple regression analysis (MRA) in the generation of activity correlations with structural descriptors, such as physicochemical parameters (log P, molar refractivity, etc.) or substituent constants such as ir, a, and Es (where these represent hydrophobic, electronic, and steric effects, respectively). The Hansch approach has been very successful in accurately predicting effects in many biological systems, some of which have been subsequently rationalized by inspection of the three-dimensional structures of receptor proteins.28 The use of log P (and its associated substituent parameter, tr) is very important in toxicity,29-32 as well as in other forms of bioactivity, because of the role of hydrophobicity in molecular transport across cell membranes and other biological barriers. 

Structure-activity correlations were carried out using least-squares regression analysis techniques on an IBM 360 computer. As in the accompanying publication (6), the data in Tables I and II were fitted to Equation 3 in stepwise fashion. Standard statistical tests were carried out at each stage of fitting to determine the over-all goodness of fit of the x and o- data to the various equational forms examined. As in our previous study (6), the most statistically significant correlations were always obtained when activity data for meta-substituted and para-sub-stituted TFMS herbicides were divided into two discrete series and fitted separately. 

I = 21 y = 0.394 r = 0.9476 F = 49.8 where log P is the hydrophobicity, bondrefr is the molecular refractivity, delta is the submolecular polarity parameter, ind indicator variable (0 for heterocyclics and 1 for benzene derivatives). Calculations indicated that PBD-coated alumina behaves as an RP stationary phase, the bulkiness and the polarity of the solute significantly influencing the retention. The separation efficiency of PBD-coated alumina was compared with those of other stationary phases for the analysis of Catharanthus alkaloids. It was established that the pH of the mobile phase, the concentration and type of the organic modifier, and the presence of salt simultaneously influence the retention. In this special case, the efficiency of PBD-coated alumina was inferior to that of ODS. The retention characteristics of polyethylene-coated alumina (PE-Alu) have been studied in detail using various nonionic surfactants as model compounds.It was found that PE-Alu behaves as an RP stationary phase and separates the surfactants according to the character of the hydrophobic moiety. The relationship between the physicochemical descriptors of 25 aromatic solutes and their retention on PE-coated silica (PE-Si) and PE-Alu was elucidated by stepwise regression analysis. 

Multiple regression is used to generate the final equation. Figure 5.17 outlines the derivation of the QSAR equation. After database assembly, potential parameters are examined using simple statistics for the detection of problematical distributions that may have to be transformed. Next, a stepwise regression analysis is performed. F-scores of at least 1.7 are necessary for the parameter to be included into the final equation. Care is taken to avoid spurious correlations or colinearity difficulties. 

As with multiple regression analysis, the most commonly used selection procedures involve stepwise methods with the F-test being applied at each stage to provide a measure of the value of the variable to be added, or removed, in the discriminant function. The procedure is discussed in detail in Chapter 6. 

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