First-order regression

Table 3. First order regression coefficients for column leaching of TOC at ambient pH (pHs7) and pore volumes...

Table II. Fitting DATASET D Data to the First Order Regression...

Table I. Algebraic Equations for First-Order Regression Calculations...

PHARMACOKINETICS The area under the plasma concentration-time curve (AUC) was identified, in a preliminary analysis, as the important exposure covariate that was predictive of the safety biomarker outcome. Consequently, it became necessary to compare the distributions of AUC values across studies and dosage regimens. Figure 47.8 illustrates distributions of the exposure parameter AUC across studies. It is evident that AUC values are higher in diseased subjects than in healthy volunteer subjects at the same dose level. To adjust for the difference between the two subpopulations, an indicator function was introduced in a first-order regression model to better characterize the dose-exposure data. Let y be the response variable (i.e., AUC), X is a predictor variable, P is the regression coefficient on x, and e is the error term, which is normally distributed with a mean of zero and variance cP. Thus,... 

Carbazole exhibited a photolysis rate comparable to that of the faster Indoles. All test compounds exhibited high correlation coefficients In the first-order regression analyses. The Indoles had a broader range of rates and generally behaved less linearly than carbazole beyond 90 min of exposure. While the photolysis of carbazole was nearly linear through 180 min, that of the Indoles generally decreased. This difference in behavior indicates that photolysis of the two classes of chemicals in this matrix occurs by different mechanisms. 

Linear Model With Intercept. There are two distinct linear first-order regression models that are generally encountered in analytical calibration. The non-zero intercept model is the most familiar, and it is given by Equation 1. 

FIGURE 3.49 Normalized mean FAIMS separation parameters for ubiquitin ions (z = 6, 7) in N2 gas over 7=35-80 °C, measured at u=15 (A), 20 ( ), and 25 ( ) kV/cm. For each D. we show the first-order regressions through all data (solid lines) and those for T — 50-80 °C (dashed lines). (From Robinson, E.W., Shvartsburg, A.A., Tang, K., Smith, R.D., Anal. Chem. 80, 7508, 2008.) The vertical displacements of datasets for adjacent d values that provide the best coincidence of regressions are labeled (in °C). 

The collected data are first analyzed through the Yate s method (analysis of the factor significance on the dependent variable) after, first-order regression models are obtained. Particular attention has been addressed in the description of the operation in terms of sensitivity maps. 

Each single combination of the factors has been replicated twiee to evaluate the significance of the factor modification on every dependent variable. Furthermore, the factorial analysis of variance (ANOVA) and the response surfaee method (RSM) have been applied, and the first-order regression models linking the dependent variable to the two control factors have been found and analyzed with an ANOVA. 

The ANOVA both of the experimental data and of the first-order regression models have been developed using the faetorial of Table 1, where the treatment combinations and the range of variation of each factor are reported ... 

Second, first-order regression models have been obtained, with the consequent response smfaces and contour plots. 

Consider the first order regression model. If interaction terms are added to a main effects or first-order model, resulting in... 

In a curve-fitting method the concentration of a reactant or product is monitored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. Eor example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentration of a product is followed as a function of time, can be determined by fitting a rearranged form of equation 13.12... 

Earlier we noted that a response surface can be described mathematically by an equation relating the response to its factors. If a series of experiments is carried out in which we measure the response for several combinations of factor levels, then linear regression can be used to fit an equation describing the response surface to the data. The calculations for a linear regression when the system is first-order in one factor (a straight line) were described in Chapter 5. A complete mathematical treatment of linear regression for systems that are second-order or that contain more than one factor is beyond the scope of this text. Nevertheless, the computations for... 

Strkcttire inflkence. The specificity of interphase transfer in the micellar-extraction systems is the independent and cooperative influence of the substrate molecular structure - the first-order molecular connectivity indexes) and hydrophobicity (log P - the distribution coefficient value in the water-octanole system) on its distribution between the water and the surfactant-rich phases. The possibility of substrates distribution and their D-values prediction in the cloud point extraction systems using regressions, which consider the log P and values was shown. Here the specificity of the micellar extraction is determined by the appearance of the host-guest phenomenon at molecular level and the high level of stmctural organization of the micellar phase itself. 

On the basis of data obtained the possibility of substrates distribution and their D-values prediction using the regressions which consider the hydrophobicity and stmcture of amines was investigated. The hydrophobicity of amines was estimated by the distribution coefficient value in the water-octanole system (Ig P). The molecular structure of aromatic amines was characterized by the first-order molecular connectivity indexes ( x)- H was shown the independent and cooperative influence of the Ig P and parameters of amines on their distribution. Evidently, this fact demonstrates the host-guest phenomenon which is inherent to the organized media. The obtained in the research data were used for optimization of the conditions of micellar-extraction preconcentrating of metal ions with amines into the NS-rich phase with the following determination by atomic-absorption method. 

Fig. 8.P3I. Plot of the pseudo-first-order rate constants for hydrolysis of thioesters A (O), B ( ), C (A), D (A) as a fiinction of pH at 50°C and ionic strength 0.1 (KCI). Lines are from fits of the data to = kon(K /H+)) + (k KJK + [//+])) where koH is the hydroxide term and is the intramolecular assistance term for B and C and from linear regression for A and D. Reproduced from problem reference 31 by permission of the American Chemical Society.

The simplest procedure is merely to assume reasonable values for A and to make plots according to Eq. (2-52). That value of A yielding the best straight line is taken as the correct value. (Notice how essential it is that the reaction be accurately first-order for this method to be reliable.) Williams and Taylor have shown that the standard deviation about the line shows a sharp minimum at the correct A . Holt and Norris describe an efficient search strategy in this procedure, using as their criterion minimization of the weighted sum of squares of residuals. (Least-squares regression is treated later in this section.)... 

In the easiest case, a first order autoregressive model, the effects of variations in the past are contained and accounted for in the most immediate value. This value becomes an independent variable in generalized regression analysis. 

Degradation rates were determined for the reported data using a nonlinear regression of conventional first-order kinetic equations. The software used for this fitting procedure was Model Manager, Version 1.0 (Cherwell Scientific, 1999). 

Fig. 8 Influence of concentration on the rate of absorption from the in situ rat intestine. The linear dependence of absorption rate on concentration suggests an apparent first-order absorption process over the range studied. Absorption rates have been calculated from the data in Ref. 15 and the straight lines are from linear regression of the data. Key (O) erythritol ( ) urea ( ) malonamide.

---