First-order regression models

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,... 

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. 

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 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. 

VHien this method is used, Table II shows the results when the regression model is the normal first order linear model. Since the maximum absolute studentized residual (Max ASR) found, 2.29, was less than the critical value relative to this model, 2.78, the conclusion is that there are no inconsistent values. 

Elimination parameters are determined by linear regression analysis of the measured plasma concentration data falling on the terminal line. As always, the first step is to calculate the natural logarithm of each of the measured plasma concentration values. The values of In(C ) are then plotted versus time (t). If the plot shows later points falling near a straight terminal line with no early points above the terminal line, then the data can be well represented by the one-compartment first-order absorption model. As with previous one-compartment models, early high points above the terminal line indicate that the one-compartment model is not the best PK model for the data, and erratic late data points could mean the values are unreliable, as illustrated in Figure 10.47. 

Estimation of multicompartment model parameters from measured plasma samples is very similar to the procedures described previously for the two-compartment first-order absorption model. The first step is to calculate bi(C ) for each of the measured plasma sample concentrations. The values of In(C ) are then plotted versus time (t), and the points on the terminal line are identified. Linear regression analysis of the terminal line provides values for B (B = c ) and In = —m). The first residual (/ i) values are then calculated as the difference between the measured plasma concentrations and the terminal line for points not used on the terminal line. A plot of ln(i i) versus t is then employed to identify points on the next terminal line, with linear regression analysis of this line used to determine and X -. Successive method of residuals analyses are then used to calculate the remaining B and A, values, with linear regression of the n-1 residual (Rn-i) values providing the values of Bi and Aj. If a first-order absorption model is being used, then one more set of residuals (R ) are calculated, and the linear regression analysis of these residuals then provides and kg. This type of analysis is typically performed by specialized PK software when the model contains more than two compartments. 

A comparison of the two-compartment first-order absorption model fit to measured plasma concentration data from a traditional method of residuals analysis and a nonlinear regression analysis is provided in Figure 10.99. This figure illustrates the fact that both methods offer a very reasonable fit to the measured data. It also demonstrates that there is not a large difference between the fit provided by the two different techniques. Close examination does reveal, however, that the nonlinear regression analysis does provide a more universal fit to all the data points. This is likely due to the fact that nonlinear regression fits all the points simultaneously, whereas the method of residuals analysis fits the data in a piecewise manner with different data points used for different regions of the curve. 

Figure 5. Decrease in (A) propargyl bromide and (B) MITC in the root zone with time. Values indicate the total volume contained under the concentration contours the volume at each time was normalized to the volume observed just prior to injection ofATS/water. Values are the mean of two mesocosms and error bars represent the standard error. Lines indicate regression to a first-order kinetic model. For MITC (Bf regression omitted spurious 52-hour data.

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). 

The authors describe the use of a Taylor expansion to negate the second and the higher order terms under specific mathematical conditions in order to make any function (i.e., our regression model) first-order (or linear). They introduce the use of the Jacobian matrix for solving nonlinear regression problems and describe the matrix mathematics in some detail (pp. 178-181). 

This is, then, the regression sum of squares due to the first-order terms of Eq. (69). Then, we calculate the regression sum of squares using the complete second-order model of Eq. (69). The difference between these two sums of squares is the extra regression sum of squares due to the second-order terms. The residual sum of squares is calculated as before using the second-order model of Eq. (69) the lack-of-fit and pure-error sums of squares are thus the same as in Table IV. The ratio contained in Eq. (68) still tests the adequacy of Eq. (69). Since the ratio of lack-of-fit to pure-error mean squares in Table VII is smaller than the F statistic, there is no evidence of lack of fit hence, the residual mean square can be considered to be an estimate of the experimental error variance. The ratio... 

The crucial point for building a prediction model with PCR (Section 4.6) is to determine the number of PCs to be used for prediction. In principle we could perform variable selection on the PCs, but for simplicity we limit ourselves to finding the appropriate number of PCs with the largest variances that allows the probably best prediction. In other words, the PCs are sorted in decreasing order according to their variance, and the prediction error for a regression model with the first a components will tell us which number of components is optimal. As discussed in Section 4.2... 

The concentration of any contaminant(s) from highway C R materials appearing in the effluent from the column was measured over time and the results of leachate desorption breakthrough curves [66, 67] are schematically shown in Fig. 10. The effluent concentrations of contaminants for three different flow rates were determined to follow a first-order model as shown in Eq. (95), with the coefficients fitted by the linear regressions given in Table 3 ... 

Figure 8. Regression of batch oxidation and precipitation experiment of Bullen et al. (2001), where Fe(II),q was oxidized, followed by precipitation of ferrihydrite, over a 24 h period. The reaction progress (F) is well-fit by a first-order rate law, where the rate constant is 0.0827 F/h, with an of 0.96. In the model illustrated by Equations (5)-(8) in the text, this rate constant would be set to k,.

They were also typical when the regression model chosen was first order. Mean-level bandwidths greater than 20-30% are probably indicative that errors have been made in the analysis process that should not be tolerated. In this case techniques would be carefully scrutinized to find errors, outliers, or changing chromatographic conditions. These should be remedied and the analysis repeated whenever possible. Certain manipulation can be done to reduce the bandwidth values. For example, they would be... 

In reference 88, response surfaces from optimization were used to obtain an initial idea about the method robustness and about the interval of the factors to be examined in a later robustness test. In the latter, regression analysis was applied and a full quadratic model was fitted to the data for each response. The method was considered robust concerning its quantitative aspect, since no statistically significant coefficients occurred. However, for qualitative responses, e.g., resolution, significant factors were found and the results were further used to calculate system suitability values. In reference 89, first a second-order polynomial model was fitted to the data and validated. Then response surfaces were drawn for... 

This section introduces the regression theory that is needed for the establishment of the calibration models in the forthcoming sections and chapters. The multivariate linear models considered in this chapter relate several independent variables (x) to one dependent variable (y) in the form of a first-order polynomial ... 

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