Model proportional residual variability

Another common residual variability model is the proportional residual variability model ... 

The combined residual variability model is another widely used residual variability model for the population approach. This residual variability model contains a proportional and an additive component ... 

This residual variability model behaves at small observation values like the additive residual variability model, for higher observation values the proportional component is dominating. 

Within NONMEM, a generalized least-squares-like (GLS-like) estimation algorithm can be developed by iterating separate, sequential models. In the first step, the model is fit using one of the estimation algorithms (FO-approximation, FOCE, etc.). The individual predicted values are saved in a data set that is formatted the same as the input data set, i.e., the output data set contains the original data set plus one more variable the individual predicted values. The second step then models the residual error based on the value of the individual predicted values given in the previous step. So, for example, suppose the residual error was modeled as a proportional error model... 

Type I error rates with FOCE-I were consistently near nominal values and were unaffected by number of subjects or number of observations per subject. With large residual variability (42%) and two observations per subject, Type I error rates for FOCE-I were higher than nominal, about 0.075 instead of 0.05. But when the number of observations was increased to four, the Type I error rate decreased to the nominal value and remained there as further increases in the number of observations were examined. Also, when the residual variance was modeled using a proportional residual error model, instead of an exponential residual variance model, the Type I error rate decreased. The major conclusion of this analysis was that FOCE-I should be preferred as an estimation method over FO-approximation and FOCE. 

In their third simulation, they examined the Type I error rate for inclusion of a false covariance term between CL and V in a 1-compartment model. FOCE-I Type I error rates were dependent on the number of samples collected per subject (more samples tended to decrease the Type I error rate), degree of residual variability (as residual variability increased so did the Type I error rate), and whether the residual error was defined using an exponential or proportional model (exponential residual error models always produced larger Type I errors than proportional error models). With 100 subjects, two observations per subject (at 1.75 and 7 h after... 

In summary, tobramycin pharmacokinetics were best characterized with a 2-compartment model where CL was proportional to CrCL and VI was proportional to body weight. BSV in CL was 29% with 13% variability across occasions. Residual variability was small, 14%, which compares well to assay variability of <7.5%. The model was robust to estimation algorithm and was shown to accurately predict an internally derived validation data set not used in the model development process. 

The Kalecki modified schema retains the key characteristics of the Grossmann model. Constant capital still grows at 10 per cent each year compared to 5 per cent for variable capital, and this requires a steady increase in the proportion of profits saved, from 25 per cent in year 1 to 65.4 per cent in year 35. Also in keeping with the Grossmann model, the rate of profit steadily falls over time, from 33.3 per cent in year 1 to 14.6 per cent in year 35. The difference, however, is that capitalist consumption is not treated as a residual, dependent upon the amount of profits that happen to remain after the prior commitments of capital accumulation. In Table 7.2, capitalist consumption is modelled as an active component in the model, providing an important driver in the generation of profits, as capitalists cast money into circulation. 

The population pharmacokinetics of enoxaparin was described by a one-compartment IV bolus model, the parameters of which are presented in Table 12.1. The interindividual variability parameters specify variances in the log-scale of the lognormaUy distributed PK parameters, and the residual error parameter specihes the variance of the proportional error. This model was implemented in ADAPT II using the Fortran code, provided in Appendix 12.1, which is identical to the ICOMPCL.FOR code provided as part of the software distribution, except for the residual error model. 

A new molecular entity exhibiting one-compartment pharmacokinetics with first-order absorption was assumed. The typical (mean) values of the population PK parameters for the NME were 1 h 17.5L/h, and SOL for absorption rate constant (Ka), apparent clearance (CLIP), and apparent volume of distribution (V/F), respectively. An intersubject variability of 45% (coefficient of variation) was assumed for each of these parameters, and this was assumed to be lognormally distributed with a mean of zero. A proportional error model was assumed for the residual error of 15%. 

One method for dealing with heteroscedastic data is to ignore the variability in Y and use unweighted OLS estimates of 0. Consider the data shown in Fig. 4.2 having a constant variance plus proportional error model. The true values were volume of distribution = 10 L, clearance = 1.5 L/h, and absorption rate constant = 0.7 per/h. The OLS estimates from fitting a 1-compartment model to the data were as follows volume of distribution = 10.3 0.15L, clearance = 1.49 0.01 L/h, and absorption rate constant =0.75 0.03 per h. The parameter estimates themselves were quite well estimated, despite the fact that the assumption of constant variance was violated. Figure 4.3 presents the residual plots discussed in the previous section. The top plot, raw residuals versus predicted values, shows that as the predicted values increase so do the variance of the residuals. This is confirmed by the bottom two plots of Fig. 4.3 which indicate that both the range of the absolute value of the residuals and squared residuals increase as the predicted values increase. 

Figure 9.10 Goodness of fit plots, residual plots, and histograms of weighted residuals and random effects under the reduced 2-compartment model with tobramycin clearance modeled using a power function of CrCL and interoccasion variability on CL. Dashed line in observed versus predicted plot is the line of unity. Solid line in the weighted residual plot is an inverse square kernel smoother with 0.4 sampling proportion.

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