Model combined residual variability

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 ... 

Wahlby et al. (2002) later expanded their previous study and used Monte Carlo simulation to examine the Type I error rate under the statistical portion of the model. In all simulations a 1-compartment model was used where both between-subject variability and residual variability were modeled using an exponential model. Various combinations were examined number of obser-... 

In summary, the Type I error rate from using the LRT to test for the inclusion of a covariate in a model was inflated when the data were heteroscedastic and an inappropriate estimation method was used. Type I error rates with FOCE-I were in general near nominal values under most conditions studied and suggest that in most cases FOCE-I should be the estimation method of choice. In contrast, Type I error rates with FO-approximation and FOCE were very dependent on and sensitive to many factors, including number of samples per subject, number of subjects, and how the residual error was defined. The combination of high residual variability with sparse sampling was a particularly disastrous combination using... 

FIGURE 9.2 ThePLSl regression method is shown for modeling one y variable from ZX variables in objects i — 1,2,...,/. The figure shows that the variability not picked up by the A first bilinear factors remain in the residual matrices E and f. Each factor ta represents a linear combination of the X variables. Each latent variable also has a loading vector pa, which show how it related to the individual X variables, and another loading vector qa that relates it to the y variable. 

The known models for describing retention factor in whole variable space ar e based on three-phase model and containing from three to six par ameters and variety combinations of two independent factors (micelle concentration, volume fraction of organic modifier). When the retention models are comparing or the accuracy of fitting establishing, the closeness of correlation coefficient to 1 and the sum of the squared residuals or the sum of absolute deviations and their relative values is taken into account. A number of problems ar e appear in this case ... 

If melt or fluid re-enrichment or some other form of disturbance affects a peridotite subsequent to melting then this should be reflected in PGE patterns and PGE-major element syste-matics. Modelling of melt re-enrichment involving the addition of new sulphides to variably depleted melt residues during magma-solid interaction produces elevation of the Pd/Ir ratio to supra-chondritic levels, and elevated P-PGE contents (Rehkamper et al. 1999 Fig. 4). Combined major element, PGE and Re-Os isotope systematics therefore provide a potential means to evaluate the validity of Re-Os isotope model ages (Trd V. Tma)-... 

Nonlinear mixed effects models are similar to linear mixed effects models with the difference being that the function under consideration f(x, 0) is nonlinear in the model parameters 0. Population pharmacokinetics (PopPK) is the study of pharmacokinetics in the population of interest and instead of modeling data from each individual separately, data from all individuals are modeled simultaneously. To account for the different levels of variability (between-subject, within-subject, interoccasion, residual, etc.), nonlinear mixed effects models are used. For the remainder of the chapter, the term PopPK will be used synonymously with nonlinear mixed effects models, even though the latter covers a richer class of models and data types. Along with PopPK is population pharmacodynamics (PopPD), which is the study of a drug s effect in the population of interest. Often PopPK and PopPD are combined into a singular PopPK-PD analysis. 

The correlation coefficient r (eq. 124) is a relative measure of the quality of fit of the model because its value depends on the overall variance of the dependent variable (this is illustrated by eqs. 58 — 60, chapter 3.8 while the correlation coefficients r of the two subsets are relatively small, the correlation coefficient derived from the combined set is much larger, due to the increase in the overall variance). The squared correlation coefficient r is a measure of the explained variance, most often presented as a percentage value. The overall (total) variance is defined by eq. 125, the unexplained variance (SSQ = sum of squared error residual variance variance not explained by the model) by eq. 126. 

Figure 14.5 depicts the predicted vs. observed permeability constants (log Kp) for all 288 treatment combinations studied without taking into account the specific mixtures at which these chemicals were dosed. The residuals of this model showed no further correlation to penetrant properties. However, when vehicle/mixture component properties were analyzed, trends in residuals became evident. An excellent single parameter explaining some variability of this residual pattern (R of 0.44) was log (1 /Henry constant) (1/HC). Figure 14.6 depicts the modified LFER model including an MF = log (1/HC). 

When there are only a few discrete-valued decision variables in a model, the most effective method of analysis is usually the most direct one total enumeration of all the possibilities. For example, a model with only eight 0-1 variables could be enumerated by trying all 2 = 256 combinations of values for the different variables. If the model is pure discrete, it is only necessary to check whether each possible assignment of values to discrete variables is feasible and to keep track of the feasible solution with best objective function value. For mixed models the process is more complicated because each choice of discrete values yields a residual optimization problem over the continuous variables. Each such continuous problem must be solved or shown infeasible to establish an optimtil solution for the full mixed problem. 

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