Covariance model

Note that if the kriging weights Xi(z) are data values-independent, the indicator estimates (11) are not hence, the final estimate of the conditional cdf Fx(z (N)) is data values-dependent. For each cut-off Z], the kriging estimator (11) requires a different indicator covariance model ... 

These various covariance models are Inferred directly from the corresponding indicator data i(3 z ), i-l,...,N. The indicator kriging approach is said to be "non-parametric, in the sense that it draws solely from the data, not from any multivariate distribution hypothesis, as was the case for the multi- -normal approach. 

These indicator covariance models allow differentiation of the spatial correlation of high-valued concentrations (cut-off high) and low to background-valued concentrations low). In the particular case study underlying the Figure 3, it was found that high value concentration data were more spatially correlated (due to the plume of pollution) than lower value data. 

To summarize the present situation in Fig. 4 the resulting density dependence of the SE for the approaches discussed above are compared (excluding the 3NF contribution). One sees that the covariant models predict a much larger increase of the SE with the density than the non-relativistic approaches. The lowest-order BHF method predicts a somewhat higher value for 04 than both the VCS and SCGF methods, which lead to very similar results whether that can be ascribed to a consistent treatment of correlations in these methods, or is fortuitous, is not clear. 

Generalized Covariance Models. When l x) is an intrinsic random function of order k, an alternative to the semi-variogram is the generalized covariance (GC) function of order k. Like the semi-variogram model, the GC model must be a conditionally positive definite function so that the variance of the linear functional of ZU) is greater than or equal to zero. The family of polynomial GC functions satisfy this requirement. The polynomial GC of order k is... 

A general analysis-of-covariance model for a stability design with several batches and packages can be expressed as... 

Identification of Analysis of Covariance Model A general procedure, based on regression analysis, to identify the analysis-of-covariance model that applies to a given set of assay results to determine the shelf life is introduced here. We call this procedure the regression model with indicator variables for testing poolability of... 

Rules for Determining Shelf Life Once the analysis-of-covariance model has been identified, a set of rules for computing the shelf life must be implemented. This section describes the rules to follow to determine the expiration dating period for each of the nine representative models described in the previous section [8,11] ... 

TABLE 23 Procedure for Identifying Analysis of Covariance Model... 

We are not certain which comorbid risk factors cause mortality independent of sleep effects, and therefore, we cannot be certain whether we controlled too much or too little for comorbidities. For example, since short sleep or long sleep may cause a person to be sick at present or to get little exercise or to have heart disease (17), diabetes (18), etc., controlling for these possible mediating variables may have incorrectly minimized the hazards associated with sleep durations. This would be overcontrol. The hazard ratios for participants who were rather healthy at the time of the initial questionnaires were unlikely to be overcontrolled for initial illness. Since the 32-covariate models and the hazard ratios for initially healthy participants were similar, this similarity reduced concern that the 32-covariate models were overcontrolled. On the other hand, there may have been residual confounding processes that caused both short or long sleep and early death that we could not adequately control in the CPSII data set, either because available control variables did not adequately measure the confound or because the disease did not yet manifest itself. Depression, sleep apnea, and dysregulation of cytokines are plausible confounders that were not adequately controlled. It may be impossible to be confident that all conceivable confounds are adequately controlled in epidemiological studies of sleep. 

In CPSII, in the 32-covariate models, reported insomnia was associated with risk ratios slightly but significantly less than 1.0, after controlling for sleep duration (3). A similar result was found in another study (52). This might imply a protective effect of insomnia. Similarly, insomnia did not predict total mortality when depression and other comorbidities were controlled in major Swedish studies (13,53). In general, studies that control well for comorbid factors do not find that insomnia predicts increased mortality independent of sleep duration and hypnotic drug use. 

Both candidate models were then re-evaluated using the final dataset of 8388 observations from 906 patients of the 19 studies. With the final dataset, the addition of the linear elimination pathway resulted in only a single point reduction in OFV compared to the model with saturable elimination only. Therefore, a two-compart-ment model with saturable elimination was considered the final structural model and was used for the development of the covariate model (see Table 14.4). 

Using the NLME, the population model contains three components the structural model, the statistical model and - if necessary - the (integrated) patient covariate model (Fig. 17.2). 

The covariate model describes the relationship between covariates and model parameters. Covariates are individual-specific variables that describe intrinsic factors like, e.g. sex, age, weight, creatinine clearance or extrinsic factors like, e.g. concomitant medication, alcohol consumption. The parameter-covariate relation should explain the variability in PK or PD model parameters to a certain extent. 

The statistical model is incorporated as described in Section 17.4. The covariate model allows a description of relationships between covariates and model parameters, explaining parts of the intersubject variability and identifying sub-populations at risk for concentrations below or above the therapeutic range. 

Parametric population methods also obtain estimates of the standard error of the coefficients, providing consistent significance tests for all proposed models. A hierarchy of successive joint runs, improving an objective criterion, leads to a final covariate model for the pharmacokinetic parameters. The latter step reduces the unexplained interindividual randomness in the parameters, achieving an extension of the deterministic component of the pharmacokinetic model at the expense of the random effects. Recently used individual empirical Bayes estimations exhibit more success in targeting a specific individual concentration after the same dose. 

To illustrate the remaining covariance modeling task required closing the model, the fairly rigorous instantaneous volume averaged equations expressed in terms of mass-weighted quantities are listed below. 

It is generally hard to let graphs like the ones in Figure 7.13 guide the covariate model building process. The problem lies in the fact that covariates tend to be correlated. This is to some extent illustrated in Figure 7.13. The parameter values... 

Figure 7.14 is based on a model in which creatinine clearance was included. The axis limits are the same as in Figure 7.13 and it is clear that the unexplained variability has decreased. At the same time it appears as if the sex relation is not as important anymore. On the other hand, had sex been included in the model instead of creatinine clearance then the picture would perhaps have looked the same. Again, this is the problem with using graphs to guide covariate model building. Clearly some other techniques are necessary (see other chapters in this book). 

Data analysis method Stating the software, model building procedures, model diagnostics, structural model, covariate model, stochastic model, and sensitivity analysis to be used and how the evaluation of the model is to be conducted... 

J. Ribbing and E. N. Jonsson, Power, selection bias and predictive performance of the population pharmacokinetic covariate model. J Pharmacokinet Pharmacodyn 31 109-134 (2004). 

E. N. Jonsson and M. O. Karlsson, Automated covariate model building within NONMEM. Pharm Res 15 1463-1468 (1998). 

Covariate Model Development. Treatment (with and without ritonavir) would be included as a potential covariate on clearance. Influence of AAG on clearance and of weight on volume and clearance were also included as potential covariate relationships. Assay site was also included as a potential covariate for residual error. 

P. Thall and S. Vail, Some covariance models for longitudinal count data with overdispersion. Biometrics 46 657-671 (1990). 

Time course and covariate model TPRE=THETA(1) /... 

Complex pharmacokinetic/pharmacodynamic (PK/PD) simulations are usually developed in a modular manner. Each component or subsystem of the overall simulation is developed one-by-one and then each component is linked to run in a continuous manner (see Figure 33.2). Simulation of clinical trials consists of a covariate model and input-output model coupled to a trial execution model (10). The covariate model defines patient-specific characteristics (e.g., age, weight, clearance, volume of distribution). The input-output model consists of all those elements that link the known inputs into the system (e.g., dose, dosing regimen, PK model, PK/PD model, covariate-PK/PD relationships, disease progression) to the outputs of the system (e.g., exposure, PD response, outcome, or survival). In a stochastic simulation, random error is introduced into the appropriate subsystems. For example, between-subject variability may be introduced among the PK parameters, like clearance. The outputs of the system are driven by the inputs... 

Both a drug s PK response and pharmacological response may be influenced by various patient characteristics. Additionally, the progression or extent of disease may be affected by comorbidities. Covariate models attempt to account for and quantify the influence of these factors. For example, a covariate model would be used in simulating PK differences between males and females. Likewise, for a disease progression model of atherosclerosis, or for the overall evaluation of... 

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