Subject variance-covariance

The first attempt at estimating interindividual pharmacokinetic variability without neglecting the difficulties (data imbalance, sparse data, subject-specific dosing history, etc.) associated with data from patients undergoing drug therapy was made by Sheiner et al. " using the Non-linear Mixed-effects Model Approach. The vector 9 of population characteristics is composed of all quantities of the first two moments of the distribution of the parameters the mean values (fixed effects), and the elements of the variance-covariance matrix that characterize random effects.f " " ... 

Individual predictions, however, require in addition to p and Z the estimated standard deviations of the random subject effects (t) and corresponding variance-covariance matrix (Q). The extraction of t and Q is less straightforward but the function getRanPars does just that. Notice that this function also extracts estimates and standard errors of other random effects parameters, for example. 

Here 0 is a vector of mean population pharmacokinetic parameters and Q is the variance-covariance matrix of between-subject random variability. Np represents a p-dimensional multivariate normal distribution, where p is the number of parameters. It is often more useful to consider the values of the parameters for the individual to be related to the population parameters via a covariate relationship, in which case the expression may be written as... 

In this notation, g(9, zj) is used to represent a function (g), perhaps a linear combination of CO variates, that describes the expectation of the /th subjects parameter vector 9i conditional on their demographic characteristics (z,) and population parameter values (0). The variance-covariance matrix (Q) therefore describes the random variability between subjects that is not able to be explained by covariates. 

Similar to the prior of the mean population parameter values, the prior for the between-subject variance can also be selected to have a more plausible range for PK/PD systems. If we consider the coefficient of variation of between-subject variability for most PK/PD parameters as being approximately <100%, then a choice of p for the Wishart distribution that provided a 97.5th percentile value of around this level would be biologically plausible. This is not quite as straightforward as for the precision of the population mean parameter values, since the minimum size of p is indexed to the minimum dimension of the variance-covariance matrix of between-subject effects, and p affects all variance parameters equally. The value of p required to provide a similar level of weak informativeness will vary with the dimension of the matrix. A series of simulations have been performed from the Wishart distribution, where the mean value of the variance of between-subject effects was set at 0.2. The value of p required to provide a similar level of weak informativeness will vary with the dimension of the matrix (Table 5.1). 

The can be viewed as observations of the individual parameters. The estimate for a subject may be biased and imprecise because of poor experimental design, poor study execution, or a high level of measurement error. The GTS approach makes extensive use of the matrices Mj, j = 1,..., N, which reflect the deviations (bias), together with the estimates j, j = 1,..., N. The expectation ( ) and the variance-covariance Var(-) of each (random) can be calculated ... 

A proportional error, a constant additive error, and a combination of both error models were evaluated for the residual error model. Between-subject random effects were explored on the clearance of parent drug and metabolite, the volume of distribution of the parent drug, and the absorption rate constant. An exponential model was preferred. Interoccasion random effects were explored on the clearance of the parent drug and of the metabolite, the volume of distribution of the parent drug, and the absorption rate constant. An exponential model was preferred. The joint distribution of the between-subject random effect, the interoccasion random effects, and the residual error were assumed normal with mean 0 and variance-covariance matrices O for the between-subject and interoccasion random effects, and I, for the residual error to be estimated. The FO method was used for the estimation of the parameters. 

The effect of the following covariates was investigated on the disposition parameter of nelfinavir, M8, and ritonavir, for which a between-subject variance was estimated ... 

Like other random effects, the residual error can be dependent on subject-specific covariates and as such, covariates can be included in the residual variance model. The classic example is when some blood samples were assayed with one analytical assay, while others were assayed with another type of assay. If an indicator variable, ASY, is defined such that every sample is assigned either a 0 or 1, depending on the reference assay, the residual error could then be modeled as... 

Suppose Y = f(x, 0, t ) + g(z, e) where nr] — (0, il), (0, ), x is the set of subject-specific covariates x, z, O is the variance-covariance matrix for the random effects in the model (t ), and X is the residual variance matrix. NONMEM (version 5 and higher) offers two general approaches towards parameter estimation with nonlinear mixed effects models first-order approximation (FO) and first-order conditional estimation (FOCE), with FOCE being more accurate and computationally difficult than FO. First-order (FO) approximation, which was the first algorithm derived to estimate parameters in a nonlinear mixed effects models, was originally developed by Sheiner and Beal (1980 1981 1983). FO-approximation expands the nonlinear mixed effects model as a first-order Taylor series approximation about t) = 0 and then estimates the model parameters based on the linear approximation to the nonlinear model. Consider the model... 

PCA components with small variances may only reflect noise in the data. Such a plot looks like the profile of a mountain after a steep slope a more flat region appears that is built by fallen, deposited stones (called scree). Therefore, this plot is often named scree plot so to say, it is investigated from the top until the debris is reached. However, the decrease of the variances has not always a clear cutoff, and selection of the optimum number of components may be somewhat subjective. Instead of variances, some authors plot the eigenvalues this comes from PCA calculations by computing the eigenvectors of the covariance matrix of X note, these eigenvalues are identical with the score variances. 

Both between-subject and interoccasion variances were estimated on clearance of nelfinavir, absorption rate constant, and clearance of M8. The residual error with a proportional error model was modeled for nelfinavir and M8 separately. The effect of ritonavir was found to have a statistically significant impact on the clearance of M8 but not on that of nelfinavir. The apparent clearance of M8 was 3.23 L/h it decreased to 1.87 L/h when nelfinavir was coadministered with ritonavir. After univariate selection, a large number of covariates were included in the full model. According to the acceptance criteria, none of the effect on clearance of nelfinavir on... 

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,... 

A variety of within-subject covariance matrices have been proposed (Table 6.2). The most common are the simple, unstructured, compound symmetry, first-order autoregressive [referred to as AR(1)], and Toeplitz covariance. The simple covariance assumes that observations within a subject are uncorrelated and have constant variance, like in Eq. (6.28). Unstructured assumes... 

The linear mixed effect model assumes that the random effects are normally distributed and that the residuals are normally distributed. Butler and Louis (1992) showed that estimation of the fixed effects and covariance parameters, as well as residual variance terms, were very robust to deviations from normality. However, the standard errors of the estimates can be affected by deviations from normality, as much as five times too large or three times too small (Verbeke and Lesaffre, 1997). In contrast to the estimation of the mean model, the estimation of the random effects (and hence, variance components) are very sensitive to the normality assumption. Verbeke and Lesaffre (1996) studied the effects of deviation from normality on the empirical Bayes estimates of the random effects. Using computer simulation they simulated 1000 subjects with five measurements per subject, where each subject had a random intercept coming from a 50 50 mixture of normal distributions, which may arise if two subpopulations were examined each having equal variability and size. By assuming a unimodal normal distribution of the random effects, a histogram of the empirical Bayes estimates revealed a unimodal distribution, not a bimodal distribution as would be expected. They showed that the correct distributional shape of the random effects may not be observed if the error variability is large compared to the between-subject variability. 

Nonlinear mixed effects models consist of two components the structural model (which may or may not contain covariates) and the statistical or variance model. The structural model describes the mean response for the population. Similar to a linear mixed effects model, nonlinear mixed effects models can be developed using a hierarchical approach. Data consist of an independent sample of n-subjects with the ith subject having -observations measured at time points t i, t 2, . t n . Let Y be the vector of observations, Y = Y1 1, Yi,2,. ..Ynjl,Yn,2,. ..Yn,ni)T and let s... 

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