Between-subject variance matrix

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). 

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. 

To obtain initial estimates, an Emax model was fit to the data set in a na ive-pooled manner, which does not take into account the within-subject correlations and assumes each observation comes from a unique individual. The final estimates from this nonlinear model, 84% maximal inhibition and 0.6 ng/mL as the IC50, were used as the initial values in the nonlinear mixed effects model. The additive variance component and between-subject variability (BSV) on Emax was modeled using an additive error models with initial values equal to 10%. BSV in IC50 was modeled using an exponential error model with an initial estimate of 10%. The model minimized successfully with R-matrix singularity and an objective function value (OFV) of 648.217. The standard deviation (square root of the variance component) associated with IC50 was 6.66E-5 ng/mL and was the likely source of the... 

Factor space may be obtained from the matrix of random balance and analysis of variance. Information on number of replications of design points-trials in the basic experiment is obtained from analysis of variance, and some proofs about linear or nonlinear relationships between variables of the research subject from correlation analysis. 

To get at the question of overall influence, the matrix of structural model parameters and variance components was subjected to principal component analysis. Principal component analysis (PCA) was introduced in the chapter on Nonlinear Mixed Effects Model Theory and transforms a matrix of values to another matrix such that the columns of the transformed matrix are uncorrelated and the first column contains the largest amount of variability, the second column contains the second largest, etc. Hopefully, just the first few principal components contain the majority of the variance in the original matrix. The outcome of PC A is to take X, a matrix of p-variables, and reduce it to a matrix of q-variables (q < p) that contain most of the information within X. In this PC A of the standardized parameters (fixed effects and all variance components), the first three principal components contained 74% of the total variability in the original matrix, so PCA was largely successfully. PCA works best when a high correlation exists between the variables in the original data set. Usually more than 80% variability in the first few components is considered a success. 

One of us (RC) has begun to explore the use of Principal Component Analysis (PCA) [74] to reduce the dimensions of the LDM and extract information from it to build robust QSPRs and QSARs. In the PCA approach, a matrix containing entries that may be statistically correlated is converted to an eigenvector, devoid of such correlation between variables, composed of what is termed the principal components (PCs) . The PCs may be equal in number to the original variables or can be lesser. The first PC is the one with maximal variance followed by PCs that maximize the variance subject to the constraint of being orthogonal to aU previous PCs. 

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