Covariance among variables

The other commonly alluded to theoretical concern with using PCA for specimen identification is the fact that PCA is predicated on a characterization of the covariance stracture among variables across the entire sample taken as a whole. This differs from the typical discriminant analysis situation where the purpose is to determine the character of structural relations between different... 

Figure 1 Relationships among variables affecting 6-month MDI/PDI, as revealed through structural equation analyses. Covariate-adjusted parameter estimates (and standardized regression coefficients) are indicated. All relationships significant at p = 0.05 or less, one-tailed tests. After Dietrich et al (1986)...

Polynomial regression with indicator variables is another recommended statistical method for analysis of fish-mercury data. This procedure, described by Tremblay et al. (1998), allows rigorous statistical comparison of mercury-to-length relations among years and is considered superior to simple hnear regression and analysis of covariance for analysis of data on mercury-length relations in fish. 

In a follow-up study, Calvert et al. (1996) examined the association between exposure to 2,3,7,8-TCDD and serum lipids. In the follow-up the authors chose not to adjust the 2,3,7,8-TCDD serum concentrations for total lipids to avoid the problems of interpretation that would arise when adjusting a covariate by the dependent variable. Consequently, the results obtained in this study cannot be compared directly with those from the Operation Ranch Hand study (see below). The median serum 2,3,7,8-TCDD concentration among the workers was 406.6 femtograms/g serum (fg/g) compared with 36.9 fg/g among the referents. 

A first PLS model was established from 124 reaction systems. To ensure that this set of reaction systems was not selected in such a way that the descriptor variables were correlated, a principal component analysis was made of the variation of the eight descriptors over the set. This analysis afforded eight significant principal components according to cross validation. This showed that the variance-covariance matrix of the descriptors was a full rank matrix and that there were no severe colinearities among the descriptors. 

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... 

Fisher suggested to transform the multivariate observations x to another coordinate system that enhances the separation of the samples belonging to each class tt [74]. Fisher s discriminant analysis (FDA) is optimal in terms of maximizing the separation among the set of classes. Suppose that there is a set of n = ni + U2 + + rig) m-dimensional (number of process variables) samples xi, , x belonging to classes tt, i = 1, , g. The total scatter of data points (St) consists of two types of scatter, within-class scatter Sw and hetween-class scatter Sb- The objective of the transformation proposed by Fisher is to maximize S while minimizing Sw Fisher s approach does not require that the populations have Normal distributions, but it implicitly assumes that the population covariance matrices are equal, because a pooled estimate of the common covariance matrix (S ) is used (Eq. 3.45). 

BMDs and BMDLs for four end points (Finger Tapping, CPT Reaction Time, Boston Naming, and CVLT Delayed Recall) based on (1) models that include log(PCB) as an additional covariate and (2) the subset of subjects in the lowest tertile of PCB exposures. Because PCBs were measmed only for children examined in 1993, only about half of the full cohort (approximately 450 children) are used for analysis 1, and only one-sixth (approximately 150 Children) are used for analysis 2. Results were provided for Hg measured in both maternal hair and cord blood (see Table 7-4). The reduced sample sizes in these additional analyses increased the variability among the results. There was no clear pattern with respect to how the PCB-adjusted analyses differed from the original results. 

When multiple covariates are included in the regression model, the possibility for collinearity, which is sometimes called multicollinearity or ill-conditioning, among the predictors arises. The term collinear implies that there is correlation or linear dependencies among the independent variable. Entire books (Belsley, Kuh,... 

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