Generalized covariance models

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... 

Figure 6.15 visualizes the different cluster models mentioned above. The left picture is the result of using a model with the same form cr2/ for all clusters. The middle picture changes the cluster size with crjl. The right picture shows the most general cluster model, each having a different covariance matrix Xj. Clearly, there exist several more different possible model classes. 

The analysis of covariance between a continuous variable (P is the curve shape parameter from the Weibull function) and a discrete variable (process) was determined using the general linear model (GLM) procedure from the Statistical Analysis System (SAS). The technique of the heterogeneity of slopes showed that there was no significant difference (Tables 5 and 6). 

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... 

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. 

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... 

A third approach proposed by Mandema et al. (16) was an improvement on the Maitre et al. (15) approach. The first step is similar to that proposed by Maitre et al. (15), but in the second step individual PK/PD parameters are regressed against covariates using generalized additive modeling (GAM). In the final step, NONMEM is used to optimize and finalize the population model. The approach does not discuss how a reduction in the dimensionahty of the covariate vector should be handled. 

Exploratory modeling using modem statistical modeling techniques such as generalized additive modeling (GAM) (15), cluster analysis, and tree-based modehng (TBM) to reveal structure in the data and initially select explanatory covariates. 

Generalized additive modeling (GAM) was applied to each of the output data sets. A selection criteria of a = 0.05 and a frequency cutoff of 0.50 was applied for continued investigation of a covariate that is, GAM had to select a covariate for inclusion in 50% or more of the models from the 100 bootstrap fits for the covariate to be considered for further investigation. 

As in simple linear regression, the same assumptions are made s is normally distributed, uncorrelated with each other and have mean zero with variance u2. In addition, the covariates are measured without error. In matrix notation then, the general linear model can be written as... 

Selecting a mixed effects model means identifying a structural or mean model, the components of variance, and the covariance matrix for the residuals. The basic rationale for model selection will be parsimony in parameters, i.e., to obtain the most efficient estimation of fixed effects, one selects the covariance model that has the most parsimonious structure that fits the data (Wol-finger, 1996). Estimation of the fixed effects is dependent on the covariance matrix and statistical significance may change if a different covariance structure is used. The general strategy to be used follows the ideas presented in... 

The next covariate screening approach would be to use a regression-based method and take a more rigorous statistical approach to the problem. Using the generalized additive model (GAM) procedure in SAS, a LOESS smooth was applied to the continuous covariates wherein the procedure was allowed to identify the optimal smoothing parameter for each covariate tested. Two dependent variables were examined t, and the EBE for CL. To avoid possible skewness in the residuals,... 

Wolfmger, R. Covariance structure selection in general mixed models. Communications in Statistics Series—Simulation A 1993 22 1079-1106. 

Covariate selection Examine covariate addition to model using generalized additive modeling (GAM) or equivalent stepwise insertion techniques. 

The Poisson regression model is an example of the generalized linear model. The maximum likelihood estimates of the coefficients of the predictors can be found by iteratively reweighted least squares. This also finds the covariance matrix of the normal distribution that matches the curvature of the likelihood... 

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