Variance model

Table I. Summary of R-Square Values for the Various Variance Models...

Stanimirova I. Michalik K. Drzazga Z. Trzeciak H. Wentzell P.D. Walczak B. Interpretation of analysis of variance models using principal component analysis to assess the effect of a maternal anticancer treatment on the mineralization of rat bones. Analytica Chimica Acta, 2011,689 (1), 1-7. 

PCA of MEK—Percent Variance (Model Diagnostic) The first through fourth PCs of the MEK data describe 99.0%, 0.4%, 0.3%, and 0.1% of the variation, respectively. Assuming that the noise is greater than 0.4% of the variation, a one-component PCA model may be appropriate. 

Jeanmougin M et al (2010) Should we abandon the r-test in the analysis of gene expression microarray data a comparison of variance modeling strategies. PLoS One 5 el2336. doi 10.1371/journal.pone.0012336... 

Wright GW, Simon R. A random variance model for detection of differential gene expression in small microarray experiments. Bioinformatics 2003 19 2448-2455. 

Distribution and density Mean Variance Model Example... 

We view the real or the simulated system as a black box that transforms inputs into outputs. Experiments with such a system are often analyzed through an approximating regression or analysis of variance model. Other types of approximating models include those for Kriging, neural nets, radial basis functions, and various types of splines. We call such approximating models metamodels other names include auxiliary models, emulators, and response surfaces. The simulation itself is a model of some real-world system. The goal is to build a parsimonious metamodel that describes the input-output relationship in simple terms. 

The statistical techniques which have been discussed to this point were primarily concerned with the testing of hypotheses. A more important and useful area of statistical analysis in engineering design is the development of mathematical models to represent physical situations. This type of analysis, called regression analysis, is concerned with the development of a specific mathematical relationship including the mathematical model and its statistical significance and reliability. It can be shown to be closely related to the Analysis of Variance model. 

ISO 5725 [7] relies on a statistical analysis of variance model with two variance components "laboratory" and "repetition". Hence, a homogeneous material is assumed. Only under this homogeneity condition the calculated precision values are true method characteristics. For a heterogeneous material, like the tested cementitious mortar, the precision values are contaminated by the variance component of the material. Therefore, the precision values represent both material and method characteristics. 

In addition, the pairs of trays were classified in six groups according to their sealing rank, I to VI (Figure 4B and C), and the sealing rank was handled as a covariate in the analysis-of-variance model. No interaction terms between the main factors or between the covariate and the main factors were defined in the model. 

An independent method to identify the stochastic errors of impedance data is described in Chapter 21. An alternative approach has been to use the method of maximum likelihood, in which the regression procedure is used to obtain a joint estimate for the parameter vector P and the error structure of the data. The maximum likelihood method is recommended under conditions where the error structure is unknown, but the error structure obtained by simultaneous regression is severely constrained by the assumed form of the error-variance model. In addition, the assumption that the error variance model can be obtained by minimizing the objective function ignores the differences eimong the contributions to the residual errors shown in Chapter 21. Finedly, the use of the regression procedure to estimate the standard deviation of the data precludes use of the statistic... 

A second approach has been to use the regression procedure to obtain an estimate for the error structure of the data. A sequential regression is employed in which the parameters for an assumed error structure model, e.g., equations (21.19) and (21.20), are obtained directly from regression to the data. In more recent work, the error variance model was replaced by... 

Another commonly used statistical model is the Linear Model, which represents a family of models of a similar structure. The most commonly employed linear model is the analysis of variance model (ANOVA). We shall illustrate this model using the simplest case, the one-way ANOVA model. 

Goodness-of-Fit. It is implied in steps 2 to 6 above that diagnostic plots (e.g., weighted residual versus time, weighted residual versus predicted observations, population observed versus predicted concentrations, individual observed versus predicted concentrations) and a test statistic such as the likelihood ratio test would be used in arriving at the base model (see Section 8.6.1.1 for goodness of fit). Once the base model (with optimized structural and variance models) has been obtained, the next step in the PM model identification process is the development of the population model. 

Thus, a PK model that describes the time course of the drug in the body in a specific individual, a model describing the relationship between patient characteristics and the PK model parameters, a variance model for residual random variability, and a population model for intersubject random variability that describes the unexplained random variabiUty of the model parameters in the population of subjects studied are essential for describing a PPK model. 

Enter of Variance Model Parameters. Enter of Secondary Parameters. 

A critical component of comodeling multiple outputs is the appropriate weighting of individual observations. The weights must be appropriate for small and large responses within an output and the relative weights must be appropriate between outputs. Failure of the former standard can lead to regions of systematic error in the fitted function and failure in the latter standard can cause some of the outputs to inappropriately dominate the determination of fitted parameters. However, error variance model selection, as for structural model development, should be guided by parsimony stay as simple as possible. 

Data in these studies were generated from a so-called giant rat study in our laboratory. Animals were sacrificed to obtain serial blood and tissue samples. Each point represents the measurement from one individual rat and data from all these different rats were analyzed together to obtain a time prohle as though it came from one giant rat. A naive pooled data analysis approach was therefore employed for all model fittings using ADAPT II software (21). The maximum likelihood method was used with the variance model specified as V(a, 6, h) = (j Y(d, where V a, 9, ti) is the variance for the ith point, Y 6, t,) is the ith predicted value from the dynamic model, 9 represents the estimated structural parameters, and oi and 02 are the variance parameters that were estimated. 

Many tools are available for analyzing experimentally designed data [Hoaglin Welsch 1978, Latorre 1984, Rao 1973, Searle et al. 1992, Weisberg 1985], Common to many of these approaches is that the estimated effects are treated as additive. This means that the effect of each factor is independent of the variation in other factors. In some situations, an additive model of main effects is not realistic because the factors do not affect the response independently. A well-working remedy to this is to allow interactions between the factors. Conceptually, traditional analysis of variance models start from main effects and seek to keep the number of interactions as low as possible and of the lowest possible order. 

Figure 10.65. Significant effects in analysis of variance model of color. Effects are shown in terms of scaled and centered factors and response.

Mandel J, A new analysis of variance model for non-additive data, Technometrics, 1971, 13, 1-18. 

The first term on the right hand side of Eq. (1.1) is the structural model having two inputs (also called independent variables), D (dose) and t (time), and one output (also called the dependent variable), C (concentration). The variables V (volume of distribution) and CL (clearance) are referred to as model parameters which must be estimated from the observed concentration data. The second term in Eq. (1.1) is the error component (also called the variance model), e represents the deviation between model predicted concentrations and observed concentrations. 

Plot of absolute or squared residuals versus predicted value. Again, no systematic trend in the residuals should be observed and the plot should appear as a shotgun blast. Heteroscedasticity or misspecification of the variance model is evident if a positive trend in squared or absolute residuals with increasing predicted... 

Many of the plots just suggested are not limited to ordinary residuals. Weighted residuals, partial residuals, studentized residuals, and others can all be used to aid in model diagnostics. Beyond residual plots, other plots are also informative and can help in detecting model inadequacies. One notable plot is a scatter plot of observed versus predicted values usually with the line of unity overlaid on the plot (Fig. 1.8). The model should show random variation around the line of unity. Systematic deviations from the line indicate model misspecification whereas if the variance of the predicted values increases as the observed values increase then the variance model may be inappropriate. 

The fact that the same result was obtained with the OLS estimates is dependent on the assumption of normality and that the residual variance does not depend on the model parameters. Different assumptions or a variance model that depends on the value of the observation would lead to different ML estimates. Least squares estimates focus completely on the structural model in finding the best parameter estimates. However, ML estimates are a compromise between finding a good fit to both the structural model and the variance model. ML estimates are desirable because they have the following properties (among others) ... 

To strike the proper balance between an overparameterized model and an underparameterized model, one must strike a balance between a biased model and an overinflated variance model. Mallows (1973) proposed his Cp criterion which is defined as... 

If the variance model is not homoscedastic, a suitable transformation needs to be found prior to performing the algorithm. Carroll et al. (1995) stress that the extrapolation step should be approached as any other modeling problem, with attention paid to the adequacy of the extrapolant based on theoretical considerations, residual analysis, and possible use of linearizing transformations and that extrapolation is risky in general even when model diagnostics fail to indicate problems. ... 

---