Variance-covariance relationships

Calculate the variance-covariance matrix associated with the straight line relationship y, = Po + PiA i, + r, for the following data (see Section 11.2 for a definition of D) ... 

Equation 7.1 is one of the most important relationships in the area of experimental design. As we have seen in this chapter, the precision of estimated parameter values is contained in the variance-covariance matrix V the smaller the elements of V, the more precise will be the parameter estimates. As we shall see in Chapter 11, the precision of estimating the response surface is also directly related to V the smaller the elements of V, the less fuzzy will be our view of the estimated surface. 

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... 

Transformation of the original data to a new coordinate system is another possibility of data pretreatment. The methods are based on principal component analysis (PCA) or factor analysis (FA). The first step for these transformations is the formation of a data matrix that is derived from the original data matrix and that reflects the relationships among the data. Such derived data matrices are the variance-covariance matrix and the correlation matrix. 

Finally variance estimates of the parameters are calculated by means of the Cj/ elements, as in the following relationships (where var and cov represent variance and covariance, respectively). 

Thus, 4> (k, t) d roughly corresponds to the amount of scalar variance located at point k in wavenumber space at time t. Similar statements can be made concerning the relationship between and the scalar flux (ut(p), and between [Pg.90]

On the other hand, factor analysis involves other manipulations of the eigen vectors and aims to gain insight into the structure of a multidimensional data set. The use of this technique was first proposed in biological structure-activity relationship (i. e., SAR) and illustrated with an analysis of the activities of 21 di-phenylaminopropanol derivatives in 11 biological tests [116-119, 289]. This method has been more commonly used to determine the intrinsic dimensionality of certain experimentally determined chemical properties which are the number of fundamental factors required to account for the variance. One of the best FA techniques is the Q-mode, which is based on grouping a multivariate data set based on the data structure defined by the similarity between samples [1, 313-316]. It is devoted exclusively to the interpretation of the inter-object relationships in a data set, rather than to the inter-variable (or covariance) relationships explored with R-mode factor analysis. The measure of similarity used is the cosine theta matrix, i. e., the matrix whose elements are the cosine of the angles between all sample pairs [1,313-316]. 

Essentially, the classical PCA, which uses the variance as the one-dimensional projective index, is a special case of the PP algorithm. By the use of this special projective index, the classical PCA can be accomplished directly by solving out the eigenvectors and eigenvalues of the covariance matrix XX. For the i-th eigenvalue of XX, and the associated eigenvector aj, one has the following relationship ... 

The eigenvalues of the covariance matrix of X dehne the corresponding amount of variance explained by each eigenvector. The projection of the measurements (observations) onto the eigenvectors define new points in the measurement space. These points constitute the score matrix, T whose columns are ti given in Eq. 3.1. The relationship between T, P, and X can also be expressed as... 

Since the relationship between a computed quantity and measured variables is often nonlinear, an approximate method is needed to calculate the variance of the computed quantity. Suppose that a quantity y is given by a nonlinear function and that the individual measured quantities (x X2,..., x ) are independent. (The assumption of independence removes the need for covariance terms.) Then the variance of y may be approximated [15] as... 

Structural model adequacy Does the structural model explain the data Application of structural model for all subjects on all occasions Are there subjects/occasions for which the model is inadequate Covariate modeling strategy Is the process adequate to identify important relationships or the magnitude of variance partitions ... 

In his 1986 text Aitchison proves (for the mathematically literate reader) that die covariance structure of log-ratios is superior to the covariance structure of a percentage array (the crude covariance structure, as it is termed in his text). The covariance structure of log-ratios is free from the problems of the negative bias and of subcompositions which bedevil percentage data. In detail he shows that there are three ways in which the compositional covariance structure can be specified. Each is illustrated in Table 2.5. Firsdy, it can be presented as a variation matrix in which the log-ratio variances are plotted for every variable ratioed to every other variable. This matrix provides a measure of the relative variation of every pair of variables and can be used in a descriptive sense to identify relationships within the data array and in a comparative mode between data arrays. 

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