There are cases when the order of assessment or the spatial location of the experimental units create patterns of variation, which are reflected by the covariance matrix. Thus, M or C can take any value (as long as it is p.d.) as is usual when analyzing multiple trait problems. If we do not impose any restriction on M, apart from being positive (p.d.) definite, we are talking about an unstructured matrix (US in ASReml parlance). Where the v are variances, the r correlations and the s standard deviations. The structures are easier to understand (at least for me) if we express a covariance matrix ( M) as the product of a correlation matrix ( C) pre- and postmultiplied by a diagonal matrix ( D) containing standard deviations for each of the traits. ASReml supports a large number of covariance structures (and I will present only a few of them), which are particularly useful for longitudinal and spatial analysis. You will see that the ASReml (and ASReml-R) notation for this type of analysis closely resembles matrix notation. Similarly, G = A * G 0 where all the matrices are as previously defined and G 0 is the additive covariance matrix for the traits. For example, R = I * R 0, where I is an identity matrix of size number of observations, * is the direct product operation (do not confuse with a plain matrix multiplication) and R 0 is the error covariance matrix for the traits involved in the analysis. Other example of a more complex covariance structure is a multivariate analysis in one site, where both the residual and additive genetic covariance matrices are constructed as the product of two matrices. For example, an analysis of data from several sites might consider different error variances for each site, that is R = Σd R i, where Σd represents a direct sum (see any matrix algebra book for an explanation) and R i is the residual matrix for site i. However, there are several situations when the analysis require a more complex covariance structure, usually a direct sum or direct product of two or more matrices. For example, the residual covariance matrix in simple examples is R = I σ e 2, or the additive genetic variance matrix is G = A σ a 2 (where A is the numerator relationship matrix). This is because ASReml assumes that, in absence of any additional information, the covariance structure is the product of a scalar (a variance component) by a design matrix. When fitting simple models (as in many examples of univariate analysis one needs to specify only the model equation (the bit that looks like y ~ mu.) but nothing about the covariances that complete the model specification.
0 kommentar(er)
