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#Asreml plus manual
ASReml allows fitting many more different structures, so see section 4.9 (variance model specification) of the manual for more details.At Bayer we’re visionaries, driven to solve the world’s toughest challenges and striving for a world where ,Health for all, Hunger for none’ is no longer a dream, but a real possibility. Here the level ofĪutocorrelation will depend on distance between trees rather than on time.Īnother structure, based on random regressions, is explained in the Longitudinal Analysis section of the cookbook. A common spatial model will consider the presence of autocorrelated residuals in both directions (rows and columns). A similar situation is considered in Spatial Analysis, where the ‘independent errors’ assumption of typical analyses is relaxed. Looking at the previous pattern it is a lot easier to understand why they are called ‘structures’. In this model M = D* C_AR_* D, where C_AR_ (for equally spaced assessments) is:Ī model including this structure will certainly be more parsimonious (economic on terms of number of parameters) than one using an unstructured approach. For example, an autoregressive model (AR in ASReml lingo), where the correlation between measurements j and k is r |j-k|. Rather than using a different correlation for each pair of ages, it is possible to postulate mechanisms which model the correlations.
#Asreml plus plus
For example, the breeding value of an individual i observedĪt time j (a ij) is a function of genes involved in expression at time j - k (a ij-k), plus the effect of genes acting in the new measurement (α j), which are considered independent of the past measurement a ij = ρ jk a ij-k + α j, where ρ jk is the additive genetic correlation between measures j and k.
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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.
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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 Anyway, 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 notation for this type of analysis closely resembles the 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. Matrix for the traits involved in the analysis. 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 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.
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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). Information, the covariance structure is the product of a scalar (a variance component) by a design matrix. This is because ASReml assumes that, in absence of any additional When fitting simple models (as in many examples of Univariate Analysis) one needs to specify only the model equation (the bit like y ~ mu…) but nothing about the covariances that complete the model specification.