For example, an autoregressive model (ar() in asreml-R 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.
For example, the breeding value of an individual i observed at 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. 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 it is usual when analyzing multiple trait problems. If we do not impose any restriction on M, apart from being positive definite (p.d.), we are talking about an unstructured matrix (us() in asreml-R parlance).
Where the v are variances, the r correlations and the s standard deviations. Some 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 post-multiplied by a diagonal matrix ( D) containing standard deviations for each of the traits. 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 Kronecker product (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 a single site (so the same individual is assessed for two or more traits), 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 and R i is the residual matrix for site i. However, there are several situations when analyses require a more complex covariance structure, usually a direct sum or a Kronecker product of two or more matrices. For example, the residual covariance matrix in simple models 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), or the covariance matrix for a random effect f with incidence matrix Z is Z‘ Z σ f 2. This is because most linear mixed model packages assume that, in absence of any additional information, the covariance structure is the product of a scalar (a variance component) by a design matrix. We explicitly say nothing about the covariances that complete the model specification. asreml, lme4, nlme, etc) one needs to specify only the model equation (the bit that looks like y ~ factors.) when fitting simple models. META-R performs multi-environment analyses by using the residual maximum likelihood (REML) method these analyses can be done by environment, across environments by grouping factors (stress conditions, nitrogen content, etc.) and across environments the analyses across environments can be done with a pre-defined degree of heritability.In most mixed linear model packages (e.g. Genetic correlations are very important for identifying environments with similar behavior or making indirect selection and identifying the most highly associated traits. The genetic correlations between environments or traits can be visualized in a biplot graph or a tree diagram (dendrogram). META-R also computes the phenotypic and genetic correlations among environments and between traits, as well as their statistical significance.
These parameters are very important in the selection of top performing genotypes in plant breeding. Additionally, it computes the variance-covariance parameters, as well as some statistical and genetic parameters such as the least significant difference (LSD) at 5% significance, the coefficient of variation in percentage (CV), the genetic variance, and the broad-sense heritability. META-R simultaneously estimates the best linear and unbiased estimators (BLUEs) and the best linear and unbiased predictors (BLUPs). The objective of META-R is to accurately analyze multi-environment plant breeding trials (METs) by fitting mixed and fixed linear models from experimental designs such as the randomized complete block design (RCBD) and the alpha-lattice/lattice designs.
META-R (multi-environment trial analysis in R) is a suite of R scripts linked by a graphical user interface (GUI) designed in Java language.