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Submitted to: World Congress of Genetics Applied in Livestock Production
Publication Type: Proceedings Publication Acceptance Date: 10/9/2017 Publication Date: N/A Citation: N/A Interpretive Summary: Technical Abstract: Transformations to multiple trait mixed model equations (MME) which are intended to improve computational efficiency in best linear unbiased prediction (BLUP) and restricted maximum likelihood (REML) are described. It is shown that traits that are expected or estimated to have zero residual variance can be included in a multiple trait model more efficiently than traits that have residual variance. Furthermore, traits with no residual variance add no equations to the MME and add insignificant computation compared with analysis of only the remaining traits, even if the trait with no residual has only a miniscule fraction of the records in the analysis. This is because such traits contribute information through adjustments to the right hand side of the MME and the genetic (co)variance parameters. Applications of this transformation to genomics, specifically in fitting molecular breeding values (MBV) or single nucleotide polymorphisms (SNP) as correlated traits are discussed; it appears that many such traits could be fit simultaneously. Traits with residual variation that is completely dependent on other traits in the model similarly need not be represented by equations in the MME; their contribution can be made solely through the right hand side and adjustment of the (co)variance parameters. Thus, singularities in the residual (co)variance matrix, whether due to zero variance or a correlation structure that results in non-positive definiteness, can be handled by reducing the order of the MME. Singularities in genetic (co)variance parameter matrices, which are very common when analysing more than a few traits, can also be handled by transformation of the MME. In this case the transformation involves modifying the Z matrix to account for correlations so that the genetic (co)variance parameter matrix is more diagonally dominant to avoid boundary issues. But, if the genetic (co)variance matrix is singular, a trait whose breeding value is completely dependent on others can have it’s equations dropped from the MME and it’s breeding value is connected to the other traits through the Z matrix, while its y vector appears in the right hand sides of the other traits. The MME are transformed such that, conditional on the assumed prior variance components, the residual parameter matrix to be optimized has a conditional value of identity and the genetic parameter matrix to be optimized has a conditional value that is diagonal. The transformed random effect parameters roughly ratios of genetic inverse (co)variance parameters relative to the residual inverse (co)variance parameters, analogous the ' or ' parameterizations in single trait models. It is demonstrated that it is unnecessary and inefficient to fit the same set of animal equations (with the numerator relationship matrix) for traits that have different sets of records. The pedigree can be trimmed to the minimal connecting set for the records of each trait without losing information or reducing sparsity. A simple and efficient algorithm for constructing G^(-1) under these conditions is presented. |
