@article {21,
title = {BLUP without (inverse) relationship matrix},
journal = {Proceedings of the World Congress on Genetics Applied to Livestock Production},
volume = {Electronic Poster Session - Theory to Application 3},
year = {2018},
pages = {21},
abstract = {Mixed Models provide the machinery for genetic evaluation in animal breeding. In fact they are the basis of all modern methods for obtaining estimates of their unknowns. One central component of Henderson{\textquoteright}s Mixed Model Equations (MME) is the relationship matrix (NRM) and more precisely its inverse. Only after Henderson found a way to directly set up the inverse, the animal model became computationally feasible. We show that there is a way to totally avoid the modelling of the NRM or its inverse, yet arriving at exactly the same BLUPs and BLUEs. This reduces the program complexity and the computational complexity. In practical modelling, it is useful to split the general model into blocks of uncorrelated model equations here called element equation. Each such element equation consists possibly of a small number of correlated equation linked by their covariance matrix, multiple trait models are an example. Standard animal models have three types of model equations: phenotype based, trivial for simple random effects, and pedigree model equations. The coefficients derived for the pedigree model equation are, indeed, the same as those given by Westell et al. (1987) without having to deal with the relationship matrix separately. BLUPs can be computed solely on the basis of these element equations: once the model equations have been set up in terms of coefficients and mixed model addresses and their covariance matrix association, further processing is oblivious to the statistical model and becomes a very simple sweep. Only a few lines of code are required for either setting up the complete MME in sparse format for a direct solution and possibly computing its inverse, or in conjugate gradients to iteratively solve for BLUEs and BLUPs. A small numerical example is given to describe the process. The model equations are well suited for coarse-grained parallelization. As implemented in PEST2, they scale well with little computing overhead, therefore making good use of multi core computers. An outlook is given for the use of the method for the handling of genomic (SNP) data in genetic evaluation. Also here, explicit treatment of the NRM and GRM is not required, leading to a joint uniform one step evaluation with pedigree and genomic data with no additional assumptions except the model and covariances. Keywords: Eildert Groeneveld, Arnold Neumaier, BLUP, gBLUP, inverse free, model equations},
author = {Eildert Groeneveld and Arnold Neumaier}
}