Inverse Of Genomic Relationship Matrix Research Articles

295 Parameter estimation under genomic selection

Abstract Genetic parameters are important in animal breeding for many tasks, including as input to a model for genetic evaluation, to estimate genetic gain due to selection, and to estimate correlated response due to selection on major traits. Before the genomic era, parameter estimation was facilitated by sparse structure of mixed model equations. Methods such as AI REML with sparse matrix inversion or MCMC via Gibbs sampling could estimate parameters for populations exceeding 1 million animals. With genomic selection (GS) and single-step GBLUP, the genomic matrices are mostly dense, and costs of parameter estimation increased dramatically. The estimation with 20K genotyped animals can take many days. Details in matching pedigree and genomic information influence estimated parameters. Estimation without the genomic information when GS is practiced leads to biases due to genomic-preselection. Truncating data to too few generations or to only genotyped animals leads to additional biases by excluding data on which the selection was practiced. Current studies indicate strong declines in heritability due to GS. Regular models for parameter estimation compute parameters only for the base population. Models that trace changes of parameters over time, such as random regression model on year of birth or a multiple trait model treating times slices as separate traits, are very expensive. A good compromise in parameter estimation under GS is to use slices of only 2–3 generations, with genotypes of young animals removed. When complete populations are genotyped, estimations with large number of genotyped animals are possible either with a SNP model or with GBLUP (inversion of genomic relationship matrix by APY algorithm). For simple models, Method R can provide estimates for any data size. An indirect indication of changing parameters over time is reduced predictivity or lower genetic trend despite increased data. Parameter estimation in GS would benefit from new, efficient tools.

0291 APY inverse of genomic relationship matrix–theory, analyses and questions

0291 APY inverse of genomic relationship matrix–theory, analyses and questions

Implementation of genomic recursions in single-step genomic best linear unbiased predictor for US Holsteins with a large number of genotyped animals

The objectives of this study were to develop and evaluate an efficient implementation in the computation of the inverse of genomic relationship matrix with the recursion algorithm, called the algorithm for proven and young (APY), in single-step genomic BLUP. We validated genomic predictions for young bulls with more than 500,000 genotyped animals in final score for US Holsteins. Phenotypic data included 11,626,576 final scores on 7,093,380 US Holstein cows, and genotypes were available for 569,404 animals. Daughter deviations for young bulls with no classified daughters in 2009, but at least 30 classified daughters in 2014 were computed using all the phenotypic data. Genomic predictions for the same bulls were calculated with single-step genomic BLUP using phenotypes up to 2009. We calculated the inverse of the genomic relationship matrix GAPY−1 based on a direct inversion of genomic relationship matrix on a small subset of genotyped animals (core animals) and extended that information to noncore animals by recursion. We tested several sets of core animals including 9,406 bulls with at least 1 classified daughter, 9,406 bulls and 1,052 classified dams of bulls, 9,406 bulls and 7,422 classified cows, and random samples of 5,000 to 30,000 animals. Validation reliability was assessed by the coefficient of determination from regression of daughter deviation on genomic predictions for the predicted young bulls. The reliabilities were 0.39 with 5,000 randomly chosen core animals, 0.45 with the 9,406 bulls, and 7,422 cows as core animals, and 0.44 with the remaining sets. With phenotypes truncated in 2009 and the preconditioned conjugate gradient to solve mixed model equations, the number of rounds to convergence for core animals defined by bulls was 1,343; defined by bulls and cows, 2,066; and defined by 10,000 random animals, at most 1,629. With complete phenotype data, the number of rounds decreased to 858, 1,299, and at most 1,092, respectively. Setting up GAPY−1 for 569,404 genotyped animals with 10,000 core animals took 1.3h and 57 GB of memory. The validation reliability with APY reaches a plateau when the number of core animals is at least 10,000. Predictions with APY have little differences in reliability among definitions of core animals. Single-step genomic BLUP with APY is applicable to millions of genotyped animals.