Is single-step genomic REML with the algorithm for proven and young more computationally efficient when less generations of data are present?
Authors: Junqueira Vinícius Silva , Lourenco Daniela , Masuda Yutaka , Cardoso Fernando Flores , Sávio Lopes Paulo , e Silva Fabyano Fonseca , Misztal Ignacy 
Published: January 2022

AbstractEfficient computing techniques allow the estimation of variance components for virtually any traditional dataset. When genomic information is available, variance components can be estimated using genomic REML (GREML). If only a portion of the animals have genotypes, single-step GREML (ssGREML) is the method of choice. The genomic relationship matrix (G) used in both cases is dense, limiting computations depending on the number of genotyped animals. The algorithm for proven and young (APY) can be used to create a sparse inverse of G with close to linear memory and computing requirements. In ssGREML, the inverse of the realized relationship matrix (H-1) also includes the inverse of the pedigree relationship matrix, which can be dense with long pedigree, but sparser with short. The main purpose of this study was to investigate whether costs of ssGREML can be reduced using APY with truncated pedigree and phenotypes. We also investigated the impact of truncation on variance components estimation when different numbers of core animals are used in APY. Simulations included 150K animals from 10 generations, with selection. Phenotypes (h2 = 0.3) were available for all animals in generations 1-9. A total of 30K animals in generations 8 and 9, and 15K validation animals in generation 10 were genotyped for 52,890 SNP. Average information REML and ssGREML with G-1 and using 1K, 5K, 9K, and 14K core animals were compared. Variance components are impacted when the core group in APY represents the number of eigenvalues explaining a small fraction of the total variation in G. The most time-consuming operation was the inversion, with more than 50% of the total time. Next, numerical factorization consumed nearly 30% of the total computing time. On average, a 7% decrease in the computing time for ordering was observed by removing each generation of data. APY can be successfully applied to create the inverse of the genomic relationship matrix used in ssGREML for estimating variance components. To ensure reliable variance component estimation, it is important to use a core size that corresponds to the number of largest eigenvalues explaining around 98% of total variation in G. When APY is used, pedigrees can be truncated to increase the sparsity of H and slightly reduce computing time for ordering and symbolic factorization, with no impact on the estimates.Lay SummaryThe estimation of variance components is computationally expensive under large-scale genetic evaluations due to several inversions of the coefficient matrix. Variance components are used as parameters for estimating breeding values in mixed model equations (MME). However, resulting breeding values are not Best Linear Unbiased Predictions (BLUP) unless the variance components approach the true parameters. The increasing availability of genomic data requires the development of new methods for improving the efficiency of variance component estimations. Therefore, this study aimed to reduce the costs of single-step genomic REML (ssGREML) with the Algorithm for Proven and Young (APY) for estimating variance components with truncated pedigree and phenotypes. In addition, we investigated the influence of truncation on variance components and genetic parameter estimates. Under APY, the size of the core group influences the similarity of breeding values and their reliability compared to the full genomic matrix. In this study, we found that to ensure reliable variance component estimation it is required to consider a core size that corresponds to the number of largest eigenvalues explaining around 98% of the total variation in G to avoid biased parameters. In terms of costs, the use of APY slightly decreased the time for ordering and symbolic factorization with no impact on estimations.Teaser TextEstimation of variance components is becoming computationally challenging due to the increasing size of genomic information. We investigated the impacts of using the algorithm for proven and young (APY) in genetic evaluations. The use of APY has no impact on variance components and genetic parameters estimation.