Technical note: Impact of pedigree depth on convergence of single-step genomic BLUP in a purebred swine population

Convergence behavior of single-step GBLUP and SNPBLUP for different termination criteria

Background

The preconditioned conjugate gradient (PCG) method is the current method of choice for iterative solving of genetic evaluations. The relative difference between two successive iterates and the relative residual of the system of equations are usually chosen as a termination criterion for the PCG method in animal breeding. However, our initial analyses showed that these two commonly used termination criteria may report that a PCG method applied to a single-step single nucleotide polymorphism best linear unbiased prediction (ssSNPBLUP) is not converged yet, whereas the solutions are accurate enough for practical use. Therefore, the aim of this study was to propose two termination criteria that have been (partly) developed in other fields, but are new in animal breeding, and to compare their behavior to that of the two termination criteria widely used in animal breeding for the PCG method applied to ssSNPBLUP. The convergence patterns of ssSNPBLUP were also compared to the convergence patterns of single-step genomic BLUP (ssGBLUP).

Results

Building upon previous work, we propose two termination criteria that take the properties of the system of equations into account. These two termination criteria are directly related to the relative error of the iterates with respect to the true solutions. Based on pig and dairy cattle datasets, we show that the preconditioned coefficient matrices of ssSNPBLUP and ssGBLUP have similar properties when using a second-level preconditioner for ssSNPBLUP. Therefore, the PCG method applied to ssSNPBLUP and ssGBLUP converged similarly based on the relative error of the iterates with respect to the true solutions. This similar convergence behavior between ssSNPBLUP and ssGBLUP was observed for both proposed termination criteria. This was, however, not the case for the termination criterion defined as the relative residual when applied to the dairy cattle evaluations.

Conclusion

Our results showed that the PCG method can converge similarly when applied to ssSNPBLUP and to ssGBLUP. The two proposed termination criteria always depicted these similar convergence behaviors, and we recommend them for comparing convergence properties of different models and for routine evaluations.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12711-021-00626-1.

Genomic Selection for Any Dairy Breeding Program via Optimized Investment in Phenotyping and Genotyping

This paper evaluates the potential of maximizing genetic gain in dairy cattle breeding by optimizing investment into phenotyping and genotyping. Conventional breeding focuses on phenotyping selection candidates or their close relatives to maximize selection accuracy for breeders and quality assurance for producers. Genomic selection decoupled phenotyping and selection and through this increased genetic gain per year compared to the conventional selection. Although genomic selection is established in well-resourced breeding programs, small populations and developing countries still struggle with the implementation. The main issues include the lack of training animals and lack of financial resources. To address this, we simulated a case-study of a small dairy population with a number of scenarios with equal available resources yet varied use of resources for phenotyping and genotyping. The conventional progeny testing scenario collected 11 phenotypic records per lactation. In genomic selection scenarios, we reduced phenotyping to between 10 and 1 phenotypic records per lactation and invested the saved resources into genotyping. We tested these scenarios at different relative prices of phenotyping to genotyping and with or without an initial training population for genomic selection. Reallocating a part of phenotyping resources for repeated milk records to genotyping increased genetic gain compared to the conventional selection scenario regardless of the amount and relative cost of phenotyping, and the availability of an initial training population. Genetic gain increased by increasing genotyping, despite reduced phenotyping. High-genotyping scenarios even saved resources. Genomic selection scenarios expectedly increased accuracy for young non-phenotyped candidate males and females, but also proven females. This study shows that breeding programs should optimize investment into phenotyping and genotyping to maximize return on investment. Our results suggest that any dairy breeding program using conventional progeny testing with repeated milk records can implement genomic selection without increasing the level of investment.

The quality of the algorithm for proven and young with various sets of core animals in a multibreed sheep population1

The inverses of the pedigree and genomic relationship matrices (A, G) are required for single-step GBLUP (ssGBLUP). While, inverting A is possible for millions of animals at a linear cost, inverting G has a cubic cost and feasible for at most 150,000 animals, using the current conventional algorithms. The algorithm for proven and young (APY) provides approximations of the regular ssGBLUP by splitting genotyped animals into core and noncore groups, with computational costs being cubic for core and linear for noncore animals. The data consisted of 9,406,096 animals in the pedigree, 6,243,753 weaning weight phenotypes, and 46,949 genotyped animals from 5 breeds, composites, and animals with missing breed information from New Zealand. Aiming to find a core sample for a multibreed sheep population that can provide evaluations similar to those from the regular ssGBLUP, different core types, and core sizes were studied. Core types random, composite, oldest, youngest, the most inbred animals in G (GINB), and in A (AINB) were studied in 5K, 10K, and 20K core sizes (K = 1,000). Romney core was studied in 5K and 10K, and Coopworth-Perendale core was studied in 5K. Correlation and regression coefficient (slope) between GEBV from the non-APY and the APY analyses, as indicators for consistency with non-APY and bias from non-APY, showed a large impact of APY on noncore and a small impact on nongenotyped animals. Breed-based 5K cores resulted in large bias from non-APY even for nongenotyped animals. Random and GINB at 20K core size resulted in the highest consistency with non-APY and the lowest bias from non-APY. However, GINB did not perform as well as Random at lower core sizes. The number of animals from a breed in the core sample was very important for the evaluation of that breed. We observed that cores without Texel or Highlander animals resulted in poor evaluations for those breeds. Solving the mixed model equations, within core type, the smallest core size, and within core size, Random core converged in the least number of iterations. However, APY per se did not necessarily reduce the solving time. Random cores performed the best, as they could give a good coverage on the generations and breeds, representative for the genotyped population. Core size 20K performed better than 5K and 10K, and the optimum core size was found to be 18.8K, according to the eigenvalue decomposition of G.