Alternative Ways of Computing the Numerator Relationship Matrix

Pedigree relationships between every pair of individuals forms the elements of the additive genetic relationship matrix (A). Calculation of A−1 does not require forming and inverting A, and it is faster and easier than the calculation of A. Although A−1 is used in best linear unbiased prediction of genetic merit, A is used in population studies and post-evaluation procedures, such as breeding programs and controlling the rate of inbreeding. Three pedigrees with 20,000 animals (20K) and different (1, 2, 4) litter sizes, and a pedigree with 180,000 animals (180K) and litter size 2 were simulated. Aiming to reduce the computation time for calculating A, new methods [Array-Tabular method, (T−1)−1 instead of T in Thompson's method, iterative updating of D in Thompson's method, and iteration by generation] were developed and compared with some existing methods. The methods were coded in the R programming language to demonstrate the algorithms, aiming for minimizing the computational time. Among 20K, computational time decreased with increasing litter size for most of the methods. Methods deriving A from A−1 were relatively slow. The other methods were either using only pedigree information or both the pedigree and inbreeding coefficients. Calculating inbreeding coefficients was extremely fast (<0.2 s for 180K). Parallel computing (15 cores) was adopted for methods that were based on solving A−1 for columns of A, as those methods allowed implicit parallelism. Optimizing the code for one of the earliest methods enabled A to be built in 13 s (faster than the 31 s for calculating A−1) for 20K and 17 min 3 s for 180K. Memory is a bottleneck for large pedigrees but attempts to reduce the memory usage increased the computational time. To reduce disk space usage, memory usage, and computational time, relationship coefficients of old animals in the pedigree can be archived and relationship coefficients for parents of the next generation can be saved in an external file for successive updates to the pedigree and the A matrix.

Reduced Animal Models Fitting Only Equations for Phenotyped Animals

Reduced models are equivalent models to the full model that enable reduction in the computational demand for solving the problem, here, mixed model equations for estimating breeding values of selection candidates. Since phenotyped animals provide data to the model, the aim of this study was to reduce animal models to those equations corresponding to phenotyped animals. Non-phenotyped ancestral animals have normally been included in analyses as they facilitate formation of the inverse numerator relationship matrix. However, a reduced model can exclude those animals and obtain identical solutions for the breeding values of the animals of interest. Solutions corresponding to non-phenotyped animals can be back-solved from the solutions of phenotyped animals and specific blocks of the inverted relationship matrix. This idea was extended to other forms of animal model and the results from each reduced model (and back-solving) were identical to the results from the corresponding full model. Previous studies have been mainly focused on reduced animal models that absorb equations corresponding to non-parents and solve equations only for parents of phenotyped animals. These two types of reduced animal model can be combined to formulate only equations corresponding to phenotyped parents of phenotyped progeny.

37 Sibs: an R toolkit for computation of relatedness measures using large pedigrees

Without appropriate relationships present in a given population, identifying dominance effects in the expression of desirable traits is challenging. Including non-additive effects is desirable to increase accuracy of breeding values. There is no current user-friendly tool package to investigate genetic relatedness in large pedigrees. The objective was to develop and implement efficient algorithms in R to calculate and visualize measures of relatedness (e.g., sibling and family structure, numerator relationship matrices) for large pedigrees. Comparisons to current R packages (Table 1) are also made. Functions to assign animals to families, summary of sibling counts, calculation of numerator relationship matrix (NRM), and NRM summary by groups were created, providing a comprehensive toolkit (Sibs package) not found in other packages. Pedigrees of various sizes (n = 20, 4,035, 120,000 and 132,833) were used to test functionality and compare to current packages. All runs were conducted on a Windows-based computer with an 8 GB RAM, 2.5 GHz Intel Core i7 processor. Other packages had no significant difference in runtime when constructing the NRM for small pedigrees (n = 20) compared to Sibs (0 to 0.05 s difference). However, packages such as ggroups, AGHmatrix, and pedigree were 10 to 15 min slower than Sibs for a 4,035-individual pedigree. Packages nadiv and pedigreemm competed with Sibs (0.30 to 60 s slower than Sibs), but no package besides Sibs was able to complete the 132,833-individual pedigree due to memory allocation issues in R. The nadiv package was closest with a pedigree of 120,000 individuals, but took 37 min to complete (13 min slower than Sibs). This package also provides easier input of pedigrees and is more encompassing of such relatedness measures than other packages (Table 1). Furthermore, it can provide an option to utilize other packages such as GCA for connectedness calculations when using large pedigrees.

An Algorithm for Genetic Analysis of Full-Sib Datasets with Mixed-Model Software Lacking a Numerator Relationship Matrix Function, and a Comparison with Results from a Dedicated Genetic Software Package

Research Highlights: An algorithm is presented that allows for the analysis of full-sib genetic datasets using generalized mixed-model software programs. The algorithm produces variance component estimates, genetic parameter estimates, and Best Linear Unbiased Prediction (BLUP) solutions for genetic values that are, for all practical purposes, identical to those produced by dedicated genetic software packages. Background and Objectives: The objective of this manuscript is to demonstrate an approach with a simulated full-sib dataset representing a typical forest tree breeding population (40 parents, 80 full-sib crosses, 4 tests, and 6000 trees) using two widely available mixed-model packages. Materials and Methods: The algorithm involves artificially doubling the dataset, so that each observation is in the dataset twice, once with the original female and male parent identification, and once with the female and male parent identities switched. Five linear models were examined: two models using a dedicated genetic software program (ASREML) with the capacity to specify A or other pedigree-related functions, and three models with the doubled dataset and a parent (or sire) linear model (ASREML, SAS Proc Mixed, and R lme4). Results: The variance components, genetic parameters, and BLUPs of the parental breeding values, progeny breeding values, and full-sib family-specific combining abilities were compared. Genetic parameter estimates were essentially the same across all the analyses (e.g., the heritability ranged from h2 = 0.220 to 0.223, and the proportion of dominance variance ranged from d2 = 0.057 to 0.058). The correlations between the BLUPs from the baseline analysis (ASREML with an individual tree model) and the doubled-dataset/parent models using SAS Proc Mixed or R lme4 were never lower than R = 0.99997. Conclusions: The algorithm can be useful for analysts who need to analyze full-sib genetic datasets and who are familiar with general-purpose statistical packages, but less familiar with or lacking access to other software.

Generalized gametic relationships for flexible analyses of parent-of-origin effects

Genomic imprinting causes alleles to influence the phenotype in a parent-of-origin-specific manner. In attempts to determine the effects of imprinted loci, gametic relationship matrices have widely been used in pedigree-based parent-of-origin analyses of population data. One drawback of this is the size of these matrices because they represent each individual by two gametic effects. Significantly fewer equations are needed if a previously published reduced imprinting model is used that relates observations from progeny without its own offspring to the transmitting abilities of their parents. This can be accomplished using a numerator relationship matrix, with only a single row and column per parent and ancestors. However, the reduced model is not applicable when the parents have records. To better handle the curse of dimensionality, we propose a combination of average gametic effects (transmitting abilities) for individuals without their own records and single gametic effects for others. The generalized gametic relationship matrix is the covariance of this mixture of genetic effects that allows for a significant reduction in the number of equations in gametic models depending on the trait, depth of pedigree, and population structure. It can also render the reduced model much more flexible by including observations from parents. Rules for setting-up its inverse from a pedigree are derived and implemented on an open-source program. The application of the same principles to phased marker data leads to a genomic version of the generalized gametic relationships. The implementation of generalized gametic models to the ASReml package is illustrated through worked examples.

63 A comparison of clustering methods for cross-validation of genomic predictors when training on phenotypes or deregressed Estimated Breeding Values

The objective of the study was to evaluate the impact of clustering methods for cross-validation on the accuracy of prediction of molecular breeding values (MBV) in Red Angus cattle (n = 9,763) and in simulation. Individuals were clustered using seven methods [k-means, k-medoids, principal component analysis on the numerator relationship matrix (A) and identical-by-state genomic matrix (G) as data and covariance matrices, and random] and two response variables [deregressed Estimated Breeding Values (DEBV) and adjusted phenotypes]. Genotypes were imputed to a 50K reference panel. Using cross-validation and a Bayes C model, MBV were estimated for traits including birth weight (BWT), marbling (MARB), rib-eye area (REA), and yearling weight (YWT) for DEBV and BWT, YWT, and ultrasonically measured intramuscular fat percentage and rib eye area for adjusted phenotypes. A bivariate animal model was used to estimate prediction accuracies calculated using the genetic correlation between estimated MBV and the associated response variable. To quantify the difference between true and estimated accuracies, a simulation mimicking a cattle population was replicated five times. The same clustering methods were used as with the Red Angus data with the addition of forward validation and two genotyping methods (random selection and selection of the top 25% of animals). Predicted accuracies were estimated similarly and true accuracies were estimated using the residual correlation of a bivariate model using MBV and true breeding values (TBV). The Rand index was used to quantify the similarity between clustering methods, showing relationship-based clusters were clearly different from random clusters. In simulation, random genotyping led to higher estimated accuracies than selection of top individuals; however, estimated accuracies over predicted true accuracies with random genotyping but under predicted true accuracies with the selection of top individuals. When forward validation was evaluated within simulation, results suggested DEBV led to less biased estimates of MBV accuracy.

Accuracy of genomic selection predictions for hip height in Brahman cattle using different relationship matrices

The objective of this work was to evaluate the effects of genomic information on the genetic evaluation of hip height in Brahman cattle using different matrices built from genomic and pedigree data. Hip height measurements from 1,695 animals, genotyped with high-density SNP chip or imputed from 50 K high-density SNP chip, were used. The numerator relationship matrix (NRM) was compared with the H matrix, which incorporated the NRM and genomic relationship (G) matrix simultaneously. The genotypes were used to estimate three versions of G: observed allele frequency (HGOF), average minor allele frequency (HGMF), and frequency of 0.5 for all markers (HG50). For matrix comparisons, animal data were either used in full or divided into calibration (80% older animals) and validation (20% younger animals) datasets. The accuracy values for the NRM, HGOF, and HG50 were 0.776, 0.813, and 0.594, respectively. The NRM and HGOF showed similar minor variances for diagonal and off-diagonal elements, as well as for estimated breeding values. The use of genomic information resulted in relationship estimates similar to those obtained based on pedigree; however, HGOF is the best option for estimating the genomic relationship matrix and results in a higher prediction accuracy. The ranking of the top 20% animals was very similar for all matrices, but the ranking within them varies depending on the method used.

Effect of selection on bias and accuracy in genomic prediction of breeding values

Reference populations for genomic selection (GS) usually involve highly selected individuals, which may result in biased prediction of estimated genomic breeding values (GEBV). In the present study, bias and accuracy of GEBV were explored for various genetic models and prediction methods when using selected individuals for a reference. Data were simulated for an animal breeding program to compare Best Linear Unbiased Prediction of breeding values using pedigree based relationships (PBLUP), genomic relationships for genotyped animals only (GBLUP) and a Single Step approach (SSGBLUP), where information on genotyped individuals was used to infer a matrix H with relationships among all available genotyped and non-genotyped individuals that were linked through pedigree. In SSGBLUP, various weights (α=0.95, 0.80, 0.50) for the genomic relationship matrix (G) relative to the numerator relationship matrix (A) were applied to construct H and in another version (SSGBLUP_F), inbreeding was accounted for while computing A-1. With GBLUP, accuracy of GEBV prediction increased linearly with an increase in the number of animals selected in reference. For the scenario with no-selection and random mating (RR) prediction was unbiased. For GBLUP, lower accuracy and bias observed in the scenarios with selection and random mating (SR) or selection and positive assortative mating (SA), in which prediction bias increased when a smaller and highly selected proportion genotyped. Bias disappeared when all individuals were genotyped. SSGBLUP_F showed higher accuracy compared to GBLUP and bias of prediction was negligible even with selective genotyping. However, PBLUP and SSGBLUP showed bias in SA owing to not fully accounting for allele frequency changes because of selection of quantitative trait loci (QTL) with larger effects and also due to high inbreeding rate. In genetic models with fewer QTL but each with larger effect, predictions were less accurate and more biased for selection scenarios. Results suggest that prediction accuracy and bias is affected by the genetic architecture of the trait. Selective genotyping lead to significant bias in GEBV prediction. SSGBLUP with appropriate scaling of A and G matrices can provide accurate and less biased prediction but scaling requires careful consideration in populations under selection and with high levels of inbreeding.