Genetic connectedness between management units quantifies the extent to which estimated breeding values (EBV) can be accurately compared across management units or subpopulations. When there is a sufficient level of genetic connectedness, best linear unbiased prediction (BLUP) of EBV enables fair comparisons across these units. Genotype-by-environment (G × E) interaction can alter an individual's expression of genetic potential across environments, thereby compromising the accuracy of selection decisions. The aim of this study was to investigate the extent of genetic connectedness and G × E interaction for a maternal pig line between countries (France, China, and Thailand) for five traits: age at 100 kg (AGE), number of functional teats (NFT), birth weight (BW), backfat thickness at 100 kg (BFT), and number of piglets born alive (NBA). We observed high connectedness across the three countries, as reflected by high coefficient of determination (CD) values (ranging from 0.61 to 0.96) across all traits and time periods. For all traits, the pair France-China had higher CD values (0.80 on average) than the pair France-Thailand (0.70 on average) between 2023 and 2024. These results were in line with Fst estimates, which were 0.015 between France and China, 0.006 between France and Thailand, and 0.020 between China and Thailand, indicating that the populations in the three countries are genetically close. Genetic correlations among the countries ranged from 0.70 to 0.79 for AGE, 0.95 to 0.98 for NFT, 0.53 to 0.83 for BW, 0.81 to 0.97 for NBA, and 0.77 for BFT. Therefore, there was no evidence of G × E interaction except for BW between France and China (0.67), and BW between China and Thailand (0.53). In conclusion, we found high genetic connectedness across countries, demonstrating that animals from different countries share sufficient genetic links to enable accurate and fair across-country genetic evaluations and comparison of EBVs. No biologically significant G × E interaction was detected (except for BW), which suggests that there is no substantial reranking of animals between the countries. Therefore, a single breeding program across these countries is appropriate and effective.
Likelihood-based inferences such as variance components estimation and hypothesis testing need logarithms of the determinant (log-determinant) of high dimensional matrices. Calculating the log-determinant is memory and time-consuming, making it impossible to perform likelihood-based inferences for large datasets. We presented a method for approximating the log-determinant of positive semi-definite matrices based on repeated matrix–vector products and complex calculus. We tested the approximation of the log-determinant in beef and dairy cattle, chicken, and pig datasets including single and multiple-trait models. Average absolute relative differences between the approximated and exact log-determinant were around 10–3. The approximation was between 2 and 500 times faster than the exact calculation for medium and large matrices. We compared the restricted likelihood with (approximated) and without (exact) the approximation of the log-determinant for different values of heritability for a single-trait model. We also compared estimated variance components using exact expectation–maximization (EM) and average information (AI) REML algorithms, against two derivative-free approaches using the restricted likelihood calculated with the log-determinant approximation. The approximated and exact restricted likelihood showed maxima at the same heritability value. Derivative-free estimation of variance components with the approximated log-determinant converged to the same values as EM and AI-REML. The proposed approach is feasible to apply to any data size. The method presented in this study allows to approximate the log-determinant of positive semi-definite matrices and, therefore, the likelihood for datasets of any size. This opens the possibility of performing likelihood-based inferences for large datasets in animal and plant breeding.
Abstract Background The magnitude of inbreeding depression depends on the recessive burden of the individual, which can be traced back to the hidden (recessive) inbreeding load among ancestors. However, these ancestors carry different alleles at potentially deleterious loci and therefore there is individual variability of this inbreeding load. Estimation of the additive genetic value for inbreeding load is possible using a decomposition of inbreeding in partial inbreeding components due to ancestors. Both the magnitude of variation in partial inbreeding components and the additive genetic variance of inbreeding loads are largely unknown. Our study had three objectives. First, based on substitution effect under non-random matings, we showed analytically that inbreeding load of an ancestor can be expressed as an additive genetic effect. Second, we analysed the structure of individual inbreeding by examining the contributions of specific ancestors/founders using the concept of partial inbreeding coefficients in three French dairy sheep populations (Basco-Béarnaise, Manech Tête Noire and Manech Tête Rousse). Third, we included these coefficients in a mixed model as random regression covariates, to predict genetic variance and breeding values of the inbreeding load for milk yield in the same breeds. Results Pedigrees included 190,276, 166,028 and 633,655 animals of Basco-Béarnaise, Manech Tête Noire and Manech Tête Rousse, respectively, born between 1985 and 2021. A fraction of 99.1% of the partial inbreeding coefficients were lower than 0.01 in all breeds, meaning that in practice inbreeding occurs in pedigree loops that span several generations backwards. Less than 5% ancestors generate inbreeding, because mating is essentially between unrelated individuals. Inbreeding load estimations involved 658,731, 541,180 and 2,168,454 records of yearly milk yield from 178,123, 151,863 and 596,586 females in Basco-Béarnaise, Manech Tête Noire and Manech Tête Rousse, respectively. Adding the inbreeding load effect to the model improved the fitting (values of the statistic Likelihood Ratio Test between 132 and 383) for milk yield in the three breeds. The inbreeding load variances were equal to 11,804 and 9435 L squared of milk yield for a fully inbred (100%) descendant in Manech Tête Noire and Manech Tête Rousse. In Basco-Béarnaise, the estimate of the inbreeding load variance (11,804) was not significantly different from zero. The correlations between (direct effect) additive genetic and inbreeding load effects were − 0.09, − 0.08 and − 0.12 in Basco-Béarnaise, Manech Tête Noire and Manech Tête Rousse. Conclusions The decomposition of inbreeding in partial coefficients in these populations shows that inbreeding is mostly due to several small contributions of ancestors (lower than 0.001) going back several generations (5 to 7 generations), which is according to the policy of avoiding close matings. There is variation of inbreeding load among animals, although its magnitude does not seem enough to warrant selection based on this criterion.
Evaluating the (restricted) log-likelihood of statistical models is widely used when estimating variance components and doing hypothesis testing in animal breeding and genetics. However, due to memory usage and extremely long computational times, performing likelihood evaluations is infeasible for large datasets. A major computational bottleneck in evaluating the likelihood is calculating the logarithm of the determinant (log-determinant) of high-dimensional matrices. This study aimed to present a method to approximate the log-determinant of positive semi-definite matrices, making likelihood evaluations feasible for very large datasets. Our method uses repeated matrix-vector products, notions of complex calculus, and contour integrals. We tested the approximation on beef and dairy cattle, chicken, and pig datasets, including both single and multiple-trait models. The average relative difference between approximated and exact log-determinant values was around 10-3. The approximation was 2 to 500 times faster than the exact calculation. Differences in computing time between the exact and approximated log-determinant increased with dataset size. For the largest dataset, with included 9,408,140 dairy cattle, 34,506 genotyped individuals, and a coefficient matrix rank of 19,170,955 from the mixed model equations, the exact calculation took 19 days, and the approximation took one hour. As an application example, we calculated the restricted likelihood and estimated variance components with expectation-maximization (EM), average information (AI) REML, and two derivative-free (DF) methods using the approximation of the log-determinant for a single-trait model. The exact and approximated log-likelihood showed maximums at the same heritability value among the REML methods. Our proposed approach lifts the computing time and memory limitations for likelihood evaluations in large datasets. The method allows approximating the log-determinant of positive semi-definite matrices, opening the possibility of performing likelihood evaluations for virtually any animal breeding dataset.
AbstractMetafounders are a useful concept to characterize relationships within and across populations, and to help genetic evaluations because they help modelling the means and variances of unknown base population animals. Current definitions of metafounder relationships are sensitive to the choice of reference alleles and have not been compared to their counterparts in population genetics—namely, heterozygosities, FST coefficients, and genetic distances. We redefine the relationships across populations with an arbitrary base of a maximum heterozygosity population in Hardy–Weinberg equilibrium. Then, the relationship between or within populations is a cross-product of the form $${\Gamma }_{\left(b,{b}^{\prime}\right)}=\left(\frac{2}{n}\right)\left(2{\mathbf{p}}_{b}-\mathbf{1}\right)\left(2{\mathbf{p}}_{{b}^{\prime}}-\mathbf{1}\right)^{\prime}$$ Γ b , b ′ = 2 n 2 p b - 1 2 p b ′ - 1 ′ with $$\mathbf{p}$$ p being vectors of allele frequencies at $$n$$ n markers in populations $$b$$ b and $$b^{\prime}$$ b ′ . This is simply the genomic relationship of two pseudo-individuals whose genotypes are equal to twice the allele frequencies. We also show that this coding is invariant to the choice of reference alleles. In addition, standard population genetics metrics (inbreeding coefficients of various forms; FST differentiation coefficients; segregation variance; and Nei’s genetic distance) can be obtained from elements of matrix $${\varvec{\Gamma}}$$ Γ .
