A computationally efficient method for approximating reliabilities in large-scale single-step genomic prediction.

BACKGROUND: In this study, computationally efficient methods to approximate the reliabilities of genomic estimated breeding values (GEBV) in a single-step genomic prediction model including a residual polygenic (RPG) effect are described. In order to calculate the reliabilities of the genotyped animals, a single nucleotide polymorphism best linear unbiased prediction (SNPBLUP) or a genomic BLUP (GBLUP), was used, where two alternatives to account for the RPG effect were tested. In the direct approach, the genomic model included the RPG effect, while in the blended method, it did not but an index was used to weight the genomic and pedigree-based BLUP (PBLUP) reliabilities. In order to calculate the single-step GBLUP reliabilities for the breeding values for the non-genotyped animals, a simplified weighted-PBLUP model that included a general mean and additive genetic effects with weights accounting for the non-genomic and genomic information was used. We compared five schemes for the weights. Two datasets, i.e., a small (Data 1) one and a large (Data 2) one were used. RESULTS: For the genotyped animals in Data 1, correlations between approximate reliabilities using the blended method and exact reliabilities ranged from 0.993 to 0.996 across three lactations. The slopes observed by regressing the reliabilities of GEBV from the exact method on those from the blended method were 1.0 for all three lactations. For Data 2, the correlations and slopes ranged, respectively, from 0.980 to 0.986 and from 0.91 to 0.96, and for the non-genotyped animals in Data 1, they ranged, respectively, from 0.987 to 0.994 and from 0.987 to 1, which indicate that the approximations were in line with the exact results. The best approach achieved correlations of 0.992 to 0.994 across lactations. CONCLUSIONS: Our results demonstrate that the approximated reliabilities calculated using our proposed approach are in good agreement with the exact reliabilities. The blended method for the genotyped animals is computationally more feasible than the direct method when RPG effects are included, particularly for large-scale datasets. The approach can serve as an effective strategy to estimate the reliabilities of GEBV in large-scale single-step genomic predictions.

Invited review: Reliability computation from the animal model era to the single-step genomic model era.

The calculation of exact reliabilities involving the inversion of mixed model equations poses a heavy computational challenge when the system of equations is large. This has prompted the development of different approximation methods. We give an overview of the various methods and computational approaches in calculating reliability from the era before the animal model to the era of single-step genomic models. The different methods are discussed in terms of modeling, development, and applicability in large dairy cattle populations. The paper also describes the problems faced in reliability computation. Many details dispersed throughout the literature are presented in this paper. It is clear that a universal solution applicable to every model and input data may not be possible, but we point out several efficient and accurate algorithms developed recently for a variety of very large genomic evaluations.

Mid-infrared spectroscopic analysis of raw milk to predict the blood nonesterified fatty acid concentrations in dairy cows.

In high-yielding dairy cattle, severe postpartum negative energy balance is often associated with metabolic and infectious disorders that negatively affect production, fertility, and welfare. Mobilization of adipose tissue associated with negative energy balance is reflected through an increased level of nonesterified fatty acids (NEFA) in the blood plasma. Earlier, identification of negative energy balance through detection of increased blood plasma NEFA concentration required laborious and stressful blood sampling. More recently, attempts have been made to predict blood NEFA concentration from milk samples. In this study, we aimed to develop and validate a model to predict blood plasma NEFA concentration using the milk mid-infrared (MIR) spectra that are routinely measured in the context of milk recording. To this end, blood plasma and milk samples were collected in wk 2, 3, and 20 postpartum for 192 lactations in 3 herds. The blood plasma samples were taken in the morning, and representative milk samples were collected during the morning and evening milk sessions on the same day. To predict plasma NEFA concentration from the milk MIR spectra, partial least squares regression models were trained on part of the observations from the first herd. The models were then thoroughly validated on all other observations of the first herd and on the observations of the 2 independent herds to explore their robustness and wide applicability. The final model could accurately predict blood plasma NEFA concentrations <0.6 mmol/L with a root mean square error of prediction of <0.143 mmol/L. However, for blood plasma with >1.2 mmol/L NEFA, the model clearly underestimated the true level. Additionally, we found that morning blood plasma NEFA levels were predicted with significantly higher accuracy using MIR spectra of evening milk samples compared with MIR spectra of morning samples, with root mean square error of prediction values of, respectively, 0.182 and 0.197 mmol/L, and R2 values of 0.613 and 0.502. These results suggest a time delay between variations in blood plasma NEFA and related milk biomarkers. Based on the MIR spectra of evening milk samples, cows at risk for negative energy status, indicated by detrimental morning blood plasma NEFA levels (>0.6 mmol/L), could be identified with a sensitivity and specificity of, respectively, 0.831 and 0.800. As this model can be applied to millions of historical and future milk MIR spectra, it opens an opportunity for regular metabolic screening and improved resilience phenotyping.

Genetic parameters for female fertility in Nordic Holstein and Red Cattle dairy breeds.

Genetic evaluation of female fertility in Danish, Finnish, and Swedish dairy cows was updated in 2015 to multiple-trait animal model evaluation, where heifer and cow fertility up to third parity are considered as separate traits. A model for conception rate was also developed, which required variance component estimation for Nordic Holstein and Nordic Red Dairy Cattle. We used a multiple-trait multiple-lactation sire model to determine variance components for interval from calving to first insemination, length of service period, and conception rate. Monte Carlo Expectation Maximization REML allowed estimation of all 11 traits simultaneously. Study data were sampled from Swedish Holstein (n = 140,040) and Red Dairy Cattle (n = 101,315) heifers and cows. Conception rate observations are binomial observations with various numbers of failures preceding an observation of success. Using a simulation study, we confirmed that including a service number effect into the conception rate model allowed us to model the change in expectation of successful AI with increasing number of services. Heifers outperformed cows in all fertility traits according to the phenotypic means in the records. Heritabilities for the traits varied from 3 to 7% for interval from calving to first insemination, from 1 to 5% for length of service period, and from 1 to 3% for conception rate. Genetic correlations within traits (i.e., between parities) were favorable, ranging from moderate to high; genetic correlations between heifer and cow traits were lower than between cow traits in different parities. Lowest genetic correlations between traits were for interval from calving to first insemination and conception rate, intermediate for interval from calving to first insemination and length of service period, and highest for length of service period and conception rate. The variance components estimated in this study have been used in Nordic fertility breeding value evaluations since 2016.

Across-country test-day model evaluations for Holstein, Nordic Red Cattle, and Jersey.

Three random regression models were developed for routine genetic evaluation of Danish, Finnish, and Swedish dairy cattle. Data included over 169 million test-day records with milk, protein, and fat yield observations from over 8.7 million dairy cows of all breeds. Variance component analyses showed significant differences in estimates between Holstein, Nordic Red Cattle, and Jersey, but only small to moderate differences within a breed across countries. The obtained variance component estimates were used to build, for each breed, their own set of covariance functions. The covariance functions describe the animal effects on milk, protein, and fat yields of the first 3 lactations as 9 different traits, assuming the same heritabilities and a genetic correlation of unity across countries. Only 15, 27, and 7 eigenfunctions with the largest eigenvalues were used to describe additive genetic animal effects and nonhereditary animal effects across lactations and within later lactations, respectively. These reduced-rank covariance functions explained 99.0 to 99.9% of the original variances but reduced the number of animal equations to be solved by 44%. Moderate rank reduction for nonhereditary animal effects and use of one-third-smaller measurement error correlations than obtained from variance component estimation made the models more robust against extreme observations. Estimation of the genetic levels of the countries' subpopulations within a breed was found sensitive to the way the breed effects were modeled, especially for the genetically heterogeneous Nordic Red Cattle. Means to ensure that only additive genetic effects entered the estimated breeding values were to describe the crossbreeding effects by fixed and random cofactors and the calving age effect by an age × breed proportion interaction, and to model phantom parent groups as random effects. To ensure that genetic variances were the same across the 3 countries in breeding value estimation, as suggested by the variance component estimates, the applied multiplicative heterogeneous variance adjustment method had to be tailored using country-specific reference measurement error variances. Results showed the feasibility of across-country genetic evaluation of cows and sires based on original test-day phenotypes. Nevertheless, applying a thorough model validation procedure is essential throughout the model building process to obtain reliable breeding values.

Employing a Monte Carlo algorithm in Newton-type methods for restricted maximum likelihood estimation of genetic parameters.

Estimation of variance components by Monte Carlo (MC) expectation maximization (EM) restricted maximum likelihood (REML) is computationally efficient for large data sets and complex linear mixed effects models. However, efficiency may be lost due to the need for a large number of iterations of the EM algorithm. To decrease the computing time we explored the use of faster converging Newton-type algorithms within MC REML implementations. The implemented algorithms were: MC Newton-Raphson (NR), where the information matrix was generated via sampling; MC average information(AI), where the information was computed as an average of observed and expected information; and MC Broyden's method, where the zero of the gradient was searched using a quasi-Newton-type algorithm. Performance of these algorithms was evaluated using simulated data. The final estimates were in good agreement with corresponding analytical ones. MC NR REML and MC AI REML enhanced convergence compared to MC EM REML and gave standard errors for the estimates as a by-product. MC NR REML required a larger number of MC samples, while each MC AI REML iteration demanded extra solving of mixed model equations by the number of parameters to be estimated. MC Broyden's method required the largest number of MC samples with our small data and did not give standard errors for the parameters directly. We studied the performance of three different convergence criteria for the MC AI REML algorithm. Our results indicate the importance of defining a suitable convergence criterion and critical value in order to obtain an efficient Newton-type method utilizing a MC algorithm. Overall, use of a MC algorithm with Newton-type methods proved feasible and the results encourage testing of these methods with different kinds of large-scale problem settings.

Principal component and factor analytic models in international sire evaluation.

BACKGROUND: Interbull is a non-profit organization that provides internationally comparable breeding values for globalized dairy cattle breeding programmes. Due to different trait definitions and models for genetic evaluation between countries, each biological trait is treated as a different trait in each of the participating countries. This yields a genetic covariance matrix of dimension equal to the number of countries which typically involves high genetic correlations between countries. This gives rise to several problems such as over-parameterized models and increased sampling variances, if genetic (co)variance matrices are considered to be unstructured. METHODS: Principal component (PC) and factor analytic (FA) models allow highly parsimonious representations of the (co)variance matrix compared to the standard multi-trait model and have, therefore, attracted considerable interest for their potential to ease the burden of the estimation process for multiple-trait across country evaluation (MACE). This study evaluated the utility of PC and FA models to estimate variance components and to predict breeding values for MACE for protein yield. This was tested using a dataset comprising Holstein bull evaluations obtained in 2007 from 25 countries. RESULTS: In total, 19 principal components or nine factors were needed to explain the genetic variation in the test dataset. Estimates of the genetic parameters under the optimal fit were almost identical for the two approaches. Furthermore, the results were in a good agreement with those obtained from the full rank model and with those provided by Interbull. The estimation time was shortest for models fitting the optimal number of parameters and prolonged when under- or over-parameterized models were applied. Correlations between estimated breeding values (EBV) from the PC19 and PC25 were unity. With few exceptions, correlations between EBV obtained using FA and PC approaches under the optimal fit were ≥ 0.99. For both approaches, EBV correlations decreased when the optimal model and models fitting too few parameters were compared. CONCLUSIONS: Genetic parameters from the PC and FA approaches were very similar when the optimal number of principal components or factors was fitted. Over-fitting increased estimation time and standard errors of the estimates but did not affect the estimates of genetic correlations or the predictions of breeding values, whereas fitting too few parameters affected bull rankings in different countries.

Principal component approach in variance component estimation for international sire evaluation.

BACKGROUND: The dairy cattle breeding industry is a highly globalized business, which needs internationally comparable and reliable breeding values of sires. The international Bull Evaluation Service, Interbull, was established in 1983 to respond to this need. Currently, Interbull performs multiple-trait across country evaluations (MACE) for several traits and breeds in dairy cattle and provides international breeding values to its member countries. Estimating parameters for MACE is challenging since the structure of datasets and conventional use of multiple-trait models easily result in over-parameterized genetic covariance matrices. The number of parameters to be estimated can be reduced by taking into account only the leading principal components of the traits considered. For MACE, this is readily implemented in a random regression model. METHODS: This article compares two principal component approaches to estimate variance components for MACE using real datasets. The methods tested were a REML approach that directly estimates the genetic principal components (direct PC) and the so-called bottom-up REML approach (bottom-up PC), in which traits are sequentially added to the analysis and the statistically significant genetic principal components are retained. Furthermore, this article evaluates the utility of the bottom-up PC approach to determine the appropriate rank of the (co)variance matrix. RESULTS: Our study demonstrates the usefulness of both approaches and shows that they can be applied to large multi-country models considering all concerned countries simultaneously. These strategies can thus replace the current practice of estimating the covariance components required through a series of analyses involving selected subsets of traits. Our results support the importance of using the appropriate rank in the genetic (co)variance matrix. Using too low a rank resulted in biased parameter estimates, whereas too high a rank did not result in bias, but increased standard errors of the estimates and notably the computing time. CONCLUSIONS: In terms of estimation's accuracy, both principal component approaches performed equally well and permitted the use of more parsimonious models through random regression MACE. The advantage of the bottom-up PC approach is that it does not need any previous knowledge on the rank. However, with a predetermined rank, the direct PC approach needs less computing time than the bottom-up PC.