Average information residual maximum likelihood in practice

Gilmour, Thompson, and Cullis (Biometrics, 1995, 51, 1440) presented the average information residual maximum likelihood (REML) algorithm for efficient variance parameter estimation in the linear mixed model. That paper dealt specifically with traditional variance component models, but the algorithm was quickly applied to more general models and implemented in several REML packages including ASReml (Gilmour et al., Biometrics, 2015, 51, 1440). This paper outlines the theory with respect to these more general models, describes the main issues encountered in fitting these models and how they have been addressed in the ASReml software. The issues covered are the basics steps in the implementation of the algorithm, keeping parameters within the parameter space, maximizing sparsity, avoiding issues associated with unstructured variance matrices by using the factor‐analytic structure and handling singularities in marker‐based relationship matrices and current work.     

Estimation of heritability and genetic correlation for behavioural responses by Gibbs sampling in the Thoroughbred racehorse

Genetic variation of the behaviour of racehorses is one of the major concerns for racehorse breeders. In this study, the heritabilities of behavioural responses to the inspections of conjunctiva, auscultation and blood sampling and the genetic correlations among them were estimated in the Thoroughbred racehorse. The estimation was done with Bayesian analysis with Gibbs sampling based on the univariate or bivariate threshold animal models. The behavioural responses were scored with four categories at the first entrance quarantine in Miho Training Center of Japan Racing Association from 1993 to 1995. The behavioural responses were treated as categorical or binary traits, with both showing similar results. The estimated heritabilities were in the range of 0.23–0.28, suggesting a genetic component in the variation on these traits. The estimated genetic correlations among the traits were very high (approximately 0.9), suggesting that these behavioural responses may be measures of the same trait. Because of the high genetic correlations, repeatability threshold model was applied assuming the responses to be a genetically identical trait measured with three different tests. The estimated heritabilities (approximately 0.23) were at the lower bound of the former estimates. The revealed high repeatabilities (0.97–0.98) suggest a strong contribution of the individual temperament on the behaviour of racehorses.     

Estimation of genetic parameters and response to selection for a continuous trait subject to culling before testing

The consequences of assuming a zero environmental covariance between a binary trait 'test-status' and a continuous trait on the estimates of genetic parameters by restricted maximum likelihood and Gibbs sampling and on response from genetic selection when the true environmental covariance deviates from zero were studied. Data were simulated for two traits (one that culling was based on and a continuous trait) using the following true parameters, on the underlying scale: h2 = 0.4; r(A) = 0.5; r(E) = 0.5, 0.0 or -0.5. The selection on the continuous trait was applied to five subsequent generations where 25 sires and 500 dams produced 1500 offspring per generation. Mass selection was applied in the analysis of the effect on estimation of genetic parameters. Estimated breeding values were used in the study of the effect of genetic selection on response and accuracy. The culling frequency was either 0.5 or 0.8 within each generation. Each of 10 replicates included 7500 records on 'test-status' and 9600 animals in the pedigree file. Results from bivariate analysis showed unbiased estimates of variance components and genetic parameters when true r(E) = 0.0. For r(E) = 0.5, variance components (13-19% bias) and especially (50-80%) were underestimated for the continuous trait, while heritability estimates were unbiased. For r(E) = -0.5, heritability estimates of test-status were unbiased, while genetic variance and heritability of the continuous trait together with were overestimated (25-50%). The bias was larger for the higher culling frequency. Culling always reduced genetic progress from selection, but the genetic progress was found to be robust to the use of wrong parameter values of the true environmental correlation between test-status and the continuous trait. Use of a bivariate linear-linear model reduced bias in genetic evaluations, when data were subject to culling.     

Estimation of genetic parameters for hip dysplasia in Czech Labrador Retrievers

The objective of this study was to estimate the genetic parameters, genetic trends and breeding values using linear model (LM) and threshold model (TM) for the development of hip dysplasia (HD) in Labrador Retrievers in the Czech Republic (n = 3151). The right and left hip joints were evaluated separately using the Fédération Cynologique Internationale scoring system. Four linear and four TMs were tested for the correct estimation of genetic parameters. All the tested models utilized fixed effects of sex, assessor, year of birth, regression of age at evaluation, random direct genetic effects and the effect of the animals' permanent environments. The models differed in the inclusion of the following effects: fixed effects of regression of inbreeding coefficient, random maternal effect and random effect of the maternal permanent environment. Compared to the TM, the LM provided lower coefficients of direct (0.25-0.29 versus 0.26-0.35) and maternal heritability (0.01-0.02 versus 0.03-0.05), repeatability (0.76-0.77 versus 0.78-0.83) and of the correlation between direct and maternal effects (-0.55 to -0.21 versus -0.80 to -0.27). In the tested models, no statistical significance was found for fixed regression of inbreeding coefficients or for the random effect of the permanent maternal environment. In spite of the similarity of the LM and TM results, the TM is recommended as the more suitable model for estimating genetic parameters and subsequent breeding values for HD in Labrador Retrievers in the Czech Republic.     

An alternative derivation method of mixed model equations from best linear unbiased prediction (BLUP) and restricted BLUP of breeding values not using maximum likelihood

Two mixed model equations (MME) for best linear unbiased prediction (BLUP) of breeding values and for restricted BLUP of breeding values were derived by maximum likelihood from the joint normal probability distribution of the observations and breeding values. As a result, MME is actually more general than maximum likelihood because we can prove that each set of solutions of MME are identical to BLUP and restricted BLUP of breeding values and then it does not depend on normality. In the present study, the author shows deriving directly each MME from BLUP and restricted BLUP equations for breeding values without assuming the joint normal distribution of the data and random effects. However, if we cannot assume the multivariate normal density distribution of the estimated aggregate breeding value and each breeding value for selected traits, the response to selection by restricted BLUP may deviate from the expected values.     

Comparison of repeatability and multiple trait threshold models for litter size in sheep using observed and simulated data in Bayesian analyses

Bayesian analyses were used to estimate genetic parameters on 5580 records of litter size in the first four parities from 1758 Mule ewes. To examine the appropriateness of fitting repeatability (RM) or multiple trait threshold models (MTM) to litter size of different parities, both models were used to estimate genetic parameters on the observed data and were thereafter compared in a simulation study. Posterior means of the heritabilities of litter size in different parities using a MTM ranged from 0.12 to 0.18 and were higher than the heritability based on the RM (0.08). Posterior means of the genetic correlations between litter sizes of different parities were positive and ranged from 0.24 to 0.71. Data sets were simulated based on the same pedigree structure and genetic parameters of the Mule ewe population obtained from both models. The simulation showed that the relative loss in accuracy and increase in mean squared error (MSE) was substantially higher when using the RM, given that the parameters estimated from the observed data using the opposite model are the true parameters. In contrast, Bayesian information criterion (BIC) selected the RM as most appropriate model given the data because of substantial penalty for the higher number of parameters to be estimated in the MTM model. In conclusion, when the relative change in accuracy and MSE is of main interest for estimation of breeding values of litter size of different parities, the MTM is recommended for the given population. When reduction in risk of using the wrong model is the main aim, the BIC suggest that the RM is the most appropriate model.     

Effects of correcting missing daily feed intake values on the genetic parameters and estimated breeding values for feeding traits in pigs

Daily feed intake (DFI) is an important consideration for improving feed efficiency, but measurements using electronic feeder systems contain many missing and incorrect values. Therefore, we evaluated three methods for correcting missing DFI data (quadratic, orthogonal polynomial, and locally weighted (Loess) regression equations) and assessed the effects of these missing values on the genetic parameters and the estimated breeding values (EBV) for feeding traits. DFI records were obtained from 1622 Duroc pigs, comprising 902 individuals without missing DFI and 720 individuals with missing DFI. The Loess equation was the most suitable method for correcting the missing DFI values in 5–50% randomly deleted datasets among the three equations. Both variance components and heritability for the average DFI (ADFI) did not change because of the missing DFI proportion and Loess correction. In terms of rank correlation and information criteria, Loess correction improved the accuracy of EBV for ADFI compared to randomly deleted cases. These findings indicate that the Loess equation is useful for correcting missing DFI values for individual pigs and that the correction of missing DFI values could be effective for the estimation of breeding values and genetic improvement using EBV for feeding traits.     

Estimation of breeding values from large‐sized routine carcass data in Japanese Black cattle using Bayesian analysis

Volumes of official data sets have been increasing rapidly in the genetic evaluation using the Japanese Black routine carcass field data. Therefore, an alternative approach with smaller memory requirement to the current one using the restricted maximum likelihood (REML) and the empirical best linear unbiased prediction (EBLUP) is desired. This study applied a Bayesian analysis using Gibbs sampling (GS) to a large data set of the routine carcass field data and practically verified its validity in the estimation of breeding values. A Bayesian analysis like REML‐EBLUP was implemented, and the posterior means were calculated using every 10th sample from 90 000 of samples after 10 000 samples discarded. Moment and rank correlations between breeding values estimated by GS and REML‐EBLUP were very close to one, and the linear regression coefficients and the intercepts of the GS on the REML‐EBLUP estimates were substantially one and zero, respectively, showing a very good agreement between breeding value estimation by the current GS and the REML‐EBLUP. The current GS required only one‐sixth of the memory space with REML‐EBLUP. It is confirmed that the current GS approach with relatively small memory requirement is valid as a genetic evaluation procedure using large routine carcass data.     

Estimation of accuracies and expected genetic change from selection for selection indexes that use multiple‐trait predictions of breeding values

Procedures are described for estimating selection index accuracies for individual animals and expected genetic change from selection for the general case where indexes of EBVs predict an aggregate breeding objective of traits that may or may not have been measured. Index accuracies for the breeding objective are shown to take an important general form, being able to be expressed as the product of the accuracy of the index function of true breeding values and the accuracy with which that function predicts the breeding objective. When the accuracies of the individual EBVs of the index are known, prediction error variances (PEVs) and covariances (PECs) for the EBVs within animal are able to be well approximated, and index accuracies and expected genetic change from selection estimated with high accuracy. The procedures are suited to routine use in estimating index accuracies in genetic evaluation, and for providing important information, without additional modelling, on the directions in which a population will move under selection.     

Estimation of genetic parameters of feed intake and daily gain in Durocs using data from electronic swine feeders

Genetic parameters for daily feed intake (DFI, g/day) and daily gain (DG, g/day) were estimated using records of 1916 Duroc boars from electronic feeder stations. Management was limited and resulted in varied ranges of age and weight on test. Boars were housed in 102 pens, each equipped with one feeder, and allowed ad libitum feeding. Weekly averages of DFI and DG were used due to large variation in daily records. Six traits were defined as DFI and DG during 85–106 (period 1), 107–128 (period 2) and 129–150 days of age (period 3). A six‐trait model included age as a linear and a quadratic covariate for DFI and a linear covariate for DG with a fixed effect of year–week–pen and random effects of litter, additive genetic animal and permanent environmental animal. Variance components were estimated by a Bayesian approach using Gibbs sampling algorithm. Estimates of heritability for respective periods were 18%, 12% and 10% for DFI and 21%, 11% and 10% for DG. Genetic correlations between DFI and DG in the same period were 0.70, 0.73 and 0.32 for the respective periods. DFI and DG obtained from automatic feeders can be analysed to reveal variation across testing periods by using weekly averages when many monthly averages are incomplete.     

Genetic evaluation of calf and heifer survival in Iranian Holstein cattle using linear and threshold models

Calf and heifer survival are important traits in dairy cattle affecting profitability. This study was carried out to estimate genetic parameters of survival traits in female calves at different age periods, until nearly the first calving. Records of 49 583 female calves born during 1998 and 2009 were considered in five age periods as days 1–30, 31–180, 181–365, 366–760 and full period (day 1–760). Genetic components were estimated based on linear and threshold sire models and linear animal models. The models included both fixed effects (month of birth, dam's parity number, calving ease and twin/single) and random effects (herd‐year, genetic effect of sire or animal and residual). Rates of death were 2.21, 3.37, 1.97, 4.14 and 12.4% for the above periods, respectively. Heritability estimates were very low ranging from 0.48 to 3.04, 0.62 to 3.51 and 0.50 to 4.24% for linear sire model, animal model and threshold sire model, respectively. Rank correlations between random effects of sires obtained with linear and threshold sire models and with linear animal and sire models were 0.82–0.95 and 0.61–0.83, respectively. The estimated genetic correlations between the five different periods were moderate and only significant for 31–180 and 181–365 (r_g = 0.59), 31–180 and 366–760 (r_g = 0.52), and 181–365 and 366–760 (r_g = 0.42). The low genetic correlations in current study would suggest that survival at different periods may be affected by the same genes with different expression or by different genes. Even though the additive genetic variations of survival traits were small, it might be possible to improve these traits by traditional or genomic selection.     

Accounting for genetic architecture in single‐ and multipopulation genomic prediction using weights from genomewide association studies in pigs

We studied the effect of including GWAS results on the accuracy of single‐ and multipopulation genomic predictions. Phenotypes (backfat thickness) and genotypes of animals from two sire lines (SL1, n = 1146 and SL3, n = 1264) were used in the analyses. First, GWAS were conducted for each line and for a combined data set (both lines together) to estimate the genetic variance explained by each SNP. These estimates were used to build matrices of weights (D), which was incorporated into a GBLUP method. Single population evaluated with traditional GBLUP had accuracies of 0.30 for SL1 and 0.31 for SL3. When weights were employed in GBLUP, the accuracies for both lines increased (0.32 for SL1 and 0.34 for SL3). When a multipopulation reference set was used in GBLUP, the accuracies were higher (0.36 for SL1 and 0.32 for SL3) than in single‐population prediction. In addition, putting together the multipopulation reference set and the weights from the combined GWAS provided even higher accuracies (0.37 for SL1, and 0.34 for SL3). The use of multipopulation predictions and weights estimated from a combined GWAS increased the accuracy of genomic predictions.     

Estimation of genetic parameters for farrowing traits in purebred Landrace and Large White pigs

Genetic parameters were estimated for six reproductive traits related to farrowing events in Landrace and Large White pigs; total number born (TNB), number born alive (NBA), number stillborn (NSB), total litter weight at birth (LWB), mean litter weight at birth (MWB), and gestation length (GL). We analyzed 62,534 farrowing records for 10,637 Landrace dams and 49,817 farrowing records for 8,649 Large White dams. Estimated heritabilities of TNB, NBA, NSB, LWB, MWB, and GL by single‐trait repeatability model analyses were 0.12, 0.12, 0.08, 0.18, 0.19, and 0.29, respectively, in Landrace, and 0.12, 0.10, 0.08, 0.18, 0.16, and 0.34, respectively, in Large White. Genetic correlation between NBA and NSB was unfavorable: 0.20 in Landrace and 0.33 in Large White. Genetic correlations of GL with the other five traits were weak: from ?0.18 with NSB to ?0.03 with NBA in Landrace, and from ?0.22 with NSB to ?0.07 with NBA in Large White. LWB had a highly favorable genetic correlation with NBA (0.74 in both breeds), indicating the possibility of using LWB for the genetic improvement of NBA.     

Estimation of variance components for carcass traits in Japanese Black cattle using 50K SNP genotype data

Genomic selection using high‐density single nucleotide polymorphism (SNP) genotype data may accelerate genetic improvements in livestock animals. In this study, we attempted to estimate the variance components of six carcass traits in fattened Japanese Black steers using SNP genotype data. Six hundred and seventy‐three steers were genotyped using an Illumina Bovine SNP50 BeadChip and phenotyped for cold carcass weight, ribeye area, rib thickness, subcutaneous fat thickness, estimated yield percent and marbling score. Additive polygenic variance and the variance attributable to a set of SNPs that had statistically significant effects on the trait were estimated via Gibbs sampling with two models: (i) a model with the chosen SNPs and the additive polygenic effects; and (ii) a model with the polygenic effects alone. The proportion of the estimated variance attributable to the SNPs became higher as the number of SNP effects that fit increased. High correlations between breeding values estimated with the model containing the polygenic effect alone and those estimated by chosen SNPs were obtained. No fraction of the total genetic variance was explained by SNPs associated with the trait at P ≥ 0.1. Our results suggest that for the carcass traits of Japanese Black cattle, a maximum of half of the total additive genetic variance may be explained by SNPs between 100 several tens to several 100s.     

Genetic parameters for the prediction of abdominal fat traits using blood biochemical indicators in broilers

1. Excessive deposition of body fat, especially abdominal fat, is detrimental in chickens and the prevention of excessive fat accumulation is an important problem. The aim of this study was to identify blood biochemical indicators that could be used as criteria to select lean Yellow-feathered chicken lines.

2. Levels of blood biochemical indicators in the fed and fasted states and the abdominal fat traits were measured in 332 Guangxi Yellow chickens. In the fed state, the genetic correlations (r_g) of triglycerides and very low density lipoprotein levels were positive for the abdominal fat traits (0.47 ≤ r_g ≤ 0.67), whereas total cholesterol, high-density lipoprotein cholesterol (HDL-C) and low-density lipoprotein cholesterol (LDL-C) showed higher negative correlations with abdominal fat traits (–0.59 ≤ r_g ≤ ?0.33). Heritabilities of these blood biochemical parameters were high, varying from 0.26 to 0.60.

3. In the fasted state, HDL-C:LDL-C level was positively correlated with abdominal fat traits (0.35 ≤ r_g ≤ 0.38), but triglycerides, total cholesterol, HDL-C, LDL-C, total protein, albumin, aspartate transaminase, uric acid and creatinine levels were negatively correlated with abdominal fat traits (–0.79 ≤ r_g ≤ ?0.35). The heritabilities of these 10 blood biochemical parameters were high (0.22 ≤ h² ≤ 0.59).

4. In the fed state, optimal multiple regression models were constructed to predict abdominal fat traits by using triglycerides and LDL-C. In the fasted state, triglycerides, total cholesterol, HDL-C, LDL-C, total protein, albumin and uric acid could be used to predict abdominal fat content.

5. It was concluded that these models in both nutritional states could be used to predict abdominal fat content in Guangxi Yellow broiler chickens.     

Pedigree analysis in the Austrian Noriker draught horse: genetic diversity and the impact of breeding for coat colour on population structure

The pedigree of the current Austrian Noriker draught horse population comprising 2808 horses was traced back to the animals considered as founders of this breed. In total, the number of founders was 1991, the maximum pedigree length was 31 generations, with an average of 12.3 complete generations. Population structure in this autochthonous Austrian draught horse breed is defined by seven breeding regions (Carinthia, Lower Austria, Salzburg, Styria, Tyrol, Upper Austria and Vorarlberg) or through six coat colour groups (Bay, Black, Chestnut, Roan, Leopard, Tobiano). Average inbreeding coefficients within the breeding regions ranged from 4.5% to 5.5%; for the colour groups, the coefficients varied from 3.5% to 5.9%. Other measures of genetic variability like the effective number of founders, ancestors and founder genomes revealed a slightly different genetic background of the subpopulations. Average coancestries between and within breeding areas showed that the Salzburg population may be considered as the nucleus or original stock whereas all other subpopulations showed high relationship to horses from Salzburg. The target of draught horse breeding in the 21st century does not meet the breeding concept of maximizing genetic gains any more. Stabilizing selection takes place. In this study, we show that demographic factors as well as structure given by different coat colours helped to maintain genetic diversity in this endangered horse breed.     

Development of genetic evaluation for milk production traits of Holsteins in Japan

The procedure used for the genetic evaluation of dairy cattle in Japan has developed from a lactation sire–MGS model to a multiple‐lactation random regression test‐day animal model. Genetic evaluation of Holstein bulls in Japan began in 1989 with the use of field‐style progeny testing; dairy herd improvement program data from all over Japan were used, along with a sire and maternal grandsire model. In 1993, an animal model was introduced to estimate breeding values for yield and type traits. A random regression test‐day model was first applied in 2010. In the business of breeding dairy cattle, it is very important to users that estimated breeding values are reliable and stable among subsequent routine evaluations. With experience in the genetic evaluation of dairy cattle in Japan, Japanese researchers have found ways to improve the stability of estimated breeding values. These modifications involve changes in data editing, development of evaluation models, changes to the structures of unknown‐parent groups, awareness of the problems of predicting lactation yield from partial test‐day records, and adjustment for heterogeneity within herd variances. Here, I introduce developments in, and our experiences with, the genetic evaluation of yield traits of Holstein cattle in Japan.     

闭锁群体BLUP选择下遗传方差和近交系数的变化

采用计算机随机模拟方法模拟了在一个闭锁群体内连续对单个性状进行 1 5个世代选择的情况。选择过程中世代不重叠 ,每个世代的种畜根据动物模型最佳线性无偏预测 (BLUP)法估计的育种值进行选留 ,并在此基础上系统地比较了不同群体规模、公母比例和性状遗传力对群体遗传方差和近交系数变化的影响。结果表明 ,扩大育种群规模、增加公畜比例以及对低遗传力性状进行选择时 ,群体遗传方差降低的速度和近交系数上升的速度会更慢 ,在长期选择时可望获得更大的持续进展和适宜的近交增量     

Estimation of variance and genomic prediction using genotypes imputed from low‐density marker subsets for carcass traits in Japanese black cattle

The influence of genotype imputation using low‐density single nucleotide polymorphism (SNP) marker subsets on the genomic relationship matrix (G matrix), genetic variance explained, and genomic prediction (GP) was investigated for carcass weight and marbling score in Japanese Black fattened steers, using genotype data of approximately 40,000 SNPs. Genotypes were imputed using equally spaced SNP subsets of different densities. Two different linear models were used. The first (model 1) incorporated one G matrix, while the second (model 2) used two different G matrices constructed using the selected and remaining SNPs. When using model 1, the estimated additive genetic variance was always larger when using all SNPs obtained via genotype imputation than when using only equally spaced SNP subsets. The correlations between the genomic estimated breeding values obtained using genotype imputation with at least 3,000 SNPs and those using all available SNPs without imputation were higher than 0.99 for both traits. While additive genetic variance was likely to be partitioned with model 2, it did not enhance the accuracy of GP compared with model 1. These results indicate that genotype imputation using an equally spaced low‐density panel of an appropriate size can be used to produce a cost‐effective, valid GP.     

Comparison of homogeneity and heterogeneity of residual variance using random regression test-day models for first lactation Japanese Holstein cows

Using a large‐scale data set that included first lactation test day records from 1975 to 2000 for Japanese Holsteins, genetic parameters for milk yield were estimated by using random regression (RR) test‐day models (TDM) with heterogeneous and homogeneous residual variances. It is necessary for the RR‐TDM to include a function that explains the shape of the lactation curve. The RR‐TDM with the LW curve, which combined Wilmink's curve and a Legendre polynomial, was used for fitting the model for milk yield. In recent years, increases in residual variance have been noted for Japanese dairy cattle. Thus, three kinds of heterogeneous residual variance over the calving year were considered: H1, H2 and HG. Linear and quadratic exponential functions for the calving year were used in H1 and H2, respectively. Residual variance of HG was divided into five groups according to calving year. Homogeneous residual variance was HO. All heterogeneous residual variances increased with calving year in an almost linear fashion. Residual variance increased over the study period. However, there is no need to consider heterogeneous residual variances in genetic evaluations, because the heterogeneity of residual variance over the years did not affect the ranking of top sires and cows.