Computing options for multiple‐trait test‐day random regression models while accounting for heat tolerance

Data included 90 242 799 test day records from first, second and third parities of 5 402 484 Holstein cows and 9 326 754 animals in the pedigree. Additionally, daily temperature humidity indexes (THI) from 202 weather stations were available. The fixed effects included herd test day, age at calving, milking frequency and days in milk classes (DIM). Random effects were additive genetic, permanent environment and herd‐year and were fit as random regressions. Covariates included linear splines with four knots at 5, 50, 200 and 305 DIM and a function of THI. Mixed model equations were solved using an iteration on data program with a preconditioned conjugate gradient algorithm. Preconditioners used were diagonal (D), block diagonal due to traits (BT) and block diagonal due to traits and correlated effects (BTCORR). One run included BT with a ‘diagonalized’ model in which the random effects were reparameterized for diagonal (co)variance matrices among traits (BTDIAG). Memory requirements were 8.7 Gb for D, 10.4 Gb for BT and BTDIAG, and 24.3 Gb for BTCORR. Computing times (rounds) were 14 days (952) for D, 10.7 days (706) for BT, 7.7 days (494) for BTDIAG and 4.6 days (289) for BTCORR. The convergence pattern was strongly influenced by the choice of fixed effects. When sufficient memory is available, the option BTCORR is the fastest and simplest to implement; the next efficient method, BTDIAG, requires additional steps for diagonalization and back‐diagonalization.     

Estimation of correlation between maternal permanent environmental effects of related dams in beef cattle

Weaning weights from Gelbvieh (GV; n = 82,138) and Limousin (LM; n = 88,639) calves were used to estimate genetic and environmental variance components with models that included different values for the correlation (lambda) between permanent environmental effects of dams and their daughters. Each analysis included fixed discrete effects of contemporary group, sex of calf, age of dam at calving, and month of calving, a fixed continuous effect of age of calf, random direct and maternal additive genetic effects, permanent environmental effects due to dams, and residual effects. The REML procedure was employed with a "grid search," in which the likelihood was computed for a series of values for lambda. For both breeds, models that included a nonzero value for lambda fitted the data significantly better than the model that did not include lambda. The maximum restricted likelihood was obtained for lambda of approximately -0.2 for both breeds. Estimates of residual and direct genetic variances were similar for all values of lambda, including zero; however, estimates of maternal genetic variance and maternal heritability increased slightly, and maternal permanent environmental variance and the proportion of the maternal variance to the total (phenotypic) variance decreased slightly, when the correlated structure for permanent environmental effects was assumed. As the value of lambda became more negative, absolute values of the direct-maternal genetic covariance and direct-maternal correlation estimates were decreased. Pearson and rank correlations for direct genetic, maternal genetic, and maternal environmental effects estimated with and without lambda were very high (>0.99). These results indicated that the linear relationship between maternal permanent environmental effects of dams and their daughters for weaning weight is negative but low in both breeds. Considering this relationship in the operational model did not significantly affect estimated breeding values, and thus, it may not be important in genetic evaluations.     

Studies on multiple trait and random regression models for genetic evaluation of beef cattle for growth

A simulation study examined issues important for genetic evaluation of growth in beef cattle by random regression models with cubic Legendre polynomials (RRML) and linear splines with three knots (RRMS) compared with multiple-trait models (MTM). Parameters for RRML were obtained by conversion from covariance functions. Parameters for MTM and RRMS were extracted from RRML at 1, 205, and 365 d; parameters for RRMS were the same as MTM for all effects except the permanent environment and the residual. Four data sets were generated assuming RRML included records at 1, 205, and 365 d; at 1, 160 to 250, and 320 to 410 d; at 1, 100, 205, 300, and 365 d; and at 1, 55 to 145, 160 to 250, 275 to 325, and 320 to 410 d. Accuracies were computed as correlations between the true (simulated) and predicted breeding values. With the first data set, excellent agreement in accuracy was obtained for all models. With the second data set, the accuracy of MTM dropped by up to 1.5% compared with the first data set, but accuracy was unchanged for both RRML and RRMS. With the third (fourth) data set, accuracies of RRML were up to 2.4% (2.5%) higher than with the first (second) data set. Small differences in accuracy between RRML and RRMS were found with the third and fourth data sets, which were traced to inflated correlations especially between 1 and 205 d in RRMS; inflation could be decreased by adding one extra knot at 100 d to RRMS. Diagonalization of random coefficients was crucial for RRML but not for RRMS, resulting in approximately six (two) times faster convergence with RRML (RRMS). Reduction of dimensionality in RRML associated with small eigenvalues caused a less accurate evaluation for birth weight. Genetic evaluation of growth by RRM requires careful implementation. The RRMS is simpler to implement than the RRML.     

Genetic evaluation of calving to first insemination using natural and artificial insemination mating data

Mating and calving records for 51,084 first-parity heifers in Australian Angus herds were used to examine the relationship between probability of calving to first insemination (CFI) in artificial insemination and natural service (NS) mating data. Calving to first insemination was defined as a binary trait for both sources of data. Two Bayesian models were employed: 1) a bivariate threshold model with CFI in AI data regarded as a trait separate from CFI in NS data and 2) a univariate threshold model with CFI regarded as the same trait for both sources of data. Posterior means (SD) of additive variance in the bivariate analysis were similar: 0.049 (0.013) and 0.075 (0.021) for CFI in AI and NS data, respectively, indicating lack of heterogeneity for this parameter. A similar trend was observed for heritability in the bivariate analysis, with posterior means (SD) of 0.025 (0.007) and 0.048 (0.012) for AI and NS data, respectively. The posterior means (SD) of the additive covariance and corresponding genetic correlation between the traits were 0.048 (0.006) and 0.821 (0.138), respectively. Differences were observed between posterior means for herd-year variance: 0.843 vs. 0.280 for AI and NS data, respectively, which may reflect the higher incidence of 100% conception rates within a herd-year class (extreme category problem) in AI data. Parameter estimates under the univariate model were close to the weighted average of the corresponding parameters under the bivariate model. Posterior means (SD) for additive, herd-year, and service sire variance and heritability under the univariate model were 0.063 (0.007), 0.56 (0.029), 0.131 (0.013), and 0.036 (0.007), respectively. These results indicate that, genetically, cows with a higher probability of CFI when mated using AI also have a high probability of CFI when mated via NS. The high correlation between the two traits, along with the lack of heterogeneity for the additive variance, implies that a common additive variance could be used for AI and NS data. A single-trait analysis of CFI with heterogeneous variances for herd-year and service sire could be implemented. The low estimates of heritability indicate that response to selection for probability of calving to first insemination would be expected to be low.     

Variance and covariance components for weaning weight for Herefords in three countries

Records from the Hereford Associations of the United States (USA), Canada, and Uruguay were used to estimate genetic and phenotypic variances and covariances for weaning weight. Estimation was done using a complete animal model, relatively large data sets, and the same methodology for the three countries in order to determine whether genetic parameters for weaning weight were homogeneous across environments. Data were composed of 2,322,722, 487,661, and 102,986 edited weaning weight records for USA, Canada, and Uruguay, respectively. Ten samples were obtained from each country by eliminating data from small herds with fewer than 500 records, selecting herds at random from the entire data set after removing the small herds, and then retaining the direct-sire-connected contemporary groups within each sample. The final sample sizes ranged from 9,832 to 46,377 records. An accelerated EM-REML algorithm was used in estimating the (co)variance components in each sample. The estimates were pooled by calculating the arithmetic mean of the 10 samples from within each country. Direct and maternal (in parentheses) heritability estimates were .24 (.16), .20 (.16), and .23 (.18) for USA, Canada, and Uruguay, respectively. Maternal heritabilities reported here are nearly 50% smaller than the values currently used in national genetic evaluation for the breed, which were estimated using sire-maternal grandsire models. Covariance between direct and maternal was negative in all countries, accounting for 6, 8, and 10% of the total phenotypic variation, and the total dam effect was 32.5, 37.0, and 34.0% in USA, Canada, and Uruguay, respectively. Total heritabilities were similar among the countries, with values of .19, .19, and .17 for the three respective countries. The similarity of genetic and environmental parameters across the three countries suggests that joint genetic evaluation is feasible across environments provided that the genotype x environment interaction is negligible and can be ignored.     

Estimation of genetic covariances with method R

Method R is a simple and computationally inexpensive method for estimating (co)variances. The objective of the study was to investigate properties of Method R for estimation of (co)variance components with emphasis on covariance estimation. Theoretical Method R formulas were developed for simplified single-variate and bivariate models. In single-trait models, the curve of the regression of Method R was continuous and monotonic and its slope depended on the amount of information on each animal and on the variance ratio. The curve became steeper as the number of records per animal decreased. For covariance, the curve of the regression was monotonic but not continuous. However, a regression coefficient of 1 still corresponded to the correct covariance. Similar curves were observed in analyses of simulated data sets. Because of the observed discontinuity, algorithms implementing Method R that require a continuous regression curve would not work in models with covariances. An alternative algorithm was based on a transformation matrix obtained by multiplying a matrix of numerators with the inverse of a matrix of denominators of the regression factors. Such an algorithm converged reliably for all models tested. Method R can be modified to estimate covariances in models too large for other methods.     

Approximation of reliabilities for multiple-trait model with maternal effects

Reliabilities for a multiple-trait maternal model were obtained by combining reliabilities obtained from single-trait models. Single-trait reliabilities were obtained using an approximation that supported models with additive and permanent environmental effects. For the direct effect, the maternal and permanent environmental variances were assigned to the residual. For the maternal effect, variance of the direct effect was assigned to the residual. Data included 10,550 birth weight, 11,819 weaning weight, and 3,617 postweaning gain records of Senepol cattle. Reliabilities were obtained by generalized inversion and by using single-trait and multiple-trait approximation methods. Some reliabilities obtained by inversion were negative because inbreeding was ignored in calculating the inverse of the relationship matrix. The multiple-trait approximation method reduced the bias of approximation when compared with the single-trait method. The correlations between reliabilities obtained by inversion and by multiple-trait procedures for the direct effect were 0.85 for birth weight, 0.94 for weaning weight, and 0.96 for postweaning gain. Correlations for maternal effects for birth weight and weaning weight were 0.96 to 0.98 for both approximations. Further improvements can be achieved by refining the single-trait procedures.     

Unknown‐parent groups in single‐step genomic evaluation

In single‐step genomic evaluation using best linear unbiased prediction (ssGBLUP), genomic predictions are calculated with a relationship matrix that combines pedigree and genomic information. For missing pedigrees, unknown selection processes, or inclusion of several populations, a BLUP model can include unknown‐parent groups (UPG) in the animal effect. For ssGBLUP, UPG equations also involve contributions from genomic relationships. When those contributions are ignored, UPG solutions and genetic predictions can be biased. Options to eliminate or reduce such bias are presented. First, mixed model equations can be modified to include contributions to UPG elements from genomic relationships (greater software complexity). Second, UPG can be implemented as separate effects (higher cost of computing and data processing). Third, contributions can be ignored when they are relatively small, but they may be small only after refinements to UPG definitions. Fourth, contributions may approximately cancel out when genomic and pedigree relationships are constructed for compatibility; however, different construction steps are required for unknown parents from the same or different populations. Finally, an additional polygenic effect that also includes UPG can be added to the model.