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.     

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.     

Estimation of genetic parameters and prediction of breeding values for multivariate threshold and continuous data in a simulated horse population using Gibbs sampling and residual maximum likelihood

Simulated horse data were used to compare multivariate estimation of genetic parameters and prediction of breeding values (BV) for categorical, continuous and molecular genetic data using linear animal models via residual maximum likelihood (REML) and best linear unbiased prediction (BLUP) and mixed linear-threshold animal models via Gibbs sampling (GS). Simulation included additive genetic values, residuals and fixed effects for one continuous trait, liabilities of four binary traits, and quantitative trait locus (QTL) effects and genetic markers with different recombination rates and polymorphism information content for one of the liabilities. Analysed data sets differed in the number of animals with trait records and availability of genetic marker information. Consideration of genetic marker information in the model resulted in marked overestimation of the heritability of the QTL trait. If information on 10,000 or 5,000 animals was used, bias of heritabilities and additive genetic correlations was mostly smaller, correlation between true and predicted BV was always higher and identification of genetically superior and inferior animals was - with regard to the moderately heritable traits, in many cases - more reliable with GS than with REML/BLUP. If information on only 1,000 animals was used, neither GS nor REML/BLUP produced genetic parameter estimates with relative bias 50% for all traits. Selection decisions for binary traits should rather be based on GS than on REML/BLUP breeding values.     

Comparison of models for genetic evaluation of number of inseminations to conception in Danish Holstein cows

Number of inseminations to conception (NINS), an important fertility trait, requires appropriate approaches for genetic evaluation due to its non‐normal distribution and censoring records. In this study, we analyzed NINS in 474 837 Danish Holstein cows at their first lactation by using seven models which deal with the categorical phenotypes and censored records in different manners, further assessed these models with regard to stability, lack of bias and accuracy of prediction. The estimated heritability from four models based on original NINS specified as a linear Gaussian model, categorical threshold model, threshold linear model and survival model were similar (0.031‐0.037). While for the other three models based on the binary response derived from NINS, referred as threshold model (TM), logistic and probit models (LOGM and PROM), the heritability were estimated as 0.027, 0.063 and 0.027, respectively. The model comparison concluded that different models could lead to slightly different sire rankings in terms of breeding values; a more complicated model led to less stability of prediction; the models based on the binary response derived from NINS (TM, LOGM and PROM) had slightly better performances in terms of unbiased and accurate prediction of breeding values.     

Efficient approximation of reliabilities for single-step genomic best linear unbiased predictor models with the Algorithm for Proven and Young

The objectives of this study were to develop an efficient algorithm for calculating prediction error variances (PEVs) for genomic best linear unbiased prediction (GBLUP) models using the Algorithm for Proven and Young (APY), extend it to single-step GBLUP (ssGBLUP), and apply this algorithm for approximating the theoretical reliabilities for single- and multiple-trait models in ssGBLUP. The PEV with APY was calculated by block sparse inversion, efficiently exploiting the sparse structure of the inverse of the genomic relationship matrix with APY. Single-step GBLUP reliabilities were approximated by combining reliabilities with and without genomic information in terms of effective record contributions. Multi-trait reliabilities relied on single-trait results adjusted using the genetic and residual covariance matrices among traits. Tests involved two datasets provided by the American Angus Association. A small dataset (Data1) was used for comparing the approximated reliabilities with the reliabilities obtained by the inversion of the left-hand side of the mixed model equations. A large dataset (Data2) was used for evaluating the computational performance of the algorithm. Analyses with both datasets used single-trait and three-trait models. The number of animals in the pedigree ranged from 167,951 in Data1 to 10,213,401 in Data2, with 50,000 and 20,000 genotyped animals for single-trait and multiple-trait analysis, respectively, in Data1 and 335,325 in Data2. Correlations between estimated and exact reliabilities obtained by inversion ranged from 0.97 to 0.99, whereas the intercept and slope of the regression of the exact on the approximated reliabilities ranged from 0.00 to 0.04 and from 0.93 to 1.05, respectively. For the three-trait model with the largest dataset (Data2), the elapsed time for the reliability estimation was 11 min. The computational complexity of the proposed algorithm increased linearly with the number of genotyped animals and with the number of traits in the model. This algorithm can efficiently approximate the theoretical reliability of genomic estimated breeding values in ssGBLUP with APY for large numbers of genotyped animals at a low cost.     

Genetic evaluation using random regression models with different covariance functions for test‐day milk yield in an admixture population of Thailand goats

The objectives of this study were to compare covariance functions (CF) and estimate the heritability of milk yield from test‐day records among exotic (Saanen, Anglo‐Nubian, Toggenburg and Alpine) and crossbred goats (Thai native and exotic breed), using a random regression model. A total of 1472 records of test‐day milk yield were used, collected from 112 does between 2003 and 2006. CF of the study were Wilmink function, second‐ and third‐order Legendre polynomials, and linear splines 4 knots located at 5, 25, 90 and 155 days in milk (SP25–90) and 5, 35, 95 and 155 of days in milk (SP35–95). Variance components were estimated by restricted maximum likelihood method (REML). Goodness of fit, Akaike information criterion (AIC), percentage of squared bias (PSB), mean square error (MSE), and empirical correlation (RHO) between the observed and predicted values were used to compare models. The results showed that CF had an impact on (co)variance estimation in random regression models (RRM). The RRM with splines 4 knots located at 5, 25, 90 and 155 of days in milk had the lowest AIC, PSB and MSE, and the highest RHO. The heritability estimated throughout lactation obtained with this model ranged from 0.13 to 0.23.