Random Regression Test-day Model Research Articles

Prediction of Daily Milk, Fat, and Protein Production by a Random Regression Test-Day Model

Test-day genetic evaluation models have many advantages compared with those based on 305-d lactations; however, the possible use of test-day model (TDM) results for herd management purposes has not been emphasized. The aim of this paper was to study the ability of a TDM to predict production for the next test day and for the entire lactation. Predictions of future production and detection of outliers are important factors for herd management (e.g., detection of health and management problems and compliance with quota). Because it is not possible to predict the herd-test-day (HTD) effect per se, the fixed HTD effect was split into 3 new effects: a fixed herd-test month-period effect, a fixed herd-year effect, and a random HTD effect. These new effects allow the prediction of future production for improvement of herd management. Predicted test-day yields were compared with observed yields, and the mean prediction error computed across herds was found to be close to zero. Predictions of performance records at the herd level were even more precise. Discarding herds enrolled in milk recording for <1 yr and animals with very few tests in the evaluation file improved correlations between predicted and observed yields at the next test day (correlation of 0.864 for milk in first-lactation cows as compared with a correlation of 0.821 with no records eliminated). Correlations with the observed 305-d production ranged from 0.575 to 1 for predictions based on 0 to 10 test-day records, respectively. Similar results were found for second and third lactation records for milk and milk components. These findings demonstrate the predictive ability of a TDM.

Genetic analysis of somatic cell score in Norwegian cattle using random regression test-day models.

The dataset used in this analysis contained a total of 341,736 test-day observations of somatic cell scores from 77,110 primiparous daughters of 1965 Norwegian Cattle sires. Initial analyses, using simple random regression models without genetic effects, indicated that use of homogeneous residual variance was appropriate. Further analyses were carried out by use of a repeatability model and 12 random regression sire models. Legendre polynomials of varying order were used to model both permanent environmental and sire effects, as did the Wilmink function, the Lidauer-Mäntysaari function, and the Ali-Schaeffer function. For all these models, heritability estimates were lowest at the beginning (0.05 to 0.07) and higher at the end (0.09 to 0.12) of lactation. Genetic correlations between somatic cell scores early and late in lactation were moderate to high (0.38 to 0.71), whereas genetic correlations for adjacent DIM were near unity. Models were compared based on likelihood ratio tests, Bayesian information criterion, Akaike information criterion, residual variance, and predictive ability. Based on prediction of randomly excluded observations, models with 4 coefficients for permanent environmental effect were preferred over simpler models. More highly parameterized models did not substantially increase predictive ability. Evaluation of the different model selection criteria indicated that a reduced order of fit for sire effects was desireable. Models with zeroth- or first-order of fit for sire effects and higher order of fit for permanent environmental effects probably underestimated sire variance. The chosen model had Legendre polynomials with 3 coefficients for sire, and 4 coefficients for permanent environmental effects. For this model, trajectories of sire variance and heritability were similar assuming either homogeneous or heterogeneous residual variance structure.

Assessment of Homogeneity vs. Heterogeneity of Residual Variance in Random Regression Test-Day Models in a Bayesian Analysis

Test-day first-lactation milk yields from Holstein cows were analyzed with a set of random regression models based on Legendre polynomials of varying order on additive genetic and permanent environmental effects. Homogeneity and heterogeneity of residual variance, assuming three and 30 arbitrary measurement error classes of different length were considered. Unknown parameters were estimated within a Bayesian framework. Bayes factors and a checking function for the cross-validation predictive densities of the data were the tools chosen for selecting among competing models. Residual variances obtained from 30 arbitrary intervals were nearly constant between d 70 and 300 and tended to increase towards the extremes of the lactation, especially at the onset. In early lactation, the temporary measurement errors were found to be larger and highly variable. A high order of the regression submodels employed for modeling the permanent environmental deviations tended to strongly correct the heterogeneity of the residual variance. Accordingly, the assumption of homogeneity of residual variance was the most plausible specification under both comparison criteria when the number of random regression coefficients was set to five. Otherwise, the heterogeneity assumption, using three or 30 error classes, was better supported, depending on the criterion and on the order of the submodel fitted for the permanent environmental effect.

Comparison of test-day models for genetic evaluation of production traits in dairy cattle

A multiple-trait reduced rank random regression test-day model was applied for the breeding value estimation for first parity milk, protein, and fat yield of Finnish dairy cattle. This model was compared with three other models: a similar multiple-trait random regression test-day model without rank reduction, a multiple-trait repeatability test-day model, and a multiple-trait 305-d lactation yield model. Required (co)variance parameters were derived from the same covariance functions for all four models. For both random regression models, standard deviations of breeding values were the same and correlations between breeding values were between 0.995 and 0.998, resulting in only slight differences in the ranking of animals. Genetic trends were identical for the random regression test-day models and very similar to those estimated by the 305-d lactation yield model. The repeatability test-day model gave a slightly different genetic trend and inflated standard deviations for breeding values of cows with lactations in progress. Reduction of rank in the random regression test-day model decreased memory requirements and improved convergence in iteration when solving the mixed model equations.

Genetic Evaluation of Bulls and Cows with Single- and Multiple-Country Test-Day Models

First-lactation milk yield test-day records on cows from Australia, Canada, Italy, and New Zealand were analyzed by single- and multiple-country random regression models. Models included fixed effects of herd-test day and breed composition—age at calving—season of calving by days in milk, and random regressions with Legendre polynomials of order four for animal genetic and permanent environmental effects. Milk yields in different countries were defined as genetically different traits for the purpose of multiple-trait model. Estimated breeding values of bulls and cows from single- and multiple-trait models were compared within and across countries for two traits: total milk yield in lactation and lactation persistency, defined as the linear coefficient of animal genetic curve. Correlations between single- and multiple-trait evaluations within country for total yield were higher than 0.95 for bulls and close to 1 for cows. Correlations for lactation persistency were lower than respective correlations for total yield. Between country correlations for lactation yield ranged from 0.93 to 0.96, indicating different ranking of bulls on different country scales under multiple-trait model. Lactation persistency had in general lower between-country correlations, with the highest values for Canada—Italy and Australia—New Zealand pairs, for both single- and multiple-country models. Although multiple-country random regression test-day model was computationally feasible for four countries, the same would not be true for routine international genetic evaluation in the near future.

Parallel Computing Applied to Breeding Value Estimation in Dairy Cattle

Continuous genetic evaluation of dairy cattle with test-day models is desired in Finland. However, the computing time for the genetic evaluation is 4 d and exceeds the minimum of a weekend. Three parallel implementations of the preconditioned conjugate gradient iterative solver were programmed and compared to identify the best strategy for solving mixed model equations using parallel computing. The programs were used to solve two random regression test-day models with approximately 7.28 and 49.9million unknowns. The latter model will be used in the Finnish dairy cattle evaluation. Computing times for the smaller model with the four processors available were 52, 32, and 27% of the single processor program when the complexity of the parallel program was increased. In practice, the best program required the most programming because the other parallel programs could not solve the larger model because of excess memory requirements. Parallel computing with four processors reduced the time to obtain solutions of Finnish dairy cattle evaluations to under 2 d. Benefit from parallel computing will be increased if amount of computing memory is increased.

Genetic Parameters of Legendre Polynomials for First Parity Lactation Curves

Variance components of the covariance function coefficients in a random regression test-day model were estimated by Legendre polynomials up to a fifth order for first-parity records of Dutch dairy cows using Gibbs sampling. Two Legendre polynomials of equal order were used to model the random part of the lactation curve, one for the genetic component and one for permanent environment. Test-day records from cows registered between 1990 to 1996 and collected by regular milk recording were available. For the data set, 23,700 complete lactations were selected from 475 herds sired by 262 sires.Because the application of a random regression model is limited by computing capacity, we investigated the minimum order needed to fit the variance structure in the data sufficiently. Predictions of genetic and permanent environmental variance structures were compared with bivariate estimates on 30-d intervals. A third-order or higher polynomial modeled the shape of variance curves over DIM with sufficient accuracy for the genetic and permanent environment part. Also, the genetic correlation structure was fitted with sufficient accuracy by a third-order polynomial, but, for the permanent environmental component, a fourth order was needed. Because equal orders are suggested in the literature, a fourth-order Legendre polynomial is recommended in this study. However, a rank of three for the genetic covariance matrix and of four for permanent environment allows a simpler covariance function with a reduced number of parameters based on the eigenvalues and eigenvectors.

Estimation of genetic parameters for daily milk yield of primiparous Ayrshire cows by random regression test-day models

Test-day data comprising of 63,331 test-day milk records of 6310 primiparous Finnish Ayrshire cows were used to estimate genetic parameters for daily milk production. Two alternative random regression (RR) sub-models were used to describe breeding values for the shape of lactation curves of individual cows: a five parameter logarithmic polynomial (ASM) or a normalised third order orthogonal polynomial. Permanent environment (PE) of a cow was described by either a common PE effect (ASM and OPM PE1) or a normalised third order orthogonal polynomial (OPM PE4). Variance components were estimated with an animal model using EM-REML. A multitrait (MT) approach together with continuous covariance function (CF) was used to derive reference for RR estimates. Heritability estimates obtained by ASM (0.41–0.60) and OPM PE1 (0.28–0.53) were higher than those derived from CF analysis (0.20–0.28). Fitting the RR sub-model for PE effects strongly influenced the magnitude of heritability estimates (0.23–0.36). Estimates of heritability were found to be highest during early and late lactation when estimated by ASM and OPM PE1 models, while the converse was true for those derived by CF. Estimates obtained by the OPM PE4 model were highest at the beginning of lactation and between 183 and 256 days in milk. Genetic correlations were high between consecutive test days, but decreased when intervals between test days increased. Where models ASM and OPM PE1 indicated a negative correlation between distant test days, OPM PE4 estimates were consistent with those of CF. Due to the statistical complexity of RR test-day models use of MT is a more feasible approach for the estimation of (co)variance components for CF coefficients.

Solving Large Test-Day Models by Iteration on Data and Preconditioned Conjugate Gradient

A preconditioned conjugate gradient method was implemented into an iteration on a program for data estimation of breeding values, and its convergence characteristics were studied. An algorithm was used as a reference in which one fixed effect was solved by Gauss-Seidel method, and other effects were solved by a second-order Jacobi method. Implementation of the preconditioned conjugate gradient required storing four vectors (size equal to number of unknowns in the mixed model equations) in random access memory and reading the data at each round of iteration. The preconditioner comprised diagonal blocks of the coefficient matrix. Comparison of algorithms was based on solutions of mixed model equations obtained by a single-trait animal model and a single-trait, random regression test-day model. Data sets for both models used milk yield records of primiparous Finnish dairy cows. Animal model data comprised 665,629 lactation milk yields and random regression test-day model data of 6,732,765 test-day milk yields. Both models included pedigree information of 1,099,622 animals. The animal model {random regression test-day model} required 122 305 rounds of iteration to converge with the reference algorithm, but only 88 {149} were required with the preconditioned conjugate gradient. To solve the random regression test-day model with the preconditioned conjugate gradient required 237 megabytes of random access memory and took 14% of the computation time needed by the reference algorithm.

Solving Large Mixed Linear Models Using Preconditioned Conjugate Gradient Iteration

Continuous evaluation of dairy cattle with a random regression test-day model requires a fast solving method and algorithm. A new computing technique feasible in Jacobi and conjugate gradient based iterative methods using iteration on data is presented. In the new computing technique, the calculations in multiplication of a vector by a matrix were reordered to three steps instead of the commonly used two steps. The three-step method was implemented in a general mixed linear model program that used preconditioned conjugate gradient iteration. Performance of this program in comparison to other general solving programs was assessed via estimation of breeding values using univariate, multivariate, and random regression test-day models. Central processing unit time per iteration with the new three-step technique was, at best, one-third that needed with the old technique. Performance was best with the test-day model, which was the largest and most complex model used. The new program did well in comparison to other general software. Programs keeping the mixed model equations in random access memory required at least 20 and 435% more time to solve the univariate and multivariate animal models, respectively. Computations of the second best iteration on data took approximately three and five times longer for the animal and test-day models, respectively, than did the new program. Good performance was due to fast computing time per iteration and quick convergence to the final solutions. Use of preconditioned conjugate gradient based methods in solving large breeding value problems is supported by our findings.

Genetic Parameters for First and Second Lactation Milk Yields of Polish Black and White Cattle with Random Regression Test-Day Models

Single- and two-trait random regression models were applied to estimate variance components of test-day records of milk, fat, and protein yields in the first and second lactation of Polish Black and White cattle. The model included fixed herd test-day effect, three covariates to describe lactation curve nested within age-season classes, and random regressions for additive genetic and permanent environmental effects. In two-parity models, each parity was treated as a separate trait. For milk and the two-parity model, heritabilities were in the range of 0.14 to 0.19 throughout first lactation and 0.10 to 0.16 throughout second lactation. For fat, heritabilities were within 0.11 to 0.16 and 0.11 to 0.22 throughout first and second lactations, respectively. For protein in the two-parity model, heritabilities were within 0.10 to 0.15 throughout most of first lactation and within 0.06 to 0.15 throughout the most of second lactation. For milk, genetic correlations between the first and second parities were 0.6 at the beginning of the lactation, rising to 0.9 in the middle, and 0.8 at the end of the lactation. For fat, the corresponding correlations were 0.6, 0.8, and 0.7, respectively, and for protein were 0.6, 0.8, and 0.8, respectively. Heritability estimates for all traits were flatter for the two-parity model. Relatively smooth genetic and permanent environmental variances with the two-parity model indicated that large swings of heritabilities could be artifacts of single-trait random regression models. High correlations between most of test day records across lactations suggested that a repeatability model could be considered as an alternative to a multiple-trait model to analyze multiple parities.