Adjustment For Heterogeneous Variance Research Articles

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.

Comparison of multiplicative heterogeneous variance adjustment models for genetic evaluations

Two heterogeneous variance adjustment methods and two variance models were compared in a simulation study. The method used for heterogeneous variance adjustment in the Nordic test-day model, which is a multiplicative method based on Meuwissen (J. Dairy Sci., 79, 1996, 310), was compared with a restricted multiplicative method where the fixed effects were not scaled. Both methods were tested with two different variance models, one with a herd-year and the other with a herd-year-month random effect. The simulation study was built on two field data sets from Swedish Red dairy cattle herds. For both data sets, 200 herds with test-day observations over a 12-year period were sampled. For one data set, herds were sampled randomly, while for the other, each herd was required to have at least 10 first-calving cows per year. The simulations supported the applicability of both methods and models, but the multiplicative mixed model was more sensitive in the case of small strata sizes. Estimation of variance components for the variance models resulted in different parameter estimates, depending on the applied heterogeneous variance adjustment method and variance model combination. Our analyses showed that the assumption of a first-order autoregressive correlation structure between random-effect levels is reasonable when within-herd heterogeneity is modelled by year classes, but less appropriate for within-herd heterogeneity by month classes. Of the studied alternatives, the multiplicative method and a variance model with a random herd-year effect were found most suitable for the Nordic test-day model for dairy cattle evaluation.

Bias in genetic evaluation using random regression test-day model

The effect of an upgraded Finnish evaluation model on bias in estimated breeding values for protein yield was investigated. Evaluations based on repeatability animal model and on random regression test-day model without and with heterogeneous variance adjustment were compared. Comparisons were based on the average difference between pedigree indices and the future estimated breeding values, based on own or on daughter performance records. This was defined as empirical bias. The pedigree indices were computed from reduced data sets where four years of the most recent data were excluded. Results showed an upward bias in the protein yield pedigree indices for Ayrshire young sires of 2.2 kg, 2.5 kg and 1.8 kg from the repeatability animal model, random regression test-day model and random regression test-day model with heterogeneous variance adjustment, respectively. Pedigree indices for daughters of young sires were upward biased, whereas pedigree indices for daughters of proven sires were slightly underestimated when heterogeneous variance was not accounted. Inclusion of test-day yields from the fourth lactation onwards increased the bias. Moving from repeatability animal model to random regression test-day model did not reduce the bias, whereas adjustment of heterogeneous variance reduced bias.;

Multiplicative random regression model for heterogeneous variance adjustment in genetic evaluation for milk yield in Simmental

A multiplicative random regression (M-RRM) test-day (TD) model was used to analyse daily milk yields from all available parities of German and Austrian Simmental dairy cattle. The method to account for heterogeneous variance (HV) was based on the multiplicative mixed model approach of Meuwissen. The variance model for the heterogeneity parameters included a fixed region x year x month x parity effect and a random herd x test-month effect with a within-herd first-order autocorrelation between test-months. Acceleration of variance model solutions after each multiplicative model cycle enabled fast convergence of adjustment factors and reduced total computing time significantly. Maximum Likelihood estimation of within-strata residual variances was enhanced by inclusion of approximated information on loss in degrees of freedom due to estimation of location parameters. This improved heterogeneity estimates for very small herds. The multiplicative model was compared with a model that assumed homogeneous variance. Re-estimated genetic variances, based on Mendelian sampling deviations, were homogeneous for the M-RRM TD model but heterogeneous for the homogeneous random regression TD model. Accounting for HV had large effect on cow ranking but moderate effect on bull ranking.

Genetic Parameters for a Multiple-Trait Multiple-Lactation Random Regression Test-Day Model in Italian Holsteins

The objectives of this study were to estimate variance components for test-day milk, fat, and protein yields and average daily SCS in 3 subsets of Italian Holsteins using a multiple-trait, multiple-lactation random regression test-day animal model and to determine whether a genetic heterogeneous variance adjustment was necessary. Data were test-day yields of milk, fat, and protein and SCS (on a log2 scale) from the first 3 lactations of Italian Holsteins collected from 1992 to 2002. The 3 subsets of data included 1) a random sample of Holsteins from all herds in Italy, 2) a random sample of Holsteins from herds using a minimum of 75% foreign sires, and 3) a random sample of Holsteins from herds using a maximum of 25% foreign sires. Estimations of variances and covariances for this model were achieved by Bayesian methods using the Gibbs sampler. Estimated 305-d genetic, permanent environmental, and residual variance was higher in herds using a minimum of 75% foreign sires compared with herds using a maximum of 25% foreign sires. Estimated average daily heritability of milk, fat, and protein yields did not differ among subsets. Heritability of SCS in the first lactation differed slightly among subsets and was estimated to be the highest in herds with a maximum of 25% foreign sire use (0.19±0.01). Genetic correlations across lactations for milk, fat, and protein yields were similar among subsets. Genetic correlations across lactations for SCS were 0.03 to 0.08 higher in herds using a minimum of 75% or a maximum of 25% foreign sires, compared with herds randomly sampled from the entire population. Results indicate that adjustment for heterogeneous variance at the genetic level based on the percentage of foreign sire use should not be necessary with a multiple-trait random regression test-day animal model in Italy.

Accounting for Heterogeneous Variances in Multitrait Evaluation of Jersey Type Traits

The multitrait genetic evaluation system for type traits was modified to estimate adjustments for heterogeneous variance (HV) simultaneously with estimated breeding values (EBV) for final score and 14 linear traits. Each variance within herd, year, and parity was regressed toward a predicted variance, which was determined by fitting a model with fixed effects of the mean final score for herd, size of the contemporary group, appraisal month, and year-season and a random effect for herd-appraisal date. Herd-appraisal date was included as a random effect to regress the observed heterogeneity for a given herd-appraisal date toward the fixed effects. Method R was used to estimate variances for the heterogeneity model in each EBV iteration. To evaluate the effect of the adjustment, parent averages were calculated from evaluations with recent appraisals removed. The adjustment slightly improved correlations within birth year between those parent averages and EBV from current data on bulls for most traits, but did not improve correlations for final score, strength, dairy form, teat length, or foot angle. Annual trends for EBV were lower with HV adjustment than for unadjusted EBV for all traits except final score and rump angle for cows and rump width for bulls, which were essentially unchanged. Standard deviations of Mendelian sampling (evaluation minus mean of parent evaluations) declined less over time for HV-adjusted than for unadjusted evaluations. The slope at year 2000 of Mendelian-sampling standard deviations from HV-adjusted evaluations ranged from 10.0% for udder depth to 42.7% for teat length compared with the slope for unadjusted evaluations. This HV adjustment, which was implemented for USDA evaluations in May 2001 for Jerseys and in 2002 for other breeds, improves the accuracy of evaluations, particularly comparisons over time, by accounting for the change in variation.

Adjustments for heterogeneous herd-year variances in a random regression model for genetic evaluations of Polish Black-and-White cattle

The study investigated the existence of heterogeneous variance in first-lactation daily milk yield of Polish Black-and-White cows across herds in different years. Bayesian Information Criterion was used to show that the model with unequal residual variances for different herd-years was more plausible than the model assuming equal variances. A method of adjusting phenotypic records was developed to account for unequal variability in herd-years. Factors used for the data adjustment considered variation of general residuals and residuals for specific herd-years. The size of herd-year was also taken into account. Varied power of corrections was used to analyze the effect of adjustment on estimated breeding values. The method was applied to daily milk records of 817,165 primiparous cows. The effectiveness of the data adjustment was evaluated by the analysis of differences between each bull's breeding value and its parental index. Data correction reduced the average difference and variance of differences between breeding values and parental indices. Accounting for the size of herd-year classes in correction factors improved the efficiency of heterogeneous variance adjustment.

Type trait (co)variance components for five dairy breeds.

(Co)variance components were estimated for final score and 14 or 15 linear type traits for the Ayrshire, Brown Swiss, Guernsey, Jersey, and Milking Shorthorn breeds. Appraisals from 1995 or later were used. New estimates were calculated to accommodate changes in scoring of traits and because of a change from multiplicative to additive adjustment for age and lactation stage. The adjustment method was changed for better support of the adjustment for heterogeneous variance within iteration, which was implemented in 2002. The largest changes in heritability were an increase of 0.10 for rump angle for Milking Shorthorns and a decrease of 0.11 for udder depth for Jerseys. The new estimates of (co)variance components should provide improved accuracy of type evaluations, particularly for traits that have had variance changes over time.

Equity of Elite Cow Status Across States

Cows with superior genetic merit, based on an economic index for milk, fat, and yields (milk-fat-protein dollars), have been designated as elite by USDA. Because of the concern that adjustment for heterogeneous variance may have resulted in inequitable state representation for elite cows, the percentages of US cows with elite status were compared by state using evaluations of registered Holstein cows from 1990, 1991, and 1997. The numbers and percentages of eligible cows and the proportions of those cows that were designated as elite were determined by state from May 1997 evaluations of 772,302 registered and 1,499,729 grade Holstein cows; means and standard deviations for milk-fat-protein dollars were computed. Correlations were computed among the number of cows that were eligible for elite status, the number of cows that were designated as elite, the percentage of eligible cows that were designated as elite, and the mean and the standard deviation for milk-fat-protein dollars. Models were examined to for ability to predict the percentage of elite cows by state from mean and standard deviation for milk-fat-protein dollars. The number of elite cows for a state was highly correlated to the number of cows that were eligible for elite status. States with >1.0% of eligible cows designated as elite had mean milk-fat-protein dollars that were higher than the US mean of $44, but standard deviations were equal to or slightly lower than the US standard deviation of $71. The mean value for milk-fat-protein dollars was associated with the state percentage of elite cows, but variation of the index was not related. However, the standard deviation for milk-fat-protein dollars was important in explaining the percentage of elite cows if the model also contained the mean value. Differences in the variation of lactation records across states or adjustments for those differences did not appear to cause inequity in designating elite status. Corresponding results for grade cows supported findings for registered cows.

Heterogeneity of (co)variance among herds for backfat measures of swine.

Within-herd variation for the trait of backfat was examined in swine. Backfat depth data (n=21,877) from 11 herds that had each recorded over 1,000 measures were evaluated using univariate animal model procedures in analyses for individual herds and all herds together. Variance components were estimated by a derivative-free REML algorithm, and significance tests for variance components were carried out as likelihood ratio tests. Adjustment for heterogeneous variance by correction for within-herd phenotypic SD was also evaluated. Additive genetic and residual variances were heterogeneous across herds and ranged from 1.39 to 9.78 mm2 and from .88 to 7.05 mm2, respectively. Eleven parameter estimates from individual herd analyses were significantly different (P < .05) from the analysis for all herds together. However, only small differences were observed between estimates of heritability, which ranged from .37 to .73. Scaling data to a constant within-herd SD resulted in more homogeneous variance. For additive genetic variance, only one herd differed significantly (P < .01) from the analysis for all herds together. However, five estimates of residual variance were found to be significantly different (P < .05) from the analysis for all herds together. The rank correlation between EPD predicted from the analyses with all herds together, homogeneous variance, and scaled to a constant within-herd SD was .97 (P < .01). Effects of heterogeneous variance may need to be accounted for in genetic evaluation procedures for swine.

Adjustment for Heterogeneous Variance in Genetic Evaluations for Conformation of United States Holsteins

A method was developed to account for heterogeneous phenotypic variance for final type (conformation) score within herd-year-month-classifier subclasses for national genetic evaluations of US Holsteins. Phenotypic variances decreased 3.5-fold as within-subclass means increased from 70 to 89 points, and variances increased slightly over time and with increases in subclass size and proportion of registered animals. A model containing known management characteristics was fitted to phenotypic standard deviations within subclass, and resulting solutions were used to estimate the prior variance for each subclass. Prior estimates and phenotypic variance estimates within subclass were combined using Bayesian methods, resulting in posterior estimates that contained more information than either the prior estimates or the within-subclass statistics. Observations were standardized to a common variance using the mean of the posterior density of the phenotypic standard deviation within subclass. Moderate changes in PTA occurred for low reliability bulls, bulls with many progeny in a single herd, and foreign bulls subjected to positive assortative mating. Significant increases occurred in ETA of superior cows in subclasses with high means. Incorporation of a heterogeneous variance adjustment into national genetic evaluations should increase fairness of selection among cows and young bulls and may increase the rate of genetic gain from female selection.

Effect of Sampling Status and Adjustment for Heterogeneous Variance on Bias in Bull Evaluations

Holstein bulls were assigned to sampling categories (AI stud, AI nonstud, or non-AI) based on bull code, controller number, and age at semen distribution. The AI stud bulls were sampled through traditional progeny-testing programs of 13 AI organizations; AI nonstud bulls had AI semen collection reported by another organization or by multiple organizations. The non-AI bulls had no reported AI semen collection. Actual daughter yield deviations for these three groups of bulls were compared with expected performance (parent average) to provide an indication of whether evaluations were free from bias for daughter yield deviations. Mean difference of daughter yield deviation from parent average was close to 0kg for animal model evaluations of all 22,930 bulls but was positively biased by 46kg of milk for AI nonstud bulls. Mean PTA and reliabilities for parents were highest for AI stud bulls and lowest for non-AI bulls. The AI stud bulls varied least and were intermediate for mean management, approximated as mean daughter yield minus bull ETA. Management was highest for AI nonstud bulls, which suggested that adjustment for heterogeneous variance might reduce bias. However, the effect of this adjustment on mean difference of daughter yield deviation from parent average was small.

Method and Effect of Adjustment for Heterogeneous Variance of Holstein Conformation Traits

Type classification records of Canadian Holsteins were investigated for evidence of heterogeneous variance across herds. Data consisted of records for 1,139,104 cows from 20,226 herds with classifications on 26 conformation traits collected from 1982 through 1992 and 338,046 cows from 9600 herds with classifications on 2 additional traits from 1990 through 1992. Phenotypic standard deviations of herd-round-classifier were fitted to a mixed model that included round, classifier, and region as fixed effects, herd size as a covariant, and herd as a random effect. Estimates of the variance components, solutions of fixed effects, and BLUP estimates for herd were obtained by maximum likelihood procedures. Repeatability of within-herd standard deviation across rounds ranged from 1.4 to 10.3% for the 28 traits. Type classification data were subsequently standardized for phenotypic standard deviations of herd-round-classifier that were derived from estimates of the fixed effects and the BLUP estimate of the herd effect. Genetic evaluations for cows and bulls were produced from adjusted and unadjusted data Correlations between 3754 sire and 1,142,782 cow estimated transmitting abilities obtained from unadjusted and adjusted data were essentially unity. Although some evidence of heterogeneous variance existed across herds for 28 conformation traits, standardization of the classification records had only a minor effect on genetic evaluations.

Method and Effect of Adjustment for Heterogeneous Variance

Lactation records were standardized for differing genetic and error variances across herds and over time based on phenotypic variance for each herd-year-parity group. Each herd-year-parity phenotypic variance estimate was combined with those of adjacent years and regressed toward a region-year-parity variance. Heritability was assumed to be .25 at mean variance within year and to range from .2 for herds with smallest phenotypic SD to .3 for herds with largest phenotypic SD. Lactation deviations from management group mean were adjusted by ratio of base genetic SD to genetic SD estimated from heritability and phenotypic SD. The base was defined as 1987 calvings for first parity and 1988 calvings for later parities. Records were weighted according to heritability by multiplying lactation length weight by herd error weight defined as ratio of base error variance to error variance in the adjusted record. Estimated genetic trend for milk increased by nearly 5 kg/yr for Holsteins with this adjustment, which caused predicted breeding values of oldest animals to be lower by about 100kg. Most correlations of parent and progeny information were slightly higher with adjusted data. Cows in high variance herds were most likely to have large reductions in their evaluations. Adjustment for heterogeneous variance was implemented in July 1991 for national evaluations for yield traits.