# Multibreed Sire Evaluation Procedures within a Country

01 Apr 1985-Journal of Animal Science (The American Society of Animal Science)-Vol. 60, Iss: 4, pp 942-952

About: This article is published in Journal of Animal Science.The article was published on 1985-04-01. It has received 66 citations till now. The article focuses on the topics: Sire.

##### Citations

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TL;DR: In this article, the first and second moments of a random vector of breeding values are derived from simple genetic assumptions and matrix expressions are derived for the first two moments of the vector.

Abstract: From simple genetic assumptions, matrix expressions are derived for the first and second moments of a random vector of breeding values. Emphasis is on structure in these matrix representations that can be exploited computationally in a mixed model analysis. Also derived are mixed model equations corresponding to a model in which the random vector has nonnull mean. These equations are particularly advantageous for fitting a particular kind of animal model with groups and relationships.

255 citations

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TL;DR: Inbreeding coefficients for 9.3million registered Holsteins were computed by constructing a small relationship matrix for each animal and its ancestors instead of one large matrix for the whole population, finding inbreeding was underestimated.

Abstract: Inbreeding coefficients for 9.3million registered Holsteins were computed by constructing a small relationship matrix for each animal and its ancestors instead of one large matrix for the whole population. Recent cows averaged 2.0% inbreeding if each pedigree path was extended to the most recent ancestor born before 1960. Inbreeding was underestimated because some pedigrees included unknown ancestors more recent than the defined base year. Alternative estimates of inbreeding can be derived by assigning mean relationship and inbreeding of known ancestors to unknown ancestors of the same period. Animals of different breeds are less related than animals of the same breed. Relationships and inbreeding within and across populations can be measured back to the common base population from which the breeds arose by treating earliest known ancestors within each breed as related and inbred. Increased heterozygosity and heterosis of crossbred animals are then predicted from their lower inbreeding coefficients. Relationship matrices that include related and inbred unknown-parent groups treated as random or fixed effects can be constructed and inverted quickly.

197 citations

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TL;DR: The inverse of the genotypic covariance matrix given here can be used both to obtain genetic evaluations by best linear unbiased prediction and to estimate genetic parameters by maximum likelihood in multibreed populations.

Abstract: Covariance between relatives in a multibreed population was derived for an additive model with multiple unlinked loci. An efficient algorithm to compute the inverse of the additive genetic covariance matrix is given. For an additive model, the variance for a crossbred individual is a function of the additive variances for the pure breeds, the covariance between parents, and segregation variances. Provided that the variance of a crossbred individual is computed as presented here, the covariance between crossbred relatives can be computed using formulae for purebred populations. For additive traits the inverse of the genotypic covariance matrix given here can be used both to obtain genetic evaluations by best linear unbiased prediction and to estimate genetic parameters by maximum likelihood in multibreed populations. For nonadditive traits, the procedure currently used to analyze multibreed data can be improved using the theory presented here to compute additive covariances together with a suitable approximation for nonadditive covariances.

84 citations

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TL;DR: The results showed that it is feasible to evaluate sires for additive and nonadditive genetic effects in a structured multibreed population and data from purebred breeders and commercial producers will be needed to accomplish the same goal at a national level.

Abstract: Estimates of covariances and sire expected progeny differences of additive and nonadditive direct and maternal genetic effects for birth and weaning weights were obtained using records from 1,581 straightbred and crossbred calves from the Angus-Brahman multibreed herd at the University of Florida. Covariances were estimated by Restricted Maximum Likelihood, using a Generalized Expectation-Maximization algorithm applied to multibreed populations. Estimates of heritabilities and additive genetic correlations for straightbred and crossbred groups were within the ranges of values found in the literature for these traits. Maximum values of interactibilities (ratios of nonadditive genetic variances to phenotypic variances) and nonadditive correlations were somewhat smaller than heritabilities and additive genetic correlations. Sire additive and total direct and maternal genetic predictions for birth and weaning weight tended to increase with the fraction of Brahman alleles, whereas nonadditive direct and maternal genetic predictions were similar for sires of all Angus and Brahman fractions. These results showed that it is feasible to evaluate sires for additive and nonadditive genetic effects in a structured multibreed population. Data from purebred breeders and commercial producers will be needed to accomplish the same goal at a national level.

78 citations

### Cites background from "Multibreed Sire Evaluation Procedur..."

...A population composed of straightbred and crossbred animals that interbreed constitutes a multibreed population (Elzo, 1983, 1990b; Elzo and Famula, 1985)....

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TL;DR: It was concluded that a nonadditive model was preferable for estimation of genetic variance and prediction of breeding values in crossbred dairy populations.

Abstract: Genetic parameters were estimated using data of cows with variable proportions of genes from two breeds: Dutch Friesian and Holstein-Friesian. The data set contained 92,333 first lactation records (305-d milk production) from 675 young sires and 307,050 records from 202 proven sires. Data were analyzed using four additive mixed models with genetic groups defined according to 1) breed composition of the cow, 2) breed composition of sire and dam, 3) linear regression on the fraction and Holstein-Friesian genes of the cow, and 4) breed composition of sire. A nonadditive model included a linear regression on breed fraction, heterozygosity, and recombination in the cow's genome. Estimates for heterosis varied from 2.5% (fat yield) to 0% (protein percentage). Recombination effects varied from −1.9% (protein yield) to 1.5% (fat percentage). Additive models with progeny groups overestimated genetic variance by 6%. Models with sire groups overestimated additive genetic values of imported Holstein-Friesian sires by 33%. Using a nonadditive model, heritability estimates were .38 for milk yield, .80 for fat percentage, and .70 for protein percentage. It was concluded that a nonadditive model was preferable for estimation of genetic variance and prediction of breeding values in crossbred dairy populations.

77 citations

##### References

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TL;DR: In this paper, an estimation procedure based on adding small positive quantities to the diagonal of X′X was proposed, which is a method for showing in two dimensions the effects of nonorthogonality.

Abstract: In multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory, if not incorrect, if the prediction vectors are not orthogonal. Proposed is an estimation procedure based on adding small positive quantities to the diagonal of X′X. Introduced is the ridge trace, a method for showing in two dimensions the effects of nonorthogonality. It is then shown how to augment X′X to obtain biased estimates with smaller mean square error.

8,091 citations

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TL;DR: The theory of variance component analysis has been discussed recently by Crump (1946, 1951) and by Eisenhart (1947), and most of the published works on estimating variance components deal with the one-way classification, with nested" classifications, and with factorial classifications having equal subclass numbers.

Abstract: The theory of variance component analysis has been discussed recently by Crump (1946, 1951) and by Eisenhart (1947). These papers and, indeed, most of the published works on estimating variance components deal with the one-way classification, with "nested" classifications, and with factorial classifications having equal subclass numbers. Also most papers on this subject are concerned with what Eisenhart (1947) has called Model II; that is, all elements of the linear model save gi are regarded as random variables. In the above cases, estimation of variance components is usually accomplished by computing the mean squares in the standard analysis of variance, equating these mean squares to their expectations, and solving for the unknown variances. These techniques are described in many statistical textbooks. Unfortunately, research workers in some of those fields in which much use is made of variance component estimates are unable to obtain data which have the above described characteristics. This is particularly true in those fields in which survey data must be used or where, even in a well-planned experiment, the subclasses are of quite unequal size due, for example, to differences in litter numbers. Also,

1,170 citations

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TL;DR: In this article, the inverse of a numerator relationship matrix is needed for best linear unbiased prediction of breeding values, and a simple method for computing the elements of this inverse without computing the relationship matrix itself is presented.

Abstract: The inverse of a numerator relationship matrix is needed for best linear unbiased prediction of breeding values. The purpose of this paper to is present a rapid and simple method for computation of the elements of this inverse without computing the relationship matrix itself. The method is particularly useful in noninbred populations but is much faster than the conventional method in the presence of inbreeding.

793 citations

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TL;DR: The restricted maximum likelihood (REML) estimators as discussed by the authors have the property of invariance under translation and the additional property of reducing to the analysis variance estimators for many, if not all, cases of balanced data (equal subclass numbers).

Abstract: The maximum likelihood (ML) procedure of Hartley aud Rao [2] is modified by adapting a transformation from Pattersou and Thompson [7] which partitions the likelihood render normality into two parts, one being free of the fixed effects. Maximizing this part yields what are called restricted maximum likelihood (REML) estimators. As well as retaining the property of invariance under translation that ML estimators have, the REML estimators have the additional property of reducing to the analysis variance (ANOVA) estimators for many, if not all, cases of balanced data (equal subclass numbers). A computing algorithm is developed, adapting a transformation from Hemmerle and Hartley [6], which reduces computing requirements to dealing with matrices having order equal to the dimension of the parameter space rather than that of the sample space. These same matrices also occur in the asymptotic sampling variances of the estimators.

401 citations