This article presents a deterministic method to predict rates of inbreeding (deltaF) for typical livestock improvement schemes. The method is based on a recently developed general theory to predict rates of inbreeding, which uses the concept of long-term genetic contributions. A typical livestock breeding population was modeled, with overlapping generations, BLUP selection, and progeny testing of male selection candidates. Two types of selection were practiced: animals were either selected by truncation on estimated breeding values (EBV) across age classes, or the number of parents selected from each age class was set to a fixed value and truncation selection was practiced within age classes. Bulmer's equilibrium genetic parameters were obtained by iterating on a pseudo-BLUP selection index and deltaF was predicted for the equilibrium situation. Predictions were substantially more accurate than predictions from other available methods, which ignore the effect of selection on deltaF. Predictions were accurate for schemes with up to 20 sires. Predicted deltaF was somewhat too low for schemes with more than 20 sires, which was due to the use of simple linear models to predict genetic contributions. The present method provides a computationally feasible (i.e., deterministic) tool to consider both the rate of inbreeding and the rate of genetic gain when optimizing livestock improvement schemes.

/pdf/predicting-rates-of-inbreeding-for-livestock-improvement-4axew1bomm.pdf

Predicting rates of inbreeding for livestock improvement schemes.

/pdf/estimation-of-genetic-merit-from-bivariate-all-or-none-1yzzufrap9.pdf

Estimation of genetic merit from bivariate « all or none » responses

Dutt’s and Deak’s methods can satisfy most of the geneticist’s needs. They are presented in this paper and their precision is analysed in detail. It appears that Dutt’s method is remarkably precise for dimensions 1 to 5, except when truncation points or correlation coefficients between traits are very high in absolute value. Deak’s method, less precise, is better suited for higher dimensions (6 to 20) and more generally for all the situations where Dutt’s method is no longer adequate.

/pdf/interest-in-quantitative-genetics-of-dutt-s-and-deak-s-1m7tppqhlk.pdf

Interest in quantitative genetics of Dutt's and Deak's methods for numerical computation of multivariate normal probability integrals

A method of genetic evaluation for multiple binary responses is presented. An underlying multivariate normal distribution is rendered discrete, in m dimensions, via a set of m fixed thresholds. There are 2\"’ categories of response and the probability of response in a given category is modeled with an m-dimensional multivariate normal integral. The argument of this integral follows a multivariate mixed linear model. The randomness of some elements in the model is taken into account using a Bayesian argument. Assuming that the variance-covariance structure is known, the mode of the joint posterior distribution of the fixed and random effects is taken as a point estimator. The problem is non-linear and iteration is required. The resulting equations indicate that the approach falls in the class of generalized linear models, with additional generalization stemming from the accommodation of random effects. A remarkable similarity with multiple trait evaluation via mixed linear models is observed. Important numerical issues arise in the implementation of the procedure and these are discussed in detail. An application of the method to data on calving preparation, calving difficulties and calf viability is presented.

/pdf/genetic-evaluation-for-multiple-binary-responses-10mubbhkht.pdf

Genetic evaluation for multiple binary responses.

is often practiced in breeding programs because extreme animals for some particular traits are rejected by breeders or because records on which genetic evaluation is based are collected sequentially. Optimizing these selection procedures for a given overall breeding objective is equivalent to finding the combination of truncation thresholds or culling levels which maximizes the expected value of the overall genetic value for selected animals. A general Newton-type algorithm has been derived to perform this maximization for any number of normally distributed traits and when the overall probability of being selected is fixed. Using a powerful method for the computation of multivariate normal probability integrals, it has been possible to undertake the numerical calculation of the optimal truncation points when up to 6 correlated traits or stages of selection are considered simultaneously. The extension of this algorithm to the more complex situation of maximizing annual genetic response subject to nonlinear constraints is demonstrated using a dairy cattle model involving milk production and a secondary trait such as type. Consideration is given to three of the four pathways of selection: dams of bulls; sires of bulls; and sires of cows.

/pdf/optimum-truncation-points-for-independent-culling-level-4y1g1cp7va.pdf

Optimum truncation points for independent culling level selection on a multivariate normal distribution, with an application to dairy cattle selection

Recently in Dutt (1973, (1975), intgral representations over (0,A) were obtained for upper and lover multivariate normal and the probilities. It was pointed out that these integral representaitons when evaluated by Gauss-Hermite uadrature yield rapid and accurate numerical results. Here integral representaitons, based on an integral formula due to Gurland (1948), are indicated for arbitrary multivariate probabilities. Application of this general representaion for computing multivariate x2 probabilities is discussed and numerical results using Gaussian quadrature are given for the bivariate and equicorre lated trivariate cases. Applications to the multivariate densities studied by Miller (1965) are also included

An integral representation technique for calculating general multivariate probabilities with an application to multivariate x2

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Dutt J.E

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An integral representation technique for calculating general multivariate probabilities with an application to multivariate x2