Best linear unbiased prediction of breeding value: univariate models with one random effect.

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AbstractIn Chapter 1, the use of the selection index (best linear prediction) for genetic evaluation was examined; however, it is associated with some major disadvantages. First, records may have to be pre-adjusted for fixed or environmental factors and these are assumed to be known. These are not usually known, especially when no prior data exist for new subclasses of fixed effect or new environmental factors. Secondly, solutions to the index equations require the inverse of the covariance matrix for observations and this may not be computationally feasible for large data sets. Henderson (1949) developed a methodology called best linear unbiased prediction (BLUP), by which fixed effects and breeding values can be simultaneously estimated. The properties of the methodology are similar to those of a selection index and the methodology reduces to selection indices when no adjustments for environmental factors are needed. The properties of BLUP are more or less incorporated in the name: * Best - means it maximizes the correlation between true (a) and predicted breeding value (â) or minimizes prediction error variance (PEV) (var(a-â)). * Linear - predictors are linear functions of observations. * Unbiased - estimation of realized values for a random variable, such as animal breeding values, and of estimable functions of fixed effects are unbiased (E(a=â)). * Prediction - involves prediction of true breeding value. BLUP has found widespread usage in genetic evaluation of domestic animals because of its desirable statistical properties. This has been enhanced by the steady increase in computing power and has evolved in terms of its application to simple models, such as the sire model, in its early years, and to more complex models, such as the animal, maternal, multivariate and random regression models, in recent years. Several general purpose computer packages for BLUP evaluations, such as PEST (Groeneveld et al., 1990), BREEDPLAN and a host of others, have been written and made available. In this chapter, BLUP's theoretical background is briefly presented below, considering a univariate animal model, and its application to several univariate models in genetic evaluation is illustrated.