MIXED MODELS This chapter introduces best linear unbiased prediction (BLUP), a general method for predicting random effects, while Chapter 27 is concerned with the estima- tion of variances by restricted maximum likelihood (REML). These two methods are related in that BLUP assumes that the appropriate variance components are known, while REML procedures estimate variance components in an iterative fashion from BLUP estimates of random effects. Although the basic properties of these techniques have been known for decades, because of their computational demands, their practical application is a fairly recent phenomenon. BLUP is now by far the dominant methodology for estimating breeding values. After a brief introduction to the general mixed model, we will develop expres- sions for BLUEs (best linear unbiased estimators) of fixed effects and for BLUPs of random effects under the assumption that variances are known in the base population. THE GENERAL MIXED MODEL Consider a column vector y containing the phenotypic values for a trait measured in n individuals. We assume that these observations are described adequately by a linear model with a p × 1 vector of fixed effects (β) and a q × 1 vector of random effects (u). The first element of the vector β is typically the population mean, and other factors included may be gender, location, year of birth, experimental treatment, and so on. The elements of the vector u of random effects are usually genetic effects such as additive genetic values. In matrix form, y = Xβ + Zu + e (26.1) where X and Z are respectively n × p and n × q incidence matrices (X is also called the design matrix), and e is the n × 1 column vector of residual deviations assumed to be distributed independently of the random genetic effects. Usually, all of the elements of the incidence matrices are equal to 0 or 1, depending upon whether the relevant effect contributes to the individual’s phenotype. Because this model jointly accounts for fixed and random effects, it is generally referred to as a mixed model (Eisenhart 1947). Analysis of Equation 26.1 forms the basis for the remainder of this chapter and the next. Example 1. Suppose that three sires are chosen at random from a population, 745