 ## BLUEs, BLUPs and Breeding Values?

C. Supakorn

In a linear mixed model analysis, Best Linear Unbiased Prediction (BLUP) is used to estimate random effects and Best Linear Unbiased Estimation (BLUE) to estimate fixed effects.

 Best Solutions (i.e. predictions or estimates) have minimum residual variance among all unbiased linear estimators Linear Solutions are a linear combination of the observations Unbiased Expectations of the solutions are equal to the true values The use of BLUP to predict random effects was first described by C. R. Henderson. In 1948, Henderson’s professional career in animal breeding and genetics began with his appointment as an Associate Professor at Cornell University, where he remained for nearly 30 years. During this time, he developed Henderson’s mixed model equations (MME, see below) for linear mixed models in order to obtain BLUPs of breeding values (or any random effect) and BLUEs of fixed effects.

##### C.R. Henderson

Let’s see how predicted breeding values (i.e. BLUPs of random effects) are obtained from a linear mixed model.

The linear mixed model equation can be written in matrix notion as:

y = Xb + Zu + e                                                           (1)

where y, b, u, and e are vectors of observations, (unknown) fixed effects, (unknown) random effects, and (unknown) random residuals, respectively, and X and Z are design matrices connecting the observations to the fixed and random effects.

The central problem in predicting breeding values from observed phenotypic data is separating the genetic and environmental effects. In statistical terms, this is done by estimating the fixed environmental effects () whilst simultaneously predicting the realized values of the random breeding values for the individual animals (û). The estimated fixed () and random (û) effects solve Henderson’s MME:  Let’s see an example! We need to evaluate BLUEs for the fixed effect of sex and BLUPs of all animals.

Let’s assume that additive genetic and residual variances equal 20 and 40 kg2, respectively.

From MME in equation 2, 