ASReml-S reference manual - VSN International

2.3 What are BLUPs? 132.2.2 Fixed and Random effectsTo estimate τ and predict u the objective functionlog f Y (y | u ; τ , R) + log f U (u ; G)is used. The is the log-joint distribution of (Y , u). It is not a log-likelihood though inextensions to non-normal data it has been treated as a log-likelihood.Differentiating with respect to τ and u leads to the mixed model equations [Robinson,1991] which are given by[ X ′ R −1 X X ′ R −1 ] [ ] [Z ˆτ X ′ R −1 ]yZ ′ R −1 X Z ′ R −1 Z + G −1 =ũ Z ′ R −1 . (2.11)yThese can be written asC ˜β = W R −1 ywhere C = W ′ R −1 W + G ∗ , W = [X Z] , β = [τ ′ u ′ ] ′ and[ ]G ∗ 0 0=0 G −1 .The solution of (2.11) requires values for γ and φ. In practice we replace γ and φ bytheir REML estimates ˆγ and ˆφ.Note that ˆτ is the best linear unbiased estimator (BLUE) of τ , while ũ is the best linearunbiased predictor (BLUP) of u. for known γ and φ. We also note that[ ] ([ ] )ˆτ − τ 0 ˜β − β = ∼ N , C −1ũ − u 0.2.3 What are BLUPs?Consider a balanced one-way classification. In the following we assume, that the treatmenteffects, say, u i are random. That is, u ∼ N(Aν, σ 2 b I b), for some design matrix Aand parameter vector ν. It can be shown thatũ =bσ2 bbσ 2 b + σ2 (ȳ − 1ȳ··) +bσ 2 bσ 2+ σ2Aν (2.12)where ȳ is the vector of treatment means and ȳ·· is the grand mean. The differences of thetreatment means and the grand mean are the estimates of treatment effects if treatmenteffects are fixed. The BLUP is therefore a weighted mean of the data based estimate andthe ‘prior’ mean Aν. If ν = 0, the BLUP in (2.12) becomesũ =bσ2 bbσ 2 b + σ2 (ȳ − 1ȳ··) (2.13)and the BLUP is a so-called shrinkage estimate. As σ 2 b becomes large relative to σ2 , theBLUP tends to the fixed effect solution, while for small σ 2 b relative to σ2 the BLUP tendstowards zero, the assumed initial mean. Thus (2.13) represents a weighted mean whichinvolves the prior assumption that the u i have zero mean.Note also that the BLUPs in this simple case are constrained to sum to zero. This isessentially because the unit vector defining X can be found by summing the columnsof the Z matrix. This linear dependence of the matrices translates to dependence ofthe BLUPs and hence constraints. This aspect occurs whenever the column space of Xis contained in the column space of Z. The dependence is slightly more complex withcorrelated random effects.