Download pdf guide - VSN International

2 Some theory 152.3 What are BLUPs?Consider a balanced one-way classification. For data records ordered r repeatswithin b treatments regarded as random effects, the linear mixed model is y =Xτ + Zu + e where X = 1 b ⊗ 1 r is the design matrix for τ (the overall mean),Z = I b ⊗ 1 r is the design matrix for the b (random) treatment effects u i and eis the error vector. Assuming that the treatment effects are random implies thatu ∼ N(Aψ, σb 2I b), for some design matrix A and parameter vector ψ. It can beshown thatũ =rσ2 brσb 2 + (ȳ − 1ȳ··) σ 2+ Aψ (2.12)σ2 + σ2where ȳ is the vector of treatment means, ȳ·· is the grand mean. The differencesof the treatment means and the grand mean are the estimates of treatment effectsif treatment effects are fixed. The BLUP is therefore a weighted mean of the databased estimate and the ‘prior’ mean Aψ. If ψ = 0, the BLUP in (2.12) becomesũ =rσ 2 brσ2 brσ 2 b + σ2 (ȳ − 1ȳ··) (2.13)and the BLUP is a so-called shrinkage estimate. As rσb 2 becomes large relative toσ 2 , the BLUP tends to the fixed effect solution, while for small rσb 2 relative to σ2the BLUP tends towards zero, the assumed initial mean. Thus (2.13) represents aweighted mean which involves 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. Thisis essentially because the unit vector defining X can be found by summing thecolumns of the Z matrix. This linear dependence of the matrices translates todependence of the BLUPs and hence constraints. This aspect occurs wheneverthe column space of X is contained in the column space of Z. The dependenceis slightly more complex with correlated random effects.