The R Book

Best Linear Unbiased Predictors

In aov, the effect size for treatment i is defined as _i – μ, where μ is the overall mean. In mixed-effects models, however, correlation between the pseudoreplicates within a group causes what is called shrinkage. The best linear unbiased predictors (BLUPs, denoted by a_i) are smaller than the effect sizes (_i – μ), and are given by

where σ² is the residual variance and σ²_a is the between-group variance which introduces the correlation between the pseudoreplicates within each group. Thus, the parameter estimate a_i is ‘shrunk’ compared to the fixed effect size (_i – μ). When σ²_a is estimated to be large compared with the estimate of σ²/n, then the fixed effects and the BLUP are similar (i.e. when most of the variation is between classes and there is little variation within classes). On the other hand, when σ²_a is estimated to be small compared with the estimate of σ²/n, then the fixed effects and the BLUP can be very different (p. 547).