A two-stage approach to recovery of inter-block information and shrinkage of block effect estimates

Linear mixed models often comprise several effects, and the focus is usually only on one or a few of them, while the other effects need to be fitted merely to adjust for all sources of variation. A typical example is the analysis of a blocked experiment, where the effects of interest pertain to treatments, while effects for replicates and incomplete blocks need to be taken into account in order to obtain efficient estimates of treatment effects. If fixed and random effects that are not of major interest were known beforehand, we could subtract these from the observed data, and a reduced model could then be fitted to these corrected data in order to estimate the effects of interest. In practice, this approach cannot be used directly, however, because true values of the effects are unknown. But we may replace unknown effects by their estimates. We show in this paper, that a reduced model fitted to these 'empirically' corrected data yields BLUE and BLUP of fixed and random effects of interest in the full model. Using examples, this result is demonstrated to be useful for illustrating the recovery of inter-block information and for understanding the properties of estimators obtained from mixed-model analysis.