BLUPModelle

BLUPModelle refers to statistical models that implement Best Linear Unbiased Prediction (BLUP) for predicting random effects within linear mixed models. The approach is widely used in quantitative genetics, animal and plant breeding, and related fields to obtain simultaneous estimates of fixed effects and random effects under a specified variance-covariance structure.

Statistical framework and estimation. In the typical linear mixed model, y = Xβ + Zu + ε, where y is the observation vector, Xβ contains fixed effects, Z is the design matrix for random effects, and u and ε are random and residual terms with u ~ N(0, G) and ε ~ N(0, R). BLUP provides the best linear unbiased predictions of the random effects u (and the BLUE of β) given the variance components in G and R. In practice, these components are estimated from data, often by restricted maximum likelihood (REML), and the resulting Henderson mixed model equations are solved to obtain û and β̂.

Extensions and variants. Empirical BLUP (EBLUP) uses estimated variance components. Genomic BLUP (GBLUP) replaces pedigree-based relationships with a genomic relationship matrix derived from marker data, enabling genomic selection. Related approaches may incorporate Bayesian ideas or alternative covariance structures, but BLUP remains the cornerstone for shrinkage-adjusted predictions under a mixed-model framework.

Applications and limitations. BLUPModelle are central to estimating breeding values in animal and crop genetics, evaluating genetic merit, and analyzing longitudinal data in ecologies and clinical studies. Advantages include efficiency with unbalanced data, coherence of fixed and random effects, and shrinkage that reduces overfitting. Limitations involve reliance on correctly specified variance components and distributional assumptions; mis-specification can bias predictions, particularly for individuals with limited information.