On relations between BLUPs under two transformed linear random-effects models

Abstract

A general linear random-effects model $\mathbf{y}=\mathbf{X}\mathit{\beta }+\mathit{\epsilon }$ with $\mathit{\beta }=\mathbf{Z}\mathit{\alpha }+\mathit{\gamma }$ that includes both fixed and random effects and its two transformed models $\mathcal{A}:$ $\mathbf{Ay}=\mathbf{AXZ}\mathit{\alpha }+\mathbf{AX}\mathit{\gamma }+\mathbf{A}\mathit{\epsilon }$ and $\mathcal{B}:$ $\mathbf{By}=\mathbf{BXZ}\mathit{\alpha }+\mathbf{BX}\mathit{\gamma }+\mathbf{B}\mathit{\epsilon }$ are considered without making any restrictions on correlation of random effects and any full rank assumptions. Predictors of joint unknown parameter vectors under the transformed models $\mathcal{A}$ and $\mathcal{B}$ have different algebraic expressions and different properties in the contexts of the two transformed models. In this situation, establishing results on relations and making comparisons in between predictors under the two models are the main focuses. We first investigate relationships of best linear unbiased predictors (BLUPs) of general linear functions of fixed and random effects under the models $\mathcal{A}$ and $\mathcal{B}$ and construct several equalities for the BLUPs. Then, the comparison problem of covariance matrices of BLUPs under the models is considered. We derive from matrix rank and inertia formulas the necessary and sufficient conditions for variety of equalities and inequalities of covariance matrices’ comparisons under the models $\mathcal{A}$ and $\mathcal{B}.$

Keywords:

• BLUP
• Covariance matrix
• Inertia
• Linear random-effects model
• Rank
• Transformed model

MSC:

• 62J05
• 62J10
• 62H12
• 15A03

Acknowledgements

The authors are grateful to anonymous referees for their helpful comments and constructive suggestions that improved the presentation of the article.