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 $\mathcal{B}.$

摘要

一般线性随机效应模型$\mathbf{是的}=\mathbf{X}\mathit{\beta }+\mathit{\epsilon }$和$\mathit{\beta }=\mathbf{Z}\mathit{\alpha }+\mathit{\gamma }$包括固定效应和随机效应及其两个转换模型$\mathcal{一个}：$ $\mathbf{哎}=\mathbf{AXZ}\mathit{\alpha }+\mathbf{斧头}\mathit{\gamma }+\mathbf{一个}\mathit{\epsilon }$和$\mathcal{乙}：$ $\mathbf{经过}=\mathbf{BXZ}\mathit{\alpha }+\mathbf{BX}\mathit{\gamma }+\mathbf{乙}\mathit{\epsilon }$对随机效应的相关性和任何满秩假设没有任何限制。变换模型下联合未知参数向量的预测因子$\mathcal{一个}$和$\mathcal{乙}$在两个转换模型的上下文中具有不同的代数表达式和不同的属性。在这种情况下，建立关系结果并在两个模型下的预测变量之间进行比较是主要关注点。我们首先研究模型下固定效应和随机效应的一般线性函数的最佳线性无偏预测因子 (BLUP) 的关系$\mathcal{一个}$和$\mathcal{乙}$并为 BLUP 构造几个等式。然后，考虑模型下BLUPs协方差矩阵的比较问题。我们从矩阵秩和惯性公式推导出模型下协方差矩阵比较的各种等式和不等式的充要条件$\mathcal{一个}$和$\mathcal{乙}.$