Chapter 21 BLUP of Random Effects in Normal Linear Mixed Effects Model

Consider a linear mixed-effects model $y = X\beta + Zu + \epsilon, \text{ where } \begin{bmatrix} u \\ e \end{bmatrix} \sim N(\begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} G & 0 \\ 0 & R \end{bmatrix}).$ Given data $$y$$, what is our best guess for $$u$$?

In 611, we can show the BLUP of $$u$$ is $$GZ'\Sigma^{-1}(y - X\hat\beta_{\Sigma})$$ which can be viewed as an approximation of $$E(u\mid y) = GZ'\Sigma^{-1}(y - X\beta)$$.

Suppose $\begin{bmatrix} w_1 \\ w_2 \end{bmatrix} \sim N(\begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}, \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{bmatrix}),$ the conditional distribution of $$w_2$$ given $$w_1$$ is $$\left(\mathbf{w}_{2} \mid \mathbf{w}_{1}\right) \sim N\left(\mathbf{\mu}_{2}+\mathbf{\Sigma}_{21} \mathbf{\Sigma}_{11}^{-1}\left(\mathbf{w}_{1}-\mathbf{\mu}_{1}\right), \mathbf{\Sigma}_{22}-\mathbf{\Sigma}_{21} \mathbf{\Sigma}_{11}^{-1} \mathbf{\Sigma}_{12}\right)$$.

Based on the model, we have $\begin{gathered} {\left[\begin{array}{l} \mathbf{y} \\ \mathbf{u} \end{array}\right] \sim N\left(\left[\begin{array}{c} \mathbf{X} \mathbf{\beta} \\ \mathbf{0} \end{array}\right],\left[\begin{array}{cc} \mathbf{Z} & \mathbf{I} \\ \mathbf{I} & \mathbf{0} \end{array}\right]\left[\begin{array}{cc} \mathbf{G} & \mathbf{0} \\ \mathbf{0} & \mathbf{R} \end{array}\right]\left[\begin{array}{ll} \mathbf{Z}^{\prime} & \mathbf{I} \\ \mathbf{I} & \mathbf{0} \end{array}\right]\right)} \\ \stackrel{d}{=} N\left(\left[\begin{array}{c} \mathbf{X} \mathbf{\beta} \\ \mathbf{0} \end{array}\right],\left[\begin{array}{cc} \mathbf{Z} \mathbf{G} \mathbf{Z}^{\prime}+\mathbf{R} & \mathbf{Z} \mathbf{G} \\ \mathbf{G} \mathbf{Z}^{\prime} & \mathbf{G} \end{array}\right]\right) . \end{gathered}$ Therefore, $$E(u\mid y) = GZ'\Sigma^{-1}(y - X\beta)$$. To get the BLUP of $$u$$, we replace $$\beta$$ by $$\hat\beta_{\Sigma} = X(X'\Sigma^{-1}X)^-X'\Sigma^{-1}y$$.

For a usual case in which $$G$$ and $$\Sigma = ZGZ' + R$$ are unknown, we replace the matrices by estimates and approximate the BLUP of $$u$$ by $$\hat GZ'\hat\Sigma^{-1}(y - X\hat\beta_{\Sigma})$$. This approximation to the BLUP is called EBLUP.