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.