## Friday, March 15, 2013

### How are the random effects (BLUPs) predicted' in linear mixed models?

In linear mixed models, we fit models like these (the Ware-Laird formulation--see Pinheiro and Bates 2000, for example):

\begin{equation}
Y = X\beta + Zu + \epsilon
\end{equation}

Let $u\sim N(0,\sigma_u^2)$, and this is independent from $\epsilon\sim N(0,\sigma^2)$.

Given $Y$, the `minimum mean square error predictor'' of $u$ is the conditional expectation:

\begin{equation}
\hat{u} = E(u\mid Y)
\end{equation}

We can find $E(u\mid Y)$ as follows. We write the joint distribution of $Y$ and $u$ as:

\begin{equation}
\begin{pmatrix}
Y \\
u
\end{pmatrix}
=
N\left(
\begin{pmatrix}
X\beta\\
0
\end{pmatrix},
\begin{pmatrix}
V_Y & C_{Y,u}\\
C_{u,Y} & V_u \\
\end{pmatrix}
\right)
\end{equation}

$V_Y, C_{Y,u}, C_{u,Y}, V_u$ are the various variance-covariance matrices.
It is a fact (need to track this down) that

\begin{equation}
u\mid Y \sim N(C_{u,Y}V_Y^{-1}(Y-X\beta)),
Y_u - C_{u,Y} V_Y^{-1} C_{Y,u})
\end{equation}

This apparently allows you to derive the BLUPs:

\begin{equation}
\hat{u}= C_{u,Y}V_Y^{-1}(Y-X\beta))
\end{equation}

Substituting $\hat{\beta}$ for $\beta$, we get:

\begin{equation}
BLUP(u)= \hat{u}(\hat{\beta})C_{u,Y}V_Y^{-1}(Y-X\hat{\beta}))
\end{equation}

Here is a working example:

### Correlations of fixed effects in linear mixed models

Ever wondered what those correlations are in a linear mixed model? For example:

The estimated correlation between $\hat{\beta}_1$ and $\hat{\beta}_2$ is $0.988$.  Note that

$\hat{\beta}_1 = (Y_{1,1} + Y_{2,1} + \dots + Y_{10,1})/10=10.360$

and

$\hat{\beta}_2 = (Y_{1,2} + Y_{2,2} + \dots + Y_{10,2})/10 = 11.040$

From this we can recover the correlation $0.988$ as follows:

By comparison, in the linear model version of the above:

because $Var(\hat{\beta}) = \hat{\sigma}^2 (X^T X)^{-1}$.