Friday, March 15, 2013

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

2 comments:

Thomas said...

Cool!

Hilda said...

This is great!