[R-sig-ME] specifying custom random-effects structures

[Roger Levy]

Dear all,

I want to understand the range of random-effects covariance structure  
specifications that (a) can be handled by lme4, MCMCglmm, and related  
packages; and (b) that are reasonable to posit in principle as random- 
effects covariance structures.

If I understand correctly, lme4 handles random-effects structures that  
can be expressed as the direct sum of k arbitrary covariance matrices  
-- that is, something that looks like

M1  0   0   0
0   M2  0   0
0   0  ...  0
0   0   0   Mk

where each Mi is a covariance matrix without any constraints placed on  
its internal structure.

Is it possible to place constraints on the internal structure of each  
of these covariance matrices?  For example, suppose Mi is the  
covariance matrix for variables x1, x2, and x3. Is it possible to  
specify that Mi has the structure

\sigma_11 \sigma_12    0
\sigma_12 \sigma_22 \sigma_23
   0      \sigma_23 \sigma_33

?  Likewise, if Mj is the covariance matrix for variables x4 and x5,  
is it possible to specify that Mj has the structure

\sigma_44    1
   1      \sigma_55

?

Additionally, regardless of technical feasibility, are these sensible  
specifications in principle?  I can imagine a circumstance in which  
the latter specification would make sense: when there is theoretical  
reason to believe that the role of x4 and x5 in determining the  
response is mediated through some inaccessible third variable that is  
a linear combination of x4 and x5, but the parameters of the linear  
combination are unknown.  I'm not so sure about the former  
specification...but for some datasets I work with, I have in fact seen  
inferred covariance structures close to this form.

Best

Roger

--

Roger Levy                      Email: rlevy at ling.ucsd.edu
Assistant Professor             Phone: 858-534-7219
Department of Linguistics       Fax:   858-534-4789
UC San Diego                    Web:   http://ling.ucsd.edu/~rlevy