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

Roger Levy rlevy at ling.ucsd.edu
Mon May 25 00:46:41 CEST 2009

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