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

[Douglas Bates]

On Mon, May 25, 2009 at 4:46 PM, Roger Levy <rlevy at ling.ucsd.edu> wrote:
> Douglas Bates wrote:
>>
>> On Sun, May 24, 2009 at 5:46 PM, Roger Levy <rlevy at ling.ucsd.edu> wrote:
>>>
>>> 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.
>>
>> I view the variance-covariance structures available in the lme4
>> package as being related to random-effects terms in the model matrix.
>> A random-effects term is of the form (LMexpr | GrpFac).  The
>> expression on the right of the vertical bar is evaluated as a factor,
>> which I call the grouping factor.  The expression on the left is
>> evaluated as a linear model expression.  The number of columns in the
>> model matrix corresponding to this expression is the number of random
>> effects per level of the grouping factor.
>>
>> The basic rules for the unconditional variance-covariance of the
>> random effects are:
>>  random effects generated from different random-effects terms are
>> independent
>>  random effects corresponding to different levels of the grouping
>> factor are independent
>>  the vector of random effects for a given level of the grouping
>> factor have a general positive semidefinite symmetric
>> variance-covariance, which is common to all the levels of the grouping
>> factor.
>>
>> In future versions of lme4 I plan to allow for extensions of the
>> unconditional variance-covariance structures.  If you look at the
>> development version in the branches/allcoef section of the SVN archive
>> at R-forge you will see that there is a virtual class called the
>> reCovFac (random-effects covariance factor) class.  If an actual class
>> is defined to extend reCovFac and certain methods (getPars, setPars,
>> getBounds, getLambda) are defined for the actual class then it can be
>> used instead of the default ST class.
>
> Hi Doug,
>
> Many thanks for your detailed response to my question -- I just want to ask
> one clarificatory follow-up.  By "*unconditional* variance-covariance
> structure" you are meaning prior to conditioning on the data, is that
> correct?

Yes.

>>
>>> 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
>>>
>>> ?
>>
>> In its current implementation, no, the lme4 package does not allow
>> general modeling of the unconditional variance-covariance structure of
>> the random effects.  I doubt that it will, just because I find it
>> difficult to understand the model in that way.  Generalizing the model
>> is not just a matter of adding hooks - you also need to decide what
>> can go wrong in the generalized structure.  I have said that the most
>> valuable character trait for programmers is unbounded pessimism
>> because you spend so much of your time trying to decide how things
>> could fail to work.  In early designs of the nlme package when we
>> created a pdMat subclass to represent positive definite matrices that
>> were not matrices I knew we were in trouble.
>>
>> You can try to extend the reCovFac class but doing so in a consistent
>> way is not always easy.
>>
>>> 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
>>>
>>> _______________________________________________
>>> R-sig-mixed-models at r-project.org mailing list
>>> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>>>
>
>