[R] GLMMs & LMEs: dispersion parameters, fixed variances, design matrices

[Douglas Bates]

<Mark.Bravington at csiro.au> writes:

> Three related questions on LMEs and GLMMs in R:
> 
> (1) Is there a way to fix the dispersion parameter (at 1) in either
> glmmPQL (MASS) or GLMM (lme4)?
> 
> Note: lme does not let you  fix any variances in advance (presumably
> because it wants to "profile out"  an overall sigma^2 parameter) and
> glmmPQL repeatedly calls lme, so I couldn't see how glmmPQL would be
> able to fix the dispersion parameter. The section on glmmPQL in V&R4
> says that the  default is to estimate  the dispersion parameter, but
> didn't seem to say how to change the default.

At the core of the lme calculations is the solution of a penalized
least squares problem defined by the relative dispersion matrix of the
random effects and the model matrices for the random effects and the
fixed effects.  In versions 0.6-1 and later of the lme4 package (the
first release candidate is available from my web site
http://www.stat.wisc.edu/~bates/) the components of the log-likelihood
or the REML criterion are available as the devComp slot of the S4
object that represents the model and that is used to solve the
penalized least squares problem.  If, using these components, you can
write the log-likelihood for the model that you wish to fit then you
can give it to an optimizer such as optim or nlm to fit.

In the notation of Bates and DebRoy (2004), "Linear mixed models and
penalized least squares" (to appear in J. of Multivariate Analysis,
available in preprint form from my web site), the components are 

log(|Z'Z + \Omega|), log(|\Omega|), log(|R_{XX}|^2), and log(r_{yy}^2)

The C code that uses these to evaluate the deviance form of the
profiled log-likelihood criterion or the profiled REML criterion from
these components is in src/ssclme.c from the Matrix package.

Modifying the criteria for a fixed dispersion parameter may be trivial
or it may not.

> (3) Are there any plans to allow some variances to be fixed in lme? 
> It would be useful e.g. for meta-analysis (and indeed for glmms with
> fixed dispersion).

The method = 'Laplacian' version of the GLMM function fixes the
dispersion parameter in those families where it should be fixed.  As
we continue to develop lme4 we will provide a further enhancement
using an adaptive Gauss-Hermite evalution of the log-likelihood for
GLMMs that will also have this property.