[R] Fitting mixed-effects models with lme with fixed error term variances

[Douglas Bates]

On 11/21/06, Gregor Gorjanc <gregor.gorjanc at bfro.uni-lj.si> wrote:
> Douglas Bates <bates <at> stat.wisc.edu> writes:
> ...>
> > Can you be more specific about which parameters you want to fix and
> > which you want to vary in the optimization?
>
> It would be nice to have the ability to fix all variances i.e. variances of
> random effects.

That gets tricky in terms of the parameterization of the
variance-covariance matrices for vector-valued random effects.  These
matrices are not expressed in the conventional parameterization of
variances and covariances or even variances and correlation because
the conditions for the resulting matrix to be positive definite are
not simple bounds or easily expressed transformations then the matrix
is larger than 2 by 2.  I suppose what I could do is to allow these
matrices to be specified in the parameterization that is used in the
optimization and provide a utility function to map from the
conventional parameters to these.  That would mean that you couldn't
fix ,say, the variance of the intercept term for vector-valued random
effects but allow the variance of a slope for the same grouping factor
to be estimated.  Well, you could but only in the fortune("Yoda")
sense.

By the way, if you fix all the variances then what are you optimizing
over?  The fixed effects?  In that case the solution can be calculated
explicitly for a linear mixed model.  The conditional estimates of the
fixed effects given the variance components are the solution to a
penalized linear least squares problem.  (Yes, the solution can also
be expressed as a generalized linear least squares problem but there
are advantages to using the penalized least squares representation.
See

@Article{bates04:_linear,
  author = 	 {Douglas M. Bates and Saikat DebRoy},
  title = 	 {Linear mixed models and penalized least squares},
  journal = 	 {Journal of Multivariate Analysis},
  year = 	 2004,
  volume =	 91,
  number =	 1,
  pages =	 {1--17}
}
)