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

[Douglas Bates]

On 11/22/06, Gregor Gorjanc <gregor.gorjanc at bfro.uni-lj.si> wrote:
> Douglas Bates wrote:
> > 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.
> >
>
> Yes, I agree here. Thank you for the detailed answer.
>
> > 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.
>
> Yup. It would really be great to be able to do that nicely in R, say use
> lmer() once and since this might take some time use estimates of
> variance components next time to get fixed and random effects. Is this
> possible with lmer or any related function - not in fortune("Yoda") sense ;)

Not quite.  There is now a capability in lmer to specify starting
estimates for the relative precision matrices which means that you can
use the estimates from one fit as starting estimates for another fit.
It looks like

fm1 <- lmer(...)
fm2 <- lmer(y ~ x + (...), start = fm1 at Omega)

I should say that in my experience this has not been as effective as I
had hoped it would be.  What I see in the optimization is that the
first few iterations reduce the deviance quite quickly but the
majority of the time is spent refining the estimates near the optimum.
 So, for example, if it took 40 iterations to converge from the rather
crude starting estimates calculated within lmer it might instead take
35 iterations if you give it good starting estimates.