[R-sig-ME] Some timings for lmer2 versus lmer

[Douglas Bates]

I enclose an R source file to do some comparative timings on lmer2
fits versus lmer fits and the output generated on the machine
R-forge.R-project.org (Opteron 280 dual-core processors, R internal
BLAS, 13 GB of memory).  You can try running the script on your
computer to get an idea of the timings.

On some machines the lmer fit to the "star" data set will converge in
considerably fewer iterations than on this machine.  There is one
point in the optimization where very small differences in the floating
point operation orders cause a much better step to be taken.

If you look at the verbose output you will see that the
parameterization for lmer2 uses the relative standard deviation (as
described in the Implementation vignette from the lme4 package)
whereas lmer used the relative variance.  Generally the relative
standard deviation is more stable for the optimization.

The other big difference in the optimization, shown in the last
example, is that lmer evaluates the relative precision matrix (the
inverse of the relative variance matrix) and therefore cannot allow
variance components to go to zero.  The value of a variance component
is bounded below at 5e-10, which is why that particular number shows
up in the verbose iterations.  As described in the vignette, the
relative variance matrix is used in lmer2 hence the lower bound on the
variance component is at zero.
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