@Article{meng:vandyk:1998,
author = "Xiao-Li Meng and David van Dyk",
title = "Fast EM-type Implementations for Mixed Effects Models",
journal = "Journal of the Royal Statistical Society,
Series B",
year = 1998,
volume = 60,
pages = "559--578"
}
where the authors present alternative fitting algorithms to the
newton-type methods for linear mixed effects models used by
lme(). In passing they found some simulation examples where
lme() appeared not to give the correct ML solution. I approached
the authors and asked for the details of the simulation examples,
which they very promptly and helpfully supplied.
As is often the case with simulation the examples are artificial
and unlikely to be close to ones met practice, but nevertheless
the authors do have a point. The data sets all consisted of 100
clusters of size just 2. There were two covariates, Z1 and Z2,
which had random coefficients with no fixed effect component, so
the specification was
fixed = Y ~ 1,
random = ~Z1 + Z2 - 1,
cluster = ~clust, # 100 levels each of size 2
est.method = "ML"
There are 10 data sets, all of this identical structure. Meng
and van Dyk reported that in 5 both lme and their method found
the same solution, but in the other 5 the lme solution had a
lower likelihood and hence was incorrect.
I have now repeated the calculations and I agree with their
findings. The failure is not something inherent in algorithm,
though, but a problem with the iteration governing mechanism.
With some tweaking of control parameters it can be made to work.
I thought I would also see how the recently released NLME3 (now
on beta test, but available) version handled the same examples.
The table below gives the (exactly comparable) maximum
log-likelihood values achieved by the three implementations.
ECME lme3 (new) lme2 (current)
------- ---------- --------
Data set 1 -429.91 -429.91031 -429.9103
Data set 2 -465.698 -465.69754 -465.698
Data set 3 -494.153 -494.15305 -494.153
Data set 4 -571.973 -571.9741 -571.973
Data set 5 -614.353 -614.35355 -614.354
Data set 6 -639.986 -639.98608 -656.28
Data set 7 -674.744 -674.74439 -680.499
Data set 8 -695.136 -695.13741 -696.912
Data set 9 -712.727 -712.72657 -714.523
Data set 10 -731.330 -731.32988 -732.077
In other words the lme3 function agrees with ECME in all 10 cases
(to iteration error), but lme2 stops iterating too early in 5 of
the 10. Both the lme2 and lme3 results were got with the default
settings of all control parameters and no tweaking.
Details of the computational procedure used by lme3 are set out
in a technical report: "Computational Methods for Multilevel
Models" by D. M. Bates and J. C. Pinheiro and is available at
http://cm.bell-labs.com/stat/project/nlme/CompMulti.pdf
or
http://franz.stat.wisc.edu/pub/NLME/CompMulti.pdf
My personal conclusions are as follows:
1. There may be a problem with lme2 for very unusual models with
a relatively low information base and flat likelihood. With
such cases you may need to try re-setting the control
parameters to see if it makes a difference, in particular to
check for early termination of the iteration procedure.
2. If you use lme (or nlme) quite a bit, the NLME3 version seems
to be computationally more robust and there may be some
unexpected bonuses in shifting to it sooner rather than later.
(There are other good reasons to do so, anyway.)
3. The ideas presented by Meng and van Dyk are very interesting
and inventive and more comparisons would be valuable.
Ultimately some hybrid or adaptive algorithm incorporating
ideas from all sides may prove to be the most effective.
In my view, though, lme2 is not seriously broken and I have seen
no evidence of it giving misleading results in realistic cases.
As with most statistical software, though, great care and some
scepticism are needed.
Finally I should say that I am offering these comments entirely
on my own initiative as a fulltime user (and sometime expositor)
of S-PLUS only. I trust they fairly represent the situation.
Bill Venables.
-- Bill Venables, Head, Dept of Statistics, Tel.: +61 8 8303 5418 University of Adelaide, Fax.: +61 8 8303 3696 South AUSTRALIA. 5005. Email: Bill.Venables@adelaide.edu.au----------------------------------------------------------------------- This message was distributed by s-news@wubios.wustl.edu. To unsubscribe send e-mail to s-news-request@wubios.wustl.edu with the BODY of the message: unsubscribe s-news