[R] Help with lme basics

Peter Dalgaard BSA p.dalgaard at biostat.ku.dk
Fri Apr 19 18:44:18 CEST 2002

Douglas Bates <bates at stat.wisc.edu> writes:

> I think this comparison shows that lme evaluates the significance of
> terms differently from models that employ error strata.  I don't think
> it shows that the lme model is "wrong".  
> I will leave it to someone who understands error strata better than I
> do to explain the rationale of the results based on error strata.

Well, it's all a matter of orthogonal decompositions and dimensions of
subspaces. If you have an orthogonal design for the error components,
and the treatment terms each fall within a single part of the
decomposition, then all the tests are exact (if the model holds!).
Mostly, it comes down to analyzing certain sets of means and
differences of means. There's only a slight approximation involved in
that the variance components are allowed to be slightly negative.

However, it doesn't generalize well to non-orthogonal designs. If you
have one of those and persist in performing the decompostion into
error strata, this will correspond to a strange model for the
covariance structure of the data set. For instance, in one-way ANOVA
with random group effects, you'll end up with a different
"between"-variance for smaller groups than for larger.

In contrast, lme() fits perfectly sensible models for these cases, but
the price of the generality is that it doesn't have the exact
distribution of the test statistics, even when data happens to be from
an orthogonal design. As far as I know, all current mixed-effects
codes "get it wrong" in slightly different ways. The results only hold
asymptotically. Heuristical arguments get employed but it is fairly
easy to construct examples where they would seem to be insufficient.

(E.g. suppose you have Y = mu + Z_g + Z_gi, with groups g = 1...8  and
varying observations within groups. The containment method suggests
that you have 7 DF for inferences about mu, but if you have 1000
observations in each of groups 1-4 and only 1 in groups 5-8, and the
Z_g variance is relatively small, then I'd expect the "correct"
number of DF to be closer to 3).

   O__  ---- Peter Dalgaard             Blegdamsvej 3  
  c/ /'_ --- Dept. of Biostatistics     2200 Cph. N   
 (*) \(*) -- University of Copenhagen   Denmark      Ph: (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)             FAX: (+45) 35327907
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