[R-sig-ME] general GLMM questions

[Nicholas Lewin-Koh]

Hi 
My one comment is below in #2

Nicholas
On Thu, 08 May 2008 10:46:45 +0200,
r-sig-mixed-models-request at r-project.org said:
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> 
> Message: 1
> Date: Wed, 07 May 2008 11:39:55 -0400
> From: Ben Bolker <bolker at ufl.edu>
> Subject: [R-sig-ME] general GLMM questions
> To: R Mixed Models <r-sig-mixed-models at r-project.org>
> Message-ID: <4821CD4B.5090701 at ufl.edu>
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> Dear sig-mixed readers,
> 
> ~  Some of my students and I are foolishly attempting to write a review
> of GLMMs for an ecology/evolution audience.  Realizing that this is a
> huge, gnarly, and not-completely-understood subject (even for the
> experts -- see fortune("mixed")), we're trying to provide as much
> non-technical background and guidance as can be squeezed into a
> reasonable sized journal article ...
> 
> ~  We've run into quite a few questions we have been unable to answer
> ... perhaps because no-one knows the answers, or because we're looking
> in the wrong places.  We thought we might impose on the generosity of
> the list: if this feels ridiculous, please just ignore this message.
> Feedback ranging from "this is true, but I don't know of a published
> source" to "this just isn't true" would be useful.  We are aware of the
> deeper problems of focusing on p-values and degrees of freedom -- we
> will encourage readers to focus on estimating effect sizes and
> confidence limits -- but we would also like to answer some of these
> questions for them, if we can.
> 
> ~  Ben Bolker
> 
> 1. Is there a published justification somewhere for Lynn Eberly's
> ( http://www.biostat.umn.edu/~lynn/ph7430/class.html ) statement that df
> adjustments are largely irrelevant if number of blocks>25 ?
> 
> 2. What determines the asymptotic performance of the LRT (likelihood
> ratio test) for comparison of fixed effects, which is known to be poor
> for "small" sample sizes?  Is it the number of random-effects levels
> (as stated by Agresti 2002, p. 520), or is it the number of levels of
> the fixed effect relative to the total number of data points (as
> stated by Pinheiro and Bates 2000, pp. 87-89)?  (The example given by
> PB2000 from Littell et al. 1996 is a test of a treatment factor with
> 15 levels, in a design with 60 observations and 15 blocks.  Agresti's
> statement would imply that one would still be in trouble if the total
> number of observations increased to 600 [because # blocks is still
> small], where PB2000 would imply that the LRT would be OK in this
> limit.  (A small experiment with the simulate.lme() example given on
> PB2000 p. 89 suggests that increasing the sample size 10-fold with the
> same number of blocks DOES make the LRT OK ... but I would need to do
> this a bit more carefully to be sure.)  (Or is this a difference
> between the linear and generalized linear case?)

I ran across this issue a long time ago when I was working on my MS.
I was working on a hierarchical Poisson regression model, 
for species area curves comparing landbridge and oceanic islands.
I had over 400 data points (Islands) but only about 20 landbridge and 
12 oceanic Island chains. However the amount of information in the
posterior, 
another question to ponder, of the slopes high, in other words
there was a strong effect on the posterior, even though
the data that level was sparse. It seems that the comment prior
to mine is spot on that it depends on the the covariance, in my case
there was a lot of information on each slope that carried upward through
the model.
So in a nutshell I think the LRT will be ok when there is enough
information, either coming from the precision of the estimates from with
in the block
or enough blocks to estimate the random effects parameter. How to
account for that
precisely I leave to better theoreticians than I. 






> 
> 3. For multi-level models (nested, certainly crossed), how would one
> count the "number of random-effects levels" to quantify the 'sample
> size' above?  With a single random effect, we can just count the
> number of levels (blocks).  What would one do with e.g. a nested or
> crossed design?  (Perhaps the answer is "don't use a likelihood ratio
> test to evaluate the significance of fixed effects".)
> 
> 4. Does anyone know of any evidence (in either direction) that the
> "boundary" problems that apply to the likelihood ratio test (e.g. Self
> and Liang 1987) also apply to information criteria comparisons of
> models with and without random effects?  I would expect so, since the
> derivations of the AIC involve Taylor expansions around the
> null-hypothesis parameters ...
> 
> 5. It's common sense that estimating the variance of a random effect
> from a small number of levels (e.g. less than 5) should be dicey, and
> that one might in this case want to treat the parameter as a fixed
> effect (regardless of its philosophical/experimental design status).
> For small numbers of levels I would expect (?) that the answers MIGHT
> be similar -- among other things the difference between df=1 and
> df=(n-1) would be small.  But ... is there a good discussion of this
> in print somewhere?  (Crawley mentions this on p. 670 of "Statistical
> Computing", but without justification.)
> 
> lme4-specific questions:
> 
> 6. Behavior of glmer: Does glmer really use AGQ, or just Laplace?
> Both?  pp. 28-32 of the "Implementation" vignette in lme4 suggest that
> a Laplace approximation is used, but I can't figure out whether this
> is an additional approximation on top of the AGQ/Laplace approximation
> of the integral over the random effects used in "ordinary" LMM.  When
> I fit a GLMM with the different methods, the fitted objects are not
> identical but all the coefficients seem to be.  (I have poked at the
> code a bit but been unable to answer this question for myself
> ... sorry ...)
> 
> (The glmmML package claims to fit via Laplace or Gauss-Hermite
> quadrature (with non-adaptive, but adjustable, number of quad points
> - -- so it's at least theoretically possible?)
> 
> library(lme4)
> set.seed(1001)
> f = factor(rep(1:10,each=10))
> zb = rnorm(1:10,sd=2) ## block effects
> x = runif(100)
> eta = 2*x+zb[f]+rnorm(100)
> y = rpois(100,exp(eta))
> 
> g1 = glmer(y~x+(1|f),family="poisson",method="Laplace")
> g2 = glmer(y~x+(1|f),family="poisson",method="AGQ")
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