[R-sig-ME] Binomial GLMM vs GLM question (Ken Beath)

[Andrew Robinson]

On Fri, May 16, 2008 at 04:53:07PM -0400, Ben Bolker wrote:
> 
> Jessica Stapley wrote:
> ~  So -- I may be wrong (others please jump in) -- but I disagree with
> several points in the most recent comment.
> 
> | In your study you have taken repeated measurements on the same ponds,
> | as such, observations from each pond across years are not independent
> | (observations in pond 1 at t+1 will be correlated with observations
> | at t).  If you were only interested in the effect of rainfall on egg
> | survival I would suggest you use a mixed model with year and pond as
> | random effects (based on a similar eg in Crawley's R Book p605). In
> | this case I assume you have measurements on every pond in every year
> | and as such these random effects are crossed (not nested).
> 
> ~  (a) I would suggest that a mixed model is NOT going to work well
> in this case, even if year and pond are "philosophically" random effects
> (i.e., you don't really care about what happens in those specific
> ponds, and you may even have chosen them with a random-number generator
> out of a list of all possible ponds -- although this is much less
> likely with years ...).  The technical problem is that estimating
> variances from 2 or 3 points is nasty.  This translates into
> inference/philosophical terms because these few points really
> don't give you the data to generalize about the population, even if
> you want to.  (I think one of the confusions is that in the classical
> method-of-moments world there's nothing that says you can't have 2
> denominator degrees of freedom -- your power will be terrible, but
> the expressions won't blow up on you [unless you get negative variance
> estimates ...])

I'm just going to square away with just one of Ben's well thought out
points.  In my opinion, the allocation of an effect to being fixed or
random depends partially upon the design and partially upon the use to
which the model will be put.  There may be reasons to make effects
random even if they have small counts.  

For example, if you make the effects random then you're effectively
marginalizing them, whereas if you make them fixed you're forced to
condition on them, which can complicate an analysis and its
presentation.  You may not wish to necessarily draw inference to the
broadest possible population, but still regardless might be interested
in averaging over the effects, whilst benefitting from their
management of the (supposed) correlation in the residuals.

I'm probably influenced in my thinking by the analysis of split-plot
designs, where inference often proceeds including random effects with
embarrassingly small counts.  I know that in general it is not
necessarily safe to import intuition from ANOVA to mixed effects
models, but there we go.

So, for what it's worth, I think that I am less concerned about this
element of the analysis than Ben is.

Cheers

Andrew

-- 
Andrew Robinson  
Department of Mathematics and Statistics            Tel: +61-3-8344-6410
University of Melbourne, VIC 3010 Australia         Fax: +61-3-8344-4599
http://www.ms.unimelb.edu.au/~andrewpr
http://blogs.mbs.edu/fishing-in-the-bay/