Dear
Alain,
 
thank
you very much for your quick reply. I include some comments below (>>).
 
I would certainly include the plot effect, and cluster
effect within plot. .....it imposes a correlation structure on your data.
Not sure whether I understand the color thing though.
>> colors are fully crossed with clusters (i.e. 3 colors á 20 fruits in
every cluster; and clusters are nested in plots with 10 clusters per plot).
Therefore I would always include a random effect of clusters nested within the
plots. Additionally, I would either include color as a fixed effect (because I
am interested in its effect) or as a separate random effect (if I am not explicitly
interested in its effect, but want to control for the difference among colors),
depending on the question asked.

If your binomial GLMM is overdispersed then you should try to figure out why
this is. Common causes are:
 
1. Outliers
>> I checked for outliers with multi-panel
Cleveland dotplots > there are none
 
2. Non-linear patterns
>> I have routinely checked the relationship
between the response and all predictors with multi-panel scatterplots and LOESS
smoothers combined with correlation coefficients > these one-way relationships
are +/- linear, if present at all (i.e. no quadratic or otherwise obvious one-way
relationships). Plotting the model residuals against the single predictor variables
does not show any pattern, which I think is a good sign, as there seems to
remain little residual variance in the model that might be better explained by
e.g. fitting a power function of the respective predictor variable.
 
3. Wrong link function
>> According to the binomial family I always
used the default logit-link;
alternatives would be the probit- and cauchit- links,
that relate to the
Gaussian and Cauchy cumulative distribution functions
respectively. Why or in which cases may they be better suited?
 
4. Wrong distribution
>> As I have an upper-bound count I surely
should choose a binomial distribution. As far as I understand, the
beta-binomial distribution, which you suggested, allows for a larger heterogeneity
in the per-trial probability, i.e. the comparatively high variance in the data
compared to the mean should be less of a problem than in the binomial
distribution, therefore reducing the problem of overdispersion, but not of
zero-inflation!? This would be fine unless zero-inflation is not a problem
itself, but only the overdispersion it causes!?
 
5. Too many zeros
>> There are definitely a lot of zeros in the response
(about 72%). But as I wrote “normal” binomial GLMMs work well, i.e. the model predictions
look ok, the overdispersion is within an acceptable limit (never larger than 1.5
in all models that I fitted so far) and there are no errors/warning messages.
So are there actually not too many zeros? How can I judge whether a “normal” binomial
glmm does the job well, when I cannot run a ZIB, so that I can compare e.g.
AICs?
 
6. Dependency structure that has not been
included...or included in the wrong way.
See my noted to point 1. I think/hope that is correct,
based on the experimental design.
 
For 2...plot your residuals vs each continuous
covariate. If 5 is the cause, then yes...a zero inflated binomial is an option.
For 4 you may want to consider the beta-binomial
distribution, see for example the book from Ben Bolker, or our 'Beginner's
Guide to GLM & GLMM'; it
contains code to fit a beta-binomial GLMM in JAGS. A
zero inflated version of a beta-binomial GLMM/GAMM requires similar code.
 
As to your question how many zeros means zero inflated
models.....it all depends. I have data sets with 70% of zeros...and a
Poisson/NB GLM still do the job...and I have data sets
where 25% of zeros already means ZIP. Same holds for ZIB.
 
I would strongly suggest to do this in JAGS.
 
Have fun
 
Alain
 
I
am inclined to try to fit a (zero-inflated) beta-binomial GLMM…but have never
used JAGS. It should also be possible in the R-INLA package, but I haven’t used
that one either.
I’ll
see whether I can find somebody who has at least some experience with it…or
otherwise look for a manual in the web or in one of the two books that you have
mentioned.
…but
I am still not sure whether I really need to do something against this zero inflation,
i.e. is the zero-inflation itself the problem, or is it the (mild) overdispersion
that results from it? In the last case I wouldn’t be too worried about simply
using the “normal” binomial GLMM as the pearson chisquare - rdf ratio is always in the range of 1-1.4.

Maybe
somebody can comment on the points above, especially point 5!?
 
Thanks again for your help!!
Stefan

-- 
Stefan Ferger 
Biodiversity and Climate Research Centre (BiK-F) 
Project Area B: Biodiversity Dynamics and Climate 
Senckenberganlage 25 
D-60325 Frankfurt am Main 
phone: +49 69 7542 1871 
fax: +49 69 7542 1800 
e-mail: stefan.ferger@senckenberg.de 
homepage: www.bik-f.de/root/index.php?page_id=433 
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