Can I ignore under-dispersion in my count data? I have under-dispersed count data. I do not want to transform them, and using a negative binomial error distribution (via glmer.nb) does not help.
My results are the same regardless of the distribution I use, and agree very well with the conclusion that you would draw by eyeballing the data (it is pretty obvious that there is no effect, see below). I can not seem to find any real way to deal with this problem, and I am left wondering if the most appropriate thing to do is acknowledge the under-dispersion, point to the result that is robust to every method I've thrown at it, and move on.
If you've been in this situation- could you share any tools/advice/philosophy that you found helpful?


Here is the output from my Poisson model:
mc_p <- glmer(tubes ~ status + (1|site/plant), family="poisson", 
         data=mc)
summary(mc_p)  

Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
 Family: poisson  ( log )
Formula: tubes ~ status + (1 | site/plant)
   Data: mc

     AIC      BIC   logLik deviance df.resid 
    713.7    725.7   -352.8    705.7      143 

    Scaled residuals: 
          Min       1Q   Median       3Q      Max 
     -2.08950 -0.38627 -0.04237  0.33830  2.87921 

    Random effects:
     Groups     Name        Variance Std.Dev.
     plant:site (Intercept) 0.01958  0.1399  
     site       (Intercept) 0.01790  0.1338  
    Number of obs: 147, groups:  plant:site, 20; site, 4

    Fixed effects:
                Estimate Std. Error z value Pr(>|z|)    
    (Intercept)  2.25665    0.08417  26.809   <2e-16 ***
    statusy      0.01356    0.05301   0.256    0.798    
    ---
    Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

    Correlation of Fixed Effects:
            (Intr)
    statusy -0.308

I am trying determine whether pollen tube counts differ between nectar-robbed and un-robbed flowers. Pollen tube counts are nested within plant (multiple flowers of each type sampled from each plant) and plants are nested within site.
I've tried using a negative binomial error distribution (as I described in the first iteration of this post), but that still gave me a very high deviance relative to my residual df.
It seems that there isn't a straight-forward way to apply a useful error distribution to my nested data. At the same time, the result (that there is no difference between groups) is the same regardless of how I arrange my model, and can be seen very clearly in the bar-graph above.
 A: If the data are overdispersed, you can estimate the same relative rates and calculate prediction intervals as in a poisson model using a quasipoisson model. Even if the data are not overdispersed, a quasipoisson model is valid and fairly efficient. A quasipoisson model just extends the poisson by estimating a dispersion parameter. 
It is, of course, important to think about the originating nature of the data, the question you're trying to answer, and to ask about why this dispersion comes about, and how it might affect your interpretation of the results.
A: A descriptively adequate treatment of under-dispersion can apparently be gotten from regression using a Conway Maxwell Poisson (COM) distribution or Consul's Generalized Poisson (GP) regression.  It seems that just modeling with a Normality assumption is inefficient.
A review paper and responses for the COM model are here and an R package to implement it is here.  A Stata package with references to the GP regression model is described here.
Full disclosure: I've never used these things in anger.
