Hello,

I have count data for 4 groups, 2 of which have a
 large number of zeroes and are overdispersed, and the other 2 
underdispersed with no zeroes. I have two questions about model fitting,
 which I am quite new to, and have been using mostly the pscl package.

1 - How do I deal with underdispersion? Almost all the published and online advice is 
regarding overdispersion, and neither the Poisson nor negative binomial 
distribution seem appropriate. The COM Poisson comes up sometimes as a 
suggestion, but it's not clear to me how I can use this, explain my choice of it, or what 
information I would report for publication purposes.

2 - For the overdispersed data with lots of zeroes, I've tried 
zero-inflated Poisson and NegBin and hurdle models, and used the Vuong 
test to compare. However, I get equal fit for two candidate models that 
produce quite different coefficient estimates for my predictor 
variables, and hence different p values. I am unsure how to proceed in 
choosing one of these models, and how I would justify one over the other given that the Vuong test seems not to discriminate.

Thank you and any advice would be much appreciated.

Mo

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