# What is the appropriate model for underdispersed count data?

I am trying to model count data in R that is apparently underdispersed (Dispersion Parameter ~ .40). This is probably why a glm with family = poisson or a negative binomial (glm.nb) model are not significant. When I look at the descriptives of my data, I don't have the typical skew of count data and the residuals in my two experimental conditions are homogeneous, too.

So my questions are:

1. Do I even have to use special regression analyses for my count data, if my count data doesn't really behave like count data? I face non-normality sometimes (usually due to the kurtosis), but I used the percentile bootstrap method for comparing trimmed means (Wilcox, 2012) to account for non-normality. Can methods for count data be substituted by any robust method suggested by Wilcox and realized in the WRS package?

2. If I have to use regression analyses for count data, how do I account for the under-dispersion? The Poisson and the negative binomial distribution assume a higher dispersion, so that shouldn't be appropriate, right? I was thinking about applying the quasi-Poisson distribution, but that's usually recommended for over-dispersion. I read about beta-binomial models which seem to be able to account for over- as well as underdispersion are availabe in the VGAM package of R. The authors however seem to recommend a tilded Poisson distribution, but I can't find it in the package.

Can anyone recommend a procedure for underdispersed data and maybe provide some example R code for it?

• How do you know your data is underdispersed? How are you calculating the dispersion parameter? – Hong Ooi Aug 14 '13 at 10:22
• It would also help to tell us more about what you are interested in. For linear predictor point estimates and prediction of values, underdispersion rarely is a problem but tests and intervals may be unnecessarily conservative (quasi families would help with that). That said, for a "normal" likelihood approach check out the COM Poisson and other generalized Poisson models. – Momo Aug 14 '13 at 11:46
• @ Hung Ooi:I tested the dispersion with dispersiontest(Poissonmodel, alternative = c("less")) and the test turned out significant. – Sil Aug 14 '13 at 11:47
• @ Momo: I want to test if negotiating dyads in two experimental conditions differ in the correct offers they make. Correct offers mean that dyads claim more issues that correspond to their teams' respective interests instead of claiming issues more valuabe for the other party. First, I wasn´t even aware that this is count data. Do you mean the Conway-Maxwell-Poisson Distribution by COM Poisson? Thanks a lot already! – Sil Aug 14 '13 at 11:58
• Thanks for the additional info. Yes, I meant the conway-maxwell poisson. Shmueli & co developed a kindbof generalized linear model for it, there also is an R package if you'd like to try. – Momo Aug 14 '13 at 13:17