Hi Sinan,

Conway-Maxwell-Poisson (COM-Poisson) distribution (Shmueli et al. 2005; Sellers and Shmueli 2010) can handle underdispersed count data. It is a flexible distribution that can account for under dispersion usually encountered in some types of count data (e.g. clutch size or breeding success -productivity- in birds). In these cases, COM-Poisson models usually fit the data significantly better than standard Poisson. You can use the COMPoissonReg (Sellers and Lotze 2015) package from R Software.

Hi Sinan,

Underdispersion is pretty rare in count data....Do you have a sense of why your data may be underdispersed? One reason may be that a Poisson process does not approximate the process of counting species through time very well, perhaps because of temporal correlation in the counts. The presence of species in a community through time is certainly a Markov process as the presence of a species at a given time step is likely dependent on whether it was present in the previous time step.

I would suggest trying a repeated measures or random effects model. Or, even better, consider a multi-species occupancy model type of approach. See the following paper for guidance on determining trends in species richness.

Article Evaluating estimators of species richness: The importance of...

Hi Gwinn,

Thank you for worthy suggestions. Actually I checked the model with several temporal correlation structures. AIC values pointed out that a model without temporal autocorrelation was the best. Zero inflation was also not the case for our data.

The multispecies occupancy models seem to fit our data. But as you stated in the study, they are not common in practical applications yet. Is there any R library or another software to fit such models? I managed to find some code but they seemed a little bit complicated. I think conducting such models needs more speciality. 

Noam Ross has been developing some methods for dealing with under-dispersed count data. Check out the following web page (it begins with a "plea" for examples of under-dispersed data): https://noamross.shinyapps.io/compois/Plotting-compoisson-distributions.Rmd

Hi Sinan, occupancy models can be easily fitted using package unmarked in R. This tutorial will help you 

https://sites.google.com/site/asrworkshop/home/schedule/r-occupancy-1

Hi Sinan,

Conway-Maxwell-Poisson (COM-Poisson) distribution (Shmueli et al. 2005; Sellers and Shmueli 2010) can handle underdispersed count data. It is a flexible distribution that can account for under dispersion usually encountered in some types of count data (e.g. clutch size or breeding success -productivity- in birds). In these cases, COM-Poisson models usually fit the data significantly better than standard Poisson. You can use the COMPoissonReg (Sellers and Lotze 2015) package from R Software.

 Hi Sebastian and Walter,

I tried to fit my data by using COM-Poisson with COMPoissonReg. The parameter estimates were quite close to the ones estimated by quasipoisson. However I had some problems with calculating standard errorrs by using parametric_bootstrap() function. So I could not obtain significance of parameters. Another problem was to finding AIC values for model selection. Working with this library requires a lot of manual calculation. I hope Noam Ross's coming library may solve these problems. Thanks for reliable answers. 

Hi all. I am modeling bird count data and started running standard GLMs and, for a few species, I got c-hat < 0, thus suggesting under-dispersion. Clearly is less common than its over-dispersed counterpart and I could not get much information about it besides the one I followed here with you, until I found paper by Wu et al. (2015) based on ecological examples. My subject species were rare and my counts exhibited way more zeros than those expected under standard distributions (i.e. P and NB), and were potential candidates, I thought, for ZI models. I thought zero-inflation could be, or not, a reason for overdispersion, but for under-dispersion? I guess I am still not clear on why my species might be exhibiting that if it's under-dispersion is real phenomena and not an artifact of some sort of my modeling. Maybe improving the structural part of the model? (unmeasured covariates), incorporating imperfect detection? For these species I am modeling the max count over temporally replicated visits and I am ready to jump to approaches incorporating the observation process and see if that helps. I ran some exploratory N-mixture models (Kéry et al. 2005) and the parametric bootstrap GOF indicates overdispersion! and lack of fit. So I am clueless on which approach to take. Clearly I should follow the path of detection-corrected approaches if I have the data structure for it, but beyond that I don't know. I am using the frequentist version of this models in unmarked R package. What Wu et al. (2015) propose, if I understand correctly, that bayesian hierarchical models and framework could be flexible enough (and if you know how to program it which I can't write now!) to incorporate COM-Poisson models for the abundance part of the models to take care of either under or over-dispersion (as many of you suggested) into these mixture models that also account for

Hey. I've just came across this article, freely available, and has R code ! Under-dispersed count data in ecology might not be as rare as it seems but surely has been overlooked. I could see more clear now how a sampling situation in which we sample critters in quadrats/points could lead to under-dispersion in the counts due to both ecological and observation processes (imperfect detection) and how this two things are actually confounded. The fact that my detection-naive models (e.g. Poisson GLM and allies) indicate under-dispersion but my detection-corrected models (N-mixture w/Poisson family) indicate the opposite (GOF analysis showed that my observed variance is greater than the expected under my best model) is maybe linked to that. Now, when I specified a Neg. Binom. of ZIP distribution for my N-mixture (just with an argument in unmarked) I still have mildly problems with lack of fit but overdispersion seems to be better accounted for, comparing my models with AIC/BIC and because of the degree of discrepancy with the bootstrapped GOFs. So, I am still unsure on many things, but I tend to think that, for my bird species, the reason for having smaller variance than mean values is more tightly linked to detection error, probably related to habitat (site) covariates or other survey-level specific covariates I did not sampled or included in my models (for detection I only used julian day, minutes after sunrise and observer ID). Hope some of this considerations help! This thread and the reading and writing of all above really helped me to understand my data and models better, and to distrust them all! Hopefully though we can get to some useful ones.

Lynch, H. J., Thorson, J. T., & Shelton, A. O. (2014). Dealing with under‐and over‐dispersed count data in life history, spatial, and community ecology. Ecology, 95(11), 3173-3180.

http://dx.doi.org/10.1890/13-1912.1

Dear Sinan, I can see this is an old post. Have you figure out this?

I am modelling richness with a mixed generalized effect model (using Poisson), and the fit is very poor compared to Gaussian! I have underdispersion too.

Dear Luciana,

I think, the solution quite depends on the data. But, firstly you should try to consider why your data under-dispersed. I checked my model against temporal autocorrelation and zero inflation. These were not the cases. Unmeasured habitat variables may be a good explanation for also my data.

The results of COM-Poisson was quite close to the Quasi-Poisson. In the end, I am still unsure if quasi-Poisson can be used. But there are several examples of COM-Poisson literature. You can see the suggestions above. The paper Juan suggested (Lynch et al) was useful.

"Saluton"!

I have been using quasi-Poisson for under-dispersed outcome variables data. I have noticed that it is more sensible for identifying exposure variables than Poisson in my actual data-work: more explaining variables shows significance with quasi-Poisson.

I have encountered troubles to plot a residual graphic. I noticed that qqnorm graphics is not a good idea, because we have no presuppose of gaussian distribution on residuals. An alternative is to plot half-normal graphics. I have been using "hnp" function, from "hnp" package, in R. It is very good for several distributions, but it is not working for my quasi-Poisson models.

Wish it can be useful.

Best wishes!