Quasi-poisson for underdispersed data Related to glm() in R, I saw a few post recommending modeling underdispersed data using the Conway–Maxwell–Poisson distribution, specifically with the R package CompGLM, however, I'm not sure I saw anybody confirming that the quasi-poisson cannot be used. Therefore, I ask: why not use quasi-poisson in glm for underdispersed data? After all, isn't the idea of quasi-poisson to go beyond the assumption that variance and mean are equal ? (and in the case of underdispersion, there are not equal).
Basically, I am running a glm(y ~ x, family=poisson) where x is a categorical variable and I am getting  
Null deviance: 67.905  on 519  degrees of freedom
Residual deviance: 59.584  on 507  degrees of freedom 

Which strongly suggest underdispersion and I am therefore leaning towards a quasi-poisson solution.   
 A: Quasi-likelihood theory is as valid with underdispersed data as it is with overdispersed data, so you could just go that way.
But, I would be careful, context matters a lot. While overdispersion is quite common, and is easily explained by simple mechanisms, that is not the case with underdispersion! For instance, extra, unmodeled (or unobserved) variation/inhomogeneities leads to overdispersion, but can never produce underdispersion. Causes for underdispersion are more difficult to come by, they usually have to do with a lack of independence. For one example see Causes for Underdispersion in Poisson Regression.  One common cause of lack of independence is competion, an example I just come by is counts of territorial birds (that was from my daughters masters thesis in ecology)!
Some posts dealing with practical matters when modeling with underdispersion is

*

*GLM for proportional data and underdispersion,


*Overdispersion and Underdispersion in Negative Binomial/Poisson Regression,


*Is there a common underdispersed discrete distribution with unbounded support for general mean and variance?,


*Are these data underdispersed? If so, what mechanisms may explain this?
