# Modelling count data with extreme underdispersion - what distribution?

Suppose we have some count data, and we want to use a model that allows for "overdispersion" or "underdispersion" in the data (i.e., higher or lower variance than the Poisson distribution). Let $$X$$ be our count variable and let $$\phi = \mathbb{E}(X)/\mathbb{S}(X)$$ denote the coefficient of variation. Overdispersed data is often modelled using the negative-binomial distribution and both over or underdispersed data can be modelled using the generalised Poisson distribution. However, both of these distributions still have a restricted range for the allowable coefficient of variation. Specifically, for a given mean $$\mu$$, the allowable values of the coefficient of variation under each distribution (allowing for corner values) are:

\begin{aligned} &\text{Poisson} & & & \phi &= 1, \\[10pt] &\text{Negative Binomial} & & & \phi &\geqslant 1, \\[6pt] &\text{Generalised Poisson} & & & \phi &\geqslant \max \big( \tfrac{1}{4}, \tfrac{(4-\mu)^2}{4} \big), \\[6pt] \end{aligned}

The generalised Poisson distribution is quite a good generalisation of the Poisson, and can be used for count data, but it still has a restricted value for the coefficient of variation that does not allow you to model count data that is highly underdispersed. If the dispersion is extremely low then even the generalised Poisson would not model it well.

Question: Is there any distribution generalising the Poisson distribution, that can reasonably be used for modelling count data, and which has an unrestricted range for the coefficient of variation (i.e., it allows $$\phi \geqslant 0$$)?

• IIRC, Gary King outlines an under-dispersed count model in Unifying Political Methodology: The Likelihood Theory of Statistical Inference. United Kingdom, University of Michigan Press, 1998. I can't find my copy at the moment to provide a more complete answer. – Sycorax Nov 17 '20 at 3:29
• Can I ask why not consider a quasi-Poisson? Seems more straightforward to me. (It was suggested as a potential solution in this forum too e.g. stats.stackexchange.com/questions/308254) – usεr11852 Feb 19 at 2:00
• Whilst the quasi-Poisson allows flexibility, it is not a distribution. I am interested here in whether there is any family of distributions that has the specified property. I certainly agree that it is worth considering for modelling purposes, but my goal is to determine if there is an alternative that is actually a proper distributional family. – Ben Feb 19 at 3:30
• Thank you for the clarification. I focused on the "reasonably be used for modelling count data" and missed that part! :) (I have already upvoted the question as being useful of course) – usεr11852 Feb 19 at 9:07

## 1 Answer

The Conway-Maxwell-Poisson model has recently been shown to handle arbitrarily small underdispersion (see Huang 2020). For example, it is possible to have a mean of 15 and a variance of 2, say, by selecting the dispersion parameter large enough. In the extreme limits, it is even possible to have a mean of 15 and 0 variance, or a mean of 15.2 and variance 0.2*0.8 = 0.16, which is the smallest variance possible for mean 15.2. The mean-parametrized Conway-Maxwell-Poisson model is implemented in R in the mpcmp package (Fung et al. 2020).

Other alternatives that have potential to be arbitrarily underdispersed include the Double Poisson of Efron (JASA, 1986) and the exponentially re-weighted Poisson of Ridout & Besbeas (2004). However, neither of these models parametrize the distribution via the mean, and so it is harder to see what happens as the dispersion gets arbitrarily small.