Explanation and simulation of under-dispersion in quasi poisson GLM

Question

I'm currently working on analysis of excess mortality during the pandemic as part of my Master's thesis. I am using UK data, from Public Health England, who use a Quasi Poisson model for expected deaths had the pandemic not occured. The methodology for the model and the data are publicly available, though the model parameters etc are not.

The methodology document mainly motivates use of a Quasi Poisson model due to overdispersion, however the fitted model has a dispersion parameter of approximately 0.3, suggesting underdispersion.

Firstly I was wondering what the explanation of this under dispersion might be? I'm fairly new to quasi poisson models and excess mortality studies.

Secondly, I'm looking to obtain some prediction intervals for the ratio of registered (actual) to expected deaths, and am doing so by simulating from the quasi poisson model as described in the previously linked methodology document. I have found this code, lines 461 to 488 to do so. I am struggling to understand the justification of why Line 480 is used in the under dispersed case. I know that a dispersion parameter less than 1 will mean the rqpois function starting on Line 461 is ill defined, but I don't understand why the dispersion parameter is then just set to 1 in the simulations to deal with this.

Answer 1

I was wondering what the explanation of this under dispersion might be

A Poisson distribution can describe events happening independently over time. The variance of Poisson-distributed counts among multiple time intervals from a Poisson is the same as the mean number of counts per interval.

With under-dispersion there is less variance among time intervals than the mean count per interval. As a limit, think what would happen if events happened regularly every second. As this thread indicates, under-dispersion usually arises from a lack of independence among events. So think about ways that there might be a lack of independence among deaths. That's perhaps not too hard to imagine in a pandemic.

The term "quasi-Poisson" is just used to describe a point process that has less variance than a Poisson. You'd have to ask the authors of the program why they might have chosen to do simulations of under-dispersed events based on Poisson statistics.

As you are engaged in a research project, you might consider alternatives, like the Conway-Maxwell-Poisson distribution that naturally handles arbitrary under-dispersion. Regression based on that distribution is implemented in the R mpcmp package, although I have no experience with it.