I am working with a dataset from my job which is the number of "events" that occur at 70 different locations on a daily basis, over 964 days. So I have univariate panel data. I imagined each location likely has their daily events poisson distributed, with some autocorrelation. Each location likely has a different value of lambda due to differences in size, staff, etc, however most days have zeros, and twos are extremely rare. So I figured lambda is low in general (and extremely low for some locations). I decided to check if there was overdispersion before I figured out how to remove the autocorrelation from my dataset (if anyone has any suggestions for how to do that, that would be great).
I got a really bizarre and unexpected result I have not been able to understand. So what I did is I graphed the means (orange below) for the 70 locations (each tick on the x axis is a different location), and the variance/mean (blue). As you can see, all 70 locations are exclusively underdispersed. What is super puzzling, is for every location, the mean + the variance/mean is nearly exactly one! If you add the mean and the mean + variance/mean for every location, the minimum of the 70 is 1.0004676334594305 and the max is 1.0010245901639392. In addition, the function of mean/variance relative to mean is perfectly monotone below.
How can this happen?
If I run the exact code resampling by week (sum of events each week), I get the second graph, which is more of what I was expecting from the daily data. In my second graph, the median of variance/mean is 1.15, and the mean of variance/mean is 1.21 across all the locations. Is that overdispersed? Is there a formal test I could do? Should I treat only a subset of the locations as overdispersed? I had someone suggest zero-inflated poisson regression, however I'm skeptical that is the best option.
I'd prefer to do my analysis with the daily data, but I'm not sure if that is a good decision. Given my goals here (detecting and removing autocorrelation from my dataset, preferably with the daily data), what approach should I take here? But I'm also extremely interested in the relationship in my first graph and how such a thing could be explained/happen.