I'm dealing with a Poisson regression model for count data. My DV is the number of Covid infections in a particular city. My explanatory variables are both categorical and continuous. One of them is the region which the city belongs to. Of course, I cannot have the same number of cities for each region. My strategy is to create a multilevel factor called "Region" and assign "1" if the i-th city belongs to the k-th region and "0" otherwise, but I wonder if this unequal distribution over the regions (consider that in a couple of cases there are just 1 or 2 cities per region while there are cases with 9-10 cities per region) may affect my results and in which way. Thank you.
If region is one of your factor of interest and you'd like to examine if cases are more frequent in one region over the other, then it's fine to have different regions presented in unequal proportion. It is true that power is maximized when groups are equal, but this is usually not achievable in observational data. If region A is bigger, it'd have more cities, there is nothing we can do about that. What's more important is that your data are comprehensive and representative.
When analyzing disease incidence, we should also be mindful of two other items: i) what is the time frame? And ii) what is the population at risk? Poisson distribution allows us to estimate the chance of seeing a certain number of events given its mean rate over a certain period of time. And in order to properly compare the disease risks across cities, we need to make sure the space and time are comparable, so that the counts would be comparable.
For time, make sure the statistics to be present agree with the data inclusion criteria. Are you looking at total prevalence (all cases ever)? Or incidence rate in June of 2020? Whatever that is, make sure the data cover the same period across the cities.
For space, make sure you have included the population at risk (in the form of natural log(population)) in the regression model as an offset, also this thread. Or you'll just be computing the ratio of raw counts rather than the ratio of rates.