tldr; Aside from equidispersion, what are the model assumptions I should be checking for in a poisson mixed effects model that has a random intercept, group mean centered transformations of its explanatory variables, and group means included as variables?
I have a county-year dataset with 21 counties that each have 8 years of data (N = 168). I am attempting to model the number of prescription opioid related hospitalizations for this data using the number of prescription opioid pills supplied to each county in a year, the prescription rate of each county in a year, and demographic and economic (unemployment rate, median household income) variables also at the county-year level.
I am using a poisson distribution (with a log link) and the lme4 package to estimate this model with a random intercept, group mean centered transformations of each variable, and the group mean of each variable in line with the specification found in Bell and Jones, 2015 . Additionally, I have been using the DHARMa package to visually examine the relationships between my explanatory variables and the randomized quantile residuals from the model and to test for overdispersion.
I am not formally trained in mixed effects modeling and want to be as sure as one can be that the estimates from my model are not biased. What are the assumptions of a poisson mixed effects model and is there a rigorous set of steps for testing these assumptions (either by looking at residuals or any other part of the model output)?
Thank you in advance for any help!