I have been working with count data (n=66) recently, trying to fit a simple model to explain distribution in an outcome whereby the count (number of successful trials in a region) is relative to an offset (the total number of trials in the region). The data is panelled, with each region (11) appearing 6 times in the dataset.
The 'best fitting' model is a Poisson Regression with the log of the total trials in the region serving as an offset. Zero-negative inflation, multilevel, and OLS don't cut the mustard or provide any significant improvements on the fit. Standard errors are clustered on the regions, and there are 5 predictor variables in total.
The fitted model produces the following residual v fitted plot:
As I understand it, this patterning is almost the reverse of what we expect to see in a poisson setting, whereby the mean residual variance should increase (rather than decrease) as we move along the fitted points?
Removing either of the clustering or the standard errors does little to change the distribution of the residuals. The zero-inflation models also did not show much different.
My question is whether or not the patterning shown in this residual plot is anything to be necessarily worried about? Do I need to go back to the drawing board and find a 'better fit', or is this always likely to happen with count data including a fairly high (but not insanely high) amount of zeros (about 20%)?
Further, can anyone recommend a DHARMa equivalent in STATA for producing reliable GLM QQ plots?