# How do I determine which count model (quasipoisson, negative binomial, or zero-inflated GLM in R) best fits my count data?

Background: I am modeling counts per square meter of an invasive species, which range from 0 to 380, as part of my master's thesis. The data set has 50 observed zeros out of 161 observations. After fitting a Poisson GLM with my ten explanatory variables, suspect results (all covariates having significant p-values of 2^-16) led me to the following series of actions:

1. Suspecting overdispersion, a quasipoisson GLM was fit. The significance of certain covariates and their coefficients was congruent with published literature (yay! a signal!). Evidence for overdispersion was present, with theta = 80.74. 2. For comparison, a negative binomial GLM was fit. The output was very similar to the quasipoisson, but several "glm.fit: algorithm did not converge" and an "alternation limit reached" warnings came with the output. I set the maximum number of iterations to 10,000 and ran it again- this time I got the convergence warning, but not the alternation limit being reached. No change in outputs with the different maxit values. 3. To compare the prediction of zeroes by my quasipoisson and negative binomial GLMs (and see if zero-inflation is an issue warranting evaluation of a zero-inflated GLM), I used the check_overdispersion() function in the "performance" R package on each to compare the number of predicted zeros to those observed in the response variable. The quasipoisson model only predicted 10% of the observed zeros (5 vs. 50), while the NB model predicted 98% of the observed zeros (49 vs. 50). A vif() check reveals no collinearity issues in either model, and calculation of Nagelkerke's r-squared using rsq() provides a value of ~0.32 for the negative binomial, while the "v" type r-squared for the quasipoisson model is about ~0.34.

My takeaway is as follows: It seems hard to justify using the quasipoisson model based on its underfitting of zeroes alone, while the negative binomial seems to be handling zeros with reasonable accuracy. Moving toward interpreting the NB model would have the twofold benefit of precluding the need for the somewhat-complicated zero-inflated GLM two-part model, while the baked-in provision of an AIC value for model selection purposes precludes the need for calculating QAIC if the quasipoisson were used.

Because my coauthors and I have reached an impasse, and our resident statistician on my small campus (with limited resources) took a job elsewhere, I am seeking advice on the following:

a) Given my findings re: predicted zeroes by the NB model, is it safe to say that a zero-inflated GLM is unnecessary?

b) If so, is further justification needed for choosing the NB GLM over the QP and what might that entail?

c) I ran a stepwise AIC analysis to get a "most parsimonious" NB model-the diagnostic plots from are attached. I have concerns, especially with the QQ-normal plot, but I am curious if anyone here can identify another type of model which might fit this data appropriately. Many thanks advance for any advice or suggests you might have! Apologies if this post is clunky-it is my first here. EDIT: My process thus far has been informed by my applied regression course text: Zuur et al. Mixed Effects Models and Extensions in Ecology with R (2009)

• 0. Welcome to CV.SE. 1. Do not run stepwise regression, it leads to biased statistical significance estimates. Pick the model that best fits your modelling assumptions and it does not corresponds with the data you collected (e.g. if the data at hand show strong overdispersion then a Poisson model, is likely inadequate for your model). Apr 12, 2022 at 16:21
• The exact discussion for QP vs NB has been followed up a couple of times in CV.SE already: stats.stackexchange.com/questions/478209, stats.stackexchange.com/questions/66412, stats.stackexchange.com/questions/83636, stats.stackexchange.com/questions/157575. In short, what you have done is right. I think the impasse between yourself and your collaborators isn't a Stats issue at this point but rather a research question applicability. We can erroneously say we pick the NB cause it is a true MLE or the QP because it converges without warning. We are non-the wiser. Apr 12, 2022 at 16:45
• Thanks much for the replies! After following up with a former statistics professor, I abandoned the QP in favor of the NB GLM due to a) large theta value (overdispersion) and b) handling of zeroes by the NB GLM, which indeed precludes the need for a zeroinfl() glm. Since we know that my residuals are not normally distributed (we designated this!), model validation was handled using the DHARMa package to plot residual vs. predicted, QQ plots as described here: middleprofessor.com/files/applied-biostatistics_bookdown/_book/… Apr 13, 2022 at 19:10