# Modelling overdispersed rate data using a negative-binomial distribution

A quick overview of the analysis I'm wanting to do: I am wanting to analyze the relationship between habitat factors and the capture of my research species over a network of traps, in order to be able to predict captures. The habitat factors include binary (e.g. presence/absence of a plant species within a plot around the trap), continuous (e.g. distance to the nearest body of water), and categorical variables (e.g. substrate type). I am wanting to analyze the captures as a rate $$\text{rate}=y/T=\text{captures}/100 ~\text{days}$$ to account for the fact that the traps have been operating for different amounts of time (due to maintenance, misfires, etc.).

Initially, the plan was to do a generalized linear model (GLM) based on the Poisson distribution since that seems to be recommended for rate data, and that my initial exploration of the response data seemed to work for it. For the model, I used $$\ln(T)$$ as an offset parameter. Unfortunately, the resulting model is overdispersed (dispersion parameter was 1.76; residual deviance was 178.59 on 84 d.o.f.).

I understand that for count data, the negative binomial distribution can be used as an alternative for a GLM. However, I'm struggling to find information about whether it can also be applied to rate data. Which, brings me to my question:

Can I construct a negative binomial GLM for rate data in the same way as I would for Poisson (i.e. using the offset parameter), and if so, are there any special considerations I need to make when using it?

• If I understand you correctly the answer is yes, you can use negative binomial regression for count data with an offset if scientifically needed. Commented Jun 2, 2021 at 14:00
• A dispersion parameter of 1.76 doesn't seem so bad to me. Commented Jun 7, 2021 at 15:32