# Are binomial regression and Poisson regression with an offset to 1 substantially different?

I know that negbin can approximate the betabin distribution, especially when the probability of hitting the max is low (events are more rare). If the offset of a negative binomial regression transforms the outcome to be between 0 and 1, is it substantially different from using beta-binomial regression?

For context: My research question is: "Are my predictors associated with the number of risk behaviors (out of a possible 22) that participants reported doing in the last thirty days?"

Therefore, my DV is a count/proportion of risk behaviors out of a possible 22 that participants report in the last month. Of the sample of 222 people, only 5 have scores higher than 17, and there is significant overdispersion. I'm deciding on whether I should use a beta-binomial model or a negative binomial model with an offset indicating the number of questions that individuals were asked (20, 21, or 22, depending on specific conditions).

Questions:

1. If the offset of a negative binomial regression transforms the outcome to be between 0 and 1, is it substantially different from using beta-binomial regression?

• Using AIC to compare, my results suggest that adding an offset to the Negbin models destroys model fit (tripling AIC). Including it as a factored predictor has little effect on the AIC, and the coefficients are not significantly different from 0.

• The beta-binomial approach and the negative binomial model without an offset provide comparable fit (AIC and Est/S.E. ratios).

2. Is it ever appropriate to use a negative binomial regression with an offset to obtain estimates of a proportion?

3. The negbin model (relative to the beta-binomial) without an offset has the best AIC, better standard errors, one less df, and can model parameters that won't work in the beta-binomial model, so I'm inclined to choose it). What would be the downside to choosing negbin without an offset as opposed to beta binomial?(bias, interpretation, etc.)?

Q1: Well, you can see beta-binomial regression as modelling proprtions in $[0,1]$, but a negative binomial regression with log of exposure is a rate, that is, events per time, and can well be larger than 1. This are very different models. Most importantly, the variance structure is different. When choosing a model you must think about what is most appropriate varaince structure.