# 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.)?

## 1 Answer

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.

Q2: To obtain estimates of a rate, yes. Rates are not proportions.

Q3: You should not choose to use offset or note based on aic (or other fit criteria). Offset or not changes what you are modeling, so this is a modeling decision. Or, stated more bluntly: including an offset or not changes what question you are asking of the data, and it does not make any sense to let the data decide which question you are asking!

(In future, please, if you have three questions, ask three questions, not one)

• Thanks for the help, @kjetil! I'll try to avoid asking multiple questions in the future--I just thought it might be avoiding more clutter since the questions were all related. – mrjaws Sep 18 '17 at 13:38