# Offset for Negative Binomial Regression

I've constructed a Negative Binomial Regression wherein I predict the number of words an individual will find from a set of seven letters based on the average frequency of words that the set of letters can produce, the total number of words that set of letters can produce, and the participant's score on a vocabulary test. In R:

WholeModel <- glm.nb(NumWordsFound ~ AvgFrequency * TotalWords + VocabScore,
data = p)


My question is in regards to the inclusion of an offset term. Right now my model does not contain an offset term as the amount of time each participant had to produce words was exactly the same. While I would like to see the effect of the total number of words the set of letters could produce on actual productivity, should I instead include that number as an offset value?

Using a variable as a regular regressor vs. as an offset is mainly a question of estimating its coefficient vs. knowing (or fixing) it to be 1. As a pragmatic solution I would look at the estimation results of WholeModel. If the coefficient of TotalWords is close to 1 you might also treat it as an offset.
More importantly, it might make sense to use log(TotalWords) rather than TotalWords (possibly similarly for AvgFrequency) so that if you use it as an offset you could really interpret it as modeling the ratio NumWordsFound/TotalWords. Again, you could also decide whether or not to do this based on the model fit.