Offset term in negative binomial regression

Question

I am fitting a simple negative binomial regression model with (Yearly cancer death ~ Offset (Size of population) + Age + Household income). I used the offset term because I want to compare the yearly cancer death risk in various cities with different population size.

However, because the Size of population is used as an offset term, only regression coefficients for Age and Household income are available in the results. If I also want to predict the number of yearly cancer death of a population size 200,000 with known age and household income, is there any way I can acquire this piece of information from the model?

Or should I fit another negative binomial regression with (Yearly cancer death ~ Size of population + Age + Household income), where the Size of population is no longer an offset term, to get the answer?

Answer 1

I suppose you are using a log link function (you didn't say), which is necessary for this to work. An offset term is a covariate with a known coefficient value (equal 1.) So would include the population value of 200000 in the newdata= argument to predict. Note that in Rpredictions with models using offsets can be involved, for instance see this SO post. But questions about that is about R usage, so off-topic here. Similar consideration will apply with other software.

Also note that it could be that population size has an effect over and besides its use in defining the rate, so you could in that case use a model of form (Yearly cancer death ~ Offset (Size of population) + Size of population + Age + Household income), that is, the same predictor can be used doubly both as offset and directly.