How to estimate and interpret an offset correctly in a Poisson regression?

Question

Assume the following easy example of a glm regression with an offset:

numberofdrugs<-rpois(84, 10)
healthvalue<-rpois(84,75)
age<-rnorm(84,50,5)
test<-glm(healthvalue~age, family=poisson, offset=log(numberofdrugs))
summary(test)
fitted(test) #how to get one of these values manually?

• How can I compute the fitted values manually?
• Also, why is there no estimation of log(numberofdrugs)?
  • In the book Generalized Linear Models on page 205-207 there is an example where the offset is estimated. It was done to see if the coefficient is close to one. It's 0.903 (see page 207 if you've this classic book) and from this follows, that there is nearly a constant rate in the number of damage incident!

Previous related questions asked:

• When to use an offset?
• Whether to use an offset when predicting hockey scores?

Answer 1

About the practical part -- outputs of glm or summary are just lists which are pretty-printed for user convenience. You can see their full structure calling unclass on them and extract single values as usual, with a help of $, [[]] and [] operators.

Answer 2

There should not be an estimate of the offset: this offset is (could be) different for every observation (the whole idea is that you monitor the number of events within a (linear) 'timemeasure' (here apparently numberofdrugs).

There is no one 'population' offset you could estimate: person 1 is going to have 5 drugs administered, person 2 maybe 10, and you assume that the number of events (healthvalue?) is linear with this person's numberofdrugs (as per the answer to your previous question).

I don't have the book at hand, and Amazon won't let me look at the pages you mention, but I suppose something else is happening there (maybe simply the average numberofdrugs in the population)?