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I'm using glm.nb to test for differences in the likelihood of accumulating overtime hours among employees across multiple departments. Department is the only information I have on which to base a comparison.

The data are overdispursed and the following negative binomial improves the model significantly oner the poisson and quasipoisson alternatives:

m1 <-glm.nb(Over ~ Dept, data = DF)

I'd like to understand the difference in interpretation between including an employee's total available hours (not on vacation, leave, etc.) in the model as an offset (an exposure variable) or as a covariate.

m1 <-glm.nb(Over ~ Dept + offset(log(HoursAvail)),data = DF)
m2 <-glm.nb(Over ~ Dept + HoursAvail,data = DF)

It seems that hours available should be accounted for in some way. If included as a covariate, should the log of HoursAvail be used instead?

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First of all it is not every variable that you can throw in as an offset variable. Why ? Lets look at this example with these questions asked: If available hours is increased by 1, can the outcome (Over) increase by more than one for different subjects and the answer is no. This does not make it a good candidate for offset. Another example lets say we are counting the number of accidents at a round about, the number hours we spend at the the round about would make a difference in the number of accidents we count for different days. Now the question if we increase number of hours spent at the round about by 1 can the outcome (No. of accidents) increase by more than 1 for different days that we were there ? Yes this makes hours a good candidate for offset. You might have to include the overtime as a fixed effects covariate. If you have to interpretation a model with offset in Poisson regression you always remember is the change in the rate of the outcome per the offset variable for changes in the independent variables. Use incident rate ratios ....IRR.

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