# How to interpret the average rate of NB-GLM when offset is involved?

My question is on how to interpret the coefficients of a negative-binomial GLM that included an offset. My dataset is a clinical trial where patients got one of two treatments (A or B) and stayed in the trial for different numbers of visits where the maximum possible number of visits is 52.

The DV is number of visits where a specific event was reported (e.g. number of such visits are either 3, ..., 52) and the outcome should be reported as a rate i.e. number of healthy visits / total number of visits.

As the "total number of visits" can differ per patient, as some patients leave the clinical trial already before the end of the trial, the offset in the model includes the (log) total number of visits per patient.

As an outcome of the model, the average rate for treatment A is 21 and the average rate for treatment B is 17.

What are these average rates referring to? Is it 21/52 and 17/52? Or is the rate rather the average of all "offset-corrected" individual rates?

[edited for clarity]

• You say the DV is the "number of healthy visits" and also "the outcome should be reported as a rate". Those two sentences contradict each other. If you want the DV to be a rate, then you can create the rate variable and use that. If you want it as a number of visits, then it isn't a rate. Commented Aug 22, 2023 at 10:09
• I also don't understand what "maximum number of visits per patient" means. Each patient has only one "number of visits". Commented Aug 22, 2023 at 10:10
• Thank you Peter, I have now edited the question and I hope it is now more clear. maximum number of visits = the (total) number of visits the patient stayed in the clinical trial. You mentioned that I can create a rate variable and then use this instead. However, what is the offset then used for?
– Jens
Commented Aug 22, 2023 at 10:45

Presuming you're using a log-link, the fit at any given set of predictor-values will be for number of healthy visits per unit of exposure (maximum number of visits per patient), which if I understand correctly what you mean by that exposure is effectively proportion of visits that are healthy.

The big question - one I would be concerned about if you didn't get it from referees - is why, since presumably you can only have 0 or 1 healthy visits per visit, you didn't use a binomial model.

• We counted the number of "healthy vists" so we had e.g. 18 healthy visits out of 52 or 52 out of 52 or 3 out of 6. These proportions are of course very much different and do not have the same numerator. And yes - negative binomial usually comes with a log-link.
– Jens
Commented Aug 22, 2023 at 10:47
• @Glen_b I think you are also a moderator so I address this to you in that role. Wouldn't this have been considered better as a comment?
– DWin
Commented Nov 18, 2023 at 1:03
• @DWin I am not a moderator. The question of whether to make a response a comment or an answers is not always 100% clear, particularly when one effectively denies a premise of the question. I think in this case there's a clear attempt to guide the OP toward a potentially better approach, albeit framed somewhat more Socratically. I can see an argument the other way as well. Commented Nov 18, 2023 at 1:14
• If it had been posed as a comment the response might have made it clear that it wasn't appropriate for a binomial model. So I suppose I will promote my earlier comment to an answer.
– DWin
Commented Nov 18, 2023 at 1:39
• @DWin I'm content to flag my answer to be made a comment - or indeed for you to do so. Or to delete it outright, if you feel that would be better. Commented Nov 18, 2023 at 1:48

Number of healthy visits per number of total visits is not a rate, since it has no time in the denominator, but rather a proportion.The distinction may be helpful in searching for an appropriate method. Your (proportion) outcome would be necessarily constrained in the range of [0-1] and you might rather have chosen beta-regression which is often used for analyzing outcomes that are considered proportions.

CV.com has quite a literature on beta-regression: https://stats.stackexchange.com/search?q=beta+regression+for+proportions

Binomial models are sometimes used for such problems, but they would not use the data in the form of proportions but would rather use the data as the numbers of cases and the totals, rather than using them to form proportions. The results would be log-odds but easily transformed into either odds ratios or prediction of proportions.