When I look up what exactly the offset is doing, I only see references to poisson or negative binomial models. Neither of those analyses work for me because I do not want to predict counts. I want to predict proportions.
My questions come down to: Can you use an offset in binomial logistic regression? If so, what is it doing mathematically? And is the variable I want to use for an offset appropriate for that purpose?
Here's an explanation in pretend terms:
I and a partner went to a bunch of sites and divided them into two zones: A and B. Within each site, I sampled to see if I could detect some species every day in zone A, while my partner sampled for the same species every day in zone B right next to me. My binomial response is the number of detections in A vs the number of detections in B, like comparing the number of heads and tails if you flip a coin 100 times. It's binomial because there are no other options. At a given site, a detection was either in A or it was in B.
The tricky part is that A and B sometimes weren't sampled equally. Let's say that we planned to sample at one of the sites for 20 days, but my partner, who was watching B, fell ill and had to leave after 10 days, leaving me to just watch A for the remainder of the time. So, at this site, A was sampled 20 times while B was sampled 10.
In the full period, I detected the species 8 times in A in 20 days, and my partner detected it 4 times in B before they had to leave after 10 days.
Now, using just the counts in A and B, it looks like we got twice as many captures in A as we did in B, and we did as far as raw numbers go (8 vs 4). But that doesn't account for that reduced sampling effort in B.
So, to account for that, I would like to use the proportion of days A was sampled relative to B as an offset. In this example, I would divide 20 by (20+10) to get an offset value of .66. It would tell the model that while there were twice as many detections in A as there were in B at this site, 66% of samples at the site were taken in A.
The reason I can't use raw sampling numbers is because the unevenly sampled zone varies site to site. At some sites, zone B was sampled more, and other sites, zone A was sampled more. My sampling unit is the site, not the zone.
The underlying problem here is that I don't want to predict the number of observations per day; I want to predict a proportion of total observations. I have my own data to calculate these proportions, but I also have a long term dataset of observations in B, but none in A. So for every one observation in B, how many were in A? That's the question I'm ultimately trying to answer with the data I collected myself.