Can offset variables be used in binomial logistic regression models? 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.

There is an "offset" argument for a call to glm(), but in a binomial model it's interpreted as the number of total trials. It's not clear that would work well with your differential observation durations. As explained in this answer, it's hard to use a regression offset term to accomplish what you wish with a logistic regression model.
This is best analyzed as a count problem. I'd start with a Poisson model (log link) that includes the log of the observation duration as a regression offset term (coefficient fixed at 1) for each observation period in each zone.
It would be ideal if you have the actual number of observations for each day and zone. You then would model the number of observations per unit observation duration for each of zones A and B, and could use that to estimate what seems to be your main interest, the ratio of observations between the 2 zones given the same observation duration for both zones. It would be less satisfying if all you have is "detected" (count of 1) or "not detected" (count of 0) as your data, but that wouldn't be far off from the ideal if the probability of more than 1 detection during any one observation period was small.
