I have a binomial response variable -proportion of removed fruits, or plant visited or not visited by a frugivore- in a GLMM. I would like to control for differences in the sampling effort (time) among plants. I know that offset terms are often used for count data as "log(offset)". However, I was not able to find the proper way for binomial data. I have seen some authors including the term just as "offset" (without log transformation, although I get an error message) and others include time as covariate in the fixed part of the model. Could someone enlighten me?
Usually in a binomial model you include the denominator of the proportion. But I guess you are asking for more than that. An offset is just another predictor variable whose regression like coefficient is constrained to 1.That is you need an expected value (of no effect) in the same metric as the response variable, so if the response is a logit, then you need a logit. I am making this up as I have never done it.! Why not use a poisson log model where the offset is quite natural. So the response is the log of The count and offset is the log of the expected count if the ’exposure’ has no effect. That is the expected count based on equal sampling rate. So add up the total count to get the average rate then multiply this rate by the sampling time of this particular stand to get the expected count thereby ‘controlling’ for time. Hilbe has written several books on the Poisson and NBD models. https://www.cambridge.org/core/books/negative-binomial-regression/12D6281A46B9A980DC6021080C9419E7
I am not sure exactly what you are after. Is it similar to the education problem when test takers answer multiple choice questions where they may have a substantial probability of guessing correctly? If so, a common approach is the 3PL (three parameter) single trait IRT (item response theory) model. I don't know your area so I am not sure if I am on the right track, but the 3PL equation might suggest a transformation (link) that you could use.
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