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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 "offset(log(time))". However, I was not able to find the proper way for binomial data. I have seen some authors including the term just as "offset(time)" (without log transformation, although I get an error message) and others include "time" just as a covariate in the fixed part of the model. Could someone enlighten me?

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    $\begingroup$ It's a bit unclear exactly how your sampling works but perhaps stats.stackexchange.com/a/217773/77222 applies analogously for your data? Or is it the case that number of fruits sampled (by the investigator) just increases with sampling effort while the expected proportion of those that you record as "removed" (by the frugivore) remain unchanged? If so I don't think it makes sense to include effort as an offset. $\endgroup$ Sep 20, 2022 at 9:17

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A Poisson model with response $y$ reduces to a Binomial model with cloglog link and the same covariate structure in the transformed response $$ z=\begin{cases} 1 & y > 0 \\ 0 & y =0. \end{cases} $$ So use log effort offset and cloglog link.

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