When I am sampling the proportion of a sub-group of animals to the total number of animals within a sample, I can feel quite confident (after taking into account environmental factors) that I have a realistic representation of the community whenever my sample is large (a bigger cross section of the community). However, if for some reason I only achieve a small sample, I imagine that the proportion of my sub-group can be quite random, and there would be more variation.
For example, I might be interested in the proportional abundance made up by certain feeding guilds within a bird community. To test this, I might go out and catch birds repeatedly on a given site. On day 1, I might catch 30 birds in total in one site, and 5 of these eat mainly insects, so my proportional abundance of insectivorous birds is 5/30=0.167.
On day 2, I repeat the experiment, but happen to catch only 5 birds, out of which 4 happen to be insectivorous, resulting in a proportional abundance of 4/5=0.8. Further repeated measures might show that the proportions are generally below 0.2, but this one outlier of 0.8 will bias the data towards higher proportions.
A model in R looking at this data might be specified something like this:
model <- lme4::glmer(insectivor_captures/total_captures ~ (1|day) + site, family = binomial (link = logit), weights = total_captures, data=df)
How would you account for those days where total samples were low, and uncertainty was high? Would it make sense to just exclude those cases?