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I have a response variable of proportions representing the number of hours an individual is performing a specific activity divided by the total number of hours in the study period. I have no 0's or 1's in this data set. I am interested in knowing the extent to which specific factors (such as season) influence this proportion. I have random effects (individual ID) and fixed effects, and so I am considering a generalized linear mixed model.

I have read that the 'glmer' function can be used with a 'binomial' family and a 'weights' argument. The following is from the help file of glm:

The help file of glm says:

 "For a binomial GLM prior weights are used to give the number of trials 
  when the response is the proportion of successes" 

I am not sure what to specify for the 'weights' argument in my case, since my proportions are not representative of successes/total or failure/total.

Other sources state that, if data is "inherently proportional" (which I would think my data is), then a beta regression is more suitable. I do not think I can incorporate random effects with a beta regression. My question is thus: which approach would be most suitable for my data?

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You could use glmmTMB to do beta regression using the argument family=beta_family. It can do either GLMs or GLMMs, so random effect can be included. I don't think the beta family has been extensively tested, but it should work as far as we know.

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