# Zero offset values, glmm poisson/negative binomial distribution with

I have a data set that consists of video observations of carpenter bee nests under three treatments: a control, mothers removed and mothers and worker removed. I have counts of twenty behaviours as response variables, and I am looking at how each differs between treatments. There are different numbers of nests under each treatment, and different numbers of observations in each nest, so I have treated nest as a random factor, e.g for biting behavior:

glmmTMB(Biting ~ treatment + (1|nest), offset=log(noBees), data=biteTest, ziformula=~1, family="poisson")

As the number of bees in an observation is predictive of the number of behaviours I would like to fit a model in glmmTMB with the number of bees in an observation as exposure/offset, and another with total behaviours as exposure/offset. Because some observations have no bees visible, and some have no activity, I have the problem of a zero offset for some observations.

My intuition is to drop the zero observations, but I am concerned that this may bias my data to treatments with more bees/activity. Shouldn't observations of zero bees/activity be important to understanding the difference between treatments? If I drop the zeros how do I justify this statistically, perhaps with a reference to a published paper. I am struggling to find similar examples in ecological or behavioural literature.

My other thought was to include + 1 in my offset, but then I'm not sure how to interpret estimated marginal effects (means).

In a Poisson (or negative binomial model) your offset would normally be on the logarithm-scale. E.g. if you want to model events per time units, then you use log(observation time) as your offset (as you actually seem to do in your code). That's because if your random variable is Y and your observation time is t, and we assume $$\log(\text{E}Y/t) = \boldsymbol{x}\boldsymbol{\beta}$$, then $$\log(\text{E}Y) = \boldsymbol{x}\boldsymbol{\beta} + \log t$$.