# Using an offset term when modeling proportions

I have an experimental setup where pairs of birds are producing offspring in different treatments. I'm trying to model the total number of offspring that hatched per total number of eggs that were laid (hatching success), which is a proportion, i.e. #Hatchlings/#EggsLaid. I have zero-inflation in the data. What are your thoughts on fitting a Poisson regression model, with the response being #Hatchlings, and an offset term of #Eggslaid. Does this approach make sense?

Thanks!!

A Poisson does not seem right for your data, you cannot hatch more eggs than there were. It is perhaps no surprise that it does not fit so well. A binomial distribution (e.g. logistic regression) would enforce that constraint. If there is clutches or different incubators, you may want to account for that using a fixed or random effect.

Use the logistic regression.

Assume that Hatchlings($Y_i$) follows binomial distribution with probability $\pi_i$ with number trials = EggsLaid ($N_i$). $i$ indicates hatch.

$Y_i$ ~ $Bin(\pi_i, N_i)$

$\log(\frac{\pi}{1-\pi}) = X_i\beta + e_i$

$e_i$ ~ $N(0, \sigma^2)$

Poisson regression is not good because in your case $Y_i \le N_i$. But Poisson distribution has no this limitation.