# Combine / analyse different exposure variables in count data model

The outcome $y$ in my dataset is count data. There are three possible exposure variables $e_1, e_2, e_3$ conceivable. These exposure variables are mutually exclusive, i.e. refer to different physical entities, they are different groups. However, the exposure variables are quite correlated.

I would like to estimate the influence of the groups on the outcome. That is, I would like make a statement like "The outcome consists of 10% $e_1$, 70% $e_2$ and 20% $e_3$."

What modelling approaches can I use to address this problem? So far, I have two ideas (the model type would be negative binomial or Poisson)

1. Model averaging. Here I would fit three models $y \;\tilde\;\; \text{offset}(\log e_i)$ and combine these models using for instance the model.avg function in the MuMIn package in R. However, I am not familiar with model averaging and would like to know if this approach is sensible before I look into it.

2. I could use the sum $e=e_1+e_2+e_3$ as exposure variable and add the fractions $f_i=e_i/e$ as additional parameters. That is, $y\;\tilde\;\; \text{offset}(\log e)+ f_1 + f_2$. Here I have the problem of interpreting the parameters to derive a statement on the groups' influence.

Any hint appreciated -- also on how to find out if the data does not help to make such a statement.

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