Gamma regression and offset term

Question

I'm testing the effect of soil treatments on root and shoot length of seeds of a species. However the seeds had different sizes, which may affect root and shoot development. I have a measure of the area (in cm²) occupied by the nut in the seed that could be considered as a proxy for seed size. I'm thinking about using the nut area as an offset in the Gamma models. I'm not sure if this would be adequate nor in which unit the response variable will come out. It would be of no use if the response come as" length per seed area", which I believe it would be the case when using the offset. Should I consider using area as a fixed term in the model or is there anyway I could account for nut area as offset and back transform the results to length units?

Edit: I'm using log(seed area) as an offset term.

Answer 1

While an offset in a Gamma GLM with log-link will yield the same coefficient estimates as if you had modelled by dividing through by the offset, the model with the offset in it models the response in the original units, so the model and its fitted values will be in terms of seed length, not seed length per unit area.

If a coefficient of 1 on area does what you want with it (making expected seed length proportional to area), you should use the offset.

However, it's not clear to me the sense in which a model that divided through by area instead would be "no use", since you can (trivially) convert between the models via a simple multiplication by area to take you back to a plain seed-length scale at the end. What's the difficulty?