How to rescale a response for Gamma regression?


I would like to run the same Gamma regression model is different data sets and I’m thinking to resale the responses to 0-1. Of course, for Gamma cannot be exactly 0 and here is my question.
How to rescale a response for Gamma regression?
Does it make sense to rescale the data into 0.1 - 1.1? So there is space even bellow the smallest value? Or rescaling to 1-2 makes more sense?
Or it’s better to forget the unit range and centre/scale the data to have mu = 10 & sd = 1? The data will stay positive (too many sd far from zero) and the unit scale will keep Stan sampling happy.

The unit range might be convenient to think about priors but my issue is that all these transformations feel a bit arbitrary.

Hi @Elef, sorry it’s taken a while for anyone to respond to your question. I think folks might be unsure of the best advice to give you because it’s not clear why you have chosen a Gamma distribution rather than something that fully supports the range of your data. Often times, I think models end up describing the data generating process better if they can avoid arbitrary re-scaling altogether. Could you tell us a bit more about the properties of the data you’re working with and why the unit interval feels like a good way to express your prior knowledge? Alternatively, if you must use a Gamma and support zero, you might consider looking into how to add zero-inflation or a hurdle to your Gamma regression.

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The objective is to perform uncertainty quantification in a process. They are several responses coming from this process which (for reasons) we should model them separately (at least for now). Each response has different units and scale but all of them is expected to be continuous, positive, and heteroscedastic, so a likelihood that allows for skewed distributions seems reasonable, so that’s the motivation for Gamma.
I would like to scale the different responses to,

  1. use similar priors for all of models,
  2. I think Stan works more efficiently if the response has a unit scale.

Note: I don’t want to support zero, that’s why I’m thinking scaling the data to 0.1-1.1

If you are aiming to have priors that are scaled multiplicatives of the parameters drawn from a skewable, continuous, positive, heteroscedastic distribution with support over (0,1), why not a beta distribution?

Sure it could be that too. Beta wasn’t my first choice just because the scaling is based on the extreme values so if you were asking why do you expect that cannot get smaller values than the min(x), I wouldn’t have any answer - But this might not be a real argument any way.
For Beta likelihood, how would you scale the data to avoid being exactly 0 and 1?