Hey

Not sure if this is the right forum for my question.

I have a number of subjects that each gets 3 treatments (A, B, C).

There are several measuments per subject.

I’m interested which pairs of treatments are significantly different within the subjects.

I know that most subjects do not respond to any treatment.

Some do respond and will probably respond differently to the 3 treatments.

I started to model this by putting a random effect on the subject-treatment interaction

y \tilde{} subject_s + (1 | subject_s:treatment_t)

There is one common hyperprior on the sigma of the random effect for all subjects.

When applying this model to ground truth data, I have the impression that I don’t capture the variance structure in my mixed effect properly.

I have the impression that there is to much shrinkage for the subjects that respond to the treatment (high variance across the 3 treatments) and not enough for the subject that do not respond (no treatment effect).

I thought of putting a different hyperprior on the sigma of the random effect for each subject.

But then I ignore the information that is present in the other subjects.

What would be the correct way to deal with this problem?

Can you do something like putting a hyperprior per subject that is shrunken to a common global hyper prior? (I guess something like a hyper hyper prior) Or is there some other way to deal with this?

I had not much luck searching for something similar in the literature.

I should also mention that in the end during inference I want to report which subjects show a difference in treatment effect between a given pair of treatments at a certain FDR level.

I also not much luck finding in the literature how to calculate this FDR in a bayesian setting.

Any pointers for this are welcome :)

Thanks in advance