Hi there,

I have a complex hierarchical model and a latent group-specific parameter \theta_j,~j=1,...,J. For this parameter, I want to test multiple associations with group-specific variables Y_{jk},~k=1,...,K. By testing, I mean a simple linear association of the form

If I throw in all Y at once (multivariate model), the \beta are stealing from each other as many of the Y are strongly correlated. In the univariate case (estimating the model separately for each Y_k) I feel like I would have to adjust for the number of associations being tested, i.e. the more associations I test the more likely I will find ``significantāā ones just by coincidence.

I am aware of frequentist adjustments for multiple hypothesis testing. In the Bayesian context, it was argued that usually no adjustments are necessary (e.g. Gelman et al.). The examples revolve around constructing hierarchical models instead of subgroup comparisons (e.g. of treatments). In these settings, one can expect to have a population-level effect that is shared by all groups. However, I feel this is not applicable in my setting because one would not expect the associations to go in the same direction for all variables. Some variables may be positively associated, some negatively. Strongly correlated variables are expected to have similar associations, which I also find when estimating separate models for each variable Y_k.

Am I correct that hierarchical modeling of the associations is not the best way to go in my setting? If necessary, what adjustments could/should I make or how can I model all aossociations?

Note that a separate thing I am trying in this setting is a latent factor model for Y.

Thanks a lot for any suggestions!

Cheers,

Nic