I have some variables that I do not believe directly affect the response variable but rather influence the amount of variance that I would expect in the random effects. In other words, I believe that the random effects are distributed \alpha_i \sim N(0, f(X_i). Actually writing this up in Stan should be no problem; however, I am not sure what the best distributional form for this would be, so I wanted to ask for some recommendations on how to handle this sort of problem. My initial thought was simply to predict the log of the variance (or perhaps the variance directly) with a linear model. If anyone has any recommendations or can point me to some previous analyses with this kind of problem that would be very helpful. Thank you!

I’m not an expert in this, but I think that’s a pretty typical approach—for example using a normal-log GLM. Then all the usual modeling advice for GLMs applies.