I have a problem I’d like to model and ultimately implement in Stan; it’s similar to a medical testing/treatment type scenario so I’ll describe it in those terms.

Imagine you have a test for some condition, e.g. how at-risk a person is to heart disease or whatever, and imagine the result of the test is a score on a scale of 0-10, with 0 being least at-risk and 10 most at-risk. (In my particular problem the test scores are discrete integer values, i.e. 0, 1, 2, etc.; not sure if that matters.) Imagine we are testing some experimental treatment intended to reduce one’s at-risk score. The treatment dosage ranges from 0-8, continuous. So the data consist of each subject’s pre-treatment score (y_pre), the treatment dosage (x), and the post-treatment score (y_post). The goals of the study are to determine if the treatment is effective, and if so what is the optimal dosage. Let’s assume the data already exist, i.e. I’m not at liberty to “design” the experiment.

I’m looking for advice on how to get started with this. (Any examples in BDA or Gellman and Hill similar to this?) My initial thought is regression, with delta_y (i.e. y_post – y_pre) as the outcome and y_pre and x as predictors. Does this make sense? Also it seems like one should somehow address the issue that delta_y depends logically on y_pre, in the sense that if a given subject’s y_pre is already 0, the treatment can only worsen their score, i.e. the possible values for delta_y in each case depend on y_pre.

Thanks!