Seeking modeling advice



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.



This looks like an ordered logistic problem. Check out section 1.8 in the Stan User’s Guide and also this rstanarm article. I’m not sure if this will cover the delta_y issue. Maybe instead you could use y_pre as a categorical covariate predictor and not to define the response variate? So something like:

y_{post,n} ~~ordered~logistic(\beta_1x_{,n} + \alpha_j, c);

where \alpha_j is the intercept for category y_{pre} = j. This approach would probably benefit from a hierarchical prior on all \alpha_j's. In fact, you could even do a hierarchical approach on the \beta slope(s) if you wanted to model each category having a different effect from the dosage.

Then the dose optimization problem would depend on the severity of the initial condition, y_{pre}, which I think makes sense. So find x such that patient in category j (with y_{pre}=j) has the expected value of their y_{post} minimized?