Hi,

For an ordered probit model (with just an intercept \beta), likelihood contributions have the form

\Phi(c_{k+1} - \beta) - \Phi(c_{k} - \beta)

For the model to be well defined, we set ordering constraints on the cutpoints so that

c_{k} \le c_{k+1}

Which we can do by defining an ordered vector and setting priors e.g. @betanalpha 's induced dirichlet

For my hierarchical model, I have a hierarchical structure on the intercepts, with cutpoints fixed between studies, so that

\Phi(c_{k+1} - \beta_{s}) - \Phi(c_{k} - \beta_{s})

With between-study model

\beta_{s} \sim N(\mu_{\beta}, \sigma_{\beta} )

This works fine in Stan. Note that the model’s I am using are multivariate mixtures ( for which I used and extended bgoori’s parameterisation for ). The model I have specified here is a little simpler than the one I am programming to simplify notation for this post. The extra detail is not necessary for the specific issue I am having at the moment (i.e. ordering constraints).

Now I am trying to extend to a generalised ordered probit model to relax the proportional odds / parallel lines assumumption - so that the \beta's themselves vary between each threshold k , so the model now looks like this

\Phi( c_{k+1} - \beta_{s, k+1} ) - \Phi(c_{k} - \beta_{s, k})

With between-study model

\beta_{s,k} \sim N(\mu_{\beta} + \nu_{k} , \sigma_{\beta} )

So we have shared s.d. (\sigma_{\beta}) between thresholds k, but different means, with \mu_{\beta} a mean parameter across the thresholds

Now, we need a more complicated ordering constraint:

c_{k} - \beta_{s, k} \le c_{k+1} - \beta_{s, k+1}

Would anybody have any suggestions as to how to enforce such a constraint in Stan (whilst still being able to interpret the \beta_{s,k}'s and the means \mu_{\beta} + \nu_{k} ), given that the \beta_{s,k}'s are hierarchical?

Something I was thinking was keeping the same constraint as before for the cutpoint parameters ( c_{k} \le c_{k+1} ), and then setting \beta_{s, k} \ge \beta_{s, k+1} for the , however I am not sure how to do the latter for a non-centred parameterisation (each \beta_{s, k} is hierarchical )