Hi,

I have fitted a multilevel latent class ordinal regression model, where the measures at cutpoint k in group g for individual n are given by the cumulative probabilities:

Pr(Y_{g,n}^{[0]} \le k) = \Phi\left( {C_{k,g}^{[0]} - \nu_{g}^{[0]}} \right) in latent class 1 (non-diseased group)

Pr(Y_{g,n}^{[1]} \le k) = \Phi\left( {C_{k,g}^{[1]} - \nu_{g}^{[1]}} \right) in latent class 2 (diseased group)

Where,

\mathbf{C}^{[d]}_{g} \sim \text{Induced-Dirichlet}( \mathbf{\alpha}^{[d]} ), d \in \{0, 1\} (using @betanalpha 's induced Dirichlet, see e.g. here and here )

\mathbf{\alpha}^{[d]} \sim N_{\ge 0 }(0, 10), d \in \{0, 1\}

Now, I want to fit another model, where the measures are given by:

Pr(Y_{g,n}^{[0]} \le k) = \Phi\left(\frac{C_{k} - \mu_{g}} {\exp(\sigma_{g})}\right ) in latent class 1 (non-diseased group)

Pr(Y_{g,n}^{[1]} \le k) = \Phi\left(\frac{C_{k} + \mu_{g} } { {\exp(-\sigma_{g} })}\right) in latent class 2 (diseased group)

I want to still use the induced dirichlet model for the cutpoints \mathbf{C}_{g} with this model, i.e.,

\mathbf{C}_{g} \sim \text{Induced-Dirichlet}( \mathbf{\alpha} )

However, in the induced dirichlet, the ordinal probabilities (conditional on an arbitrary anchor point \phi) are given by:

\text{Prob}_{k,g} = \Phi(C_{k,g} - \phi) - \Phi(C_{k-1,g} - \phi) ~~~~~ (*)

So the scale = 1. The issue with trying to adjust (*) is that the scales are different for the 2 latent classes - \exp(\sigma_{g}) in one and \exp(-\sigma_{g}) in the other, but we are using the same set of thresholds, \mathbf{C}_{g} , in both classes.

I was wondering if anyone has any tips for how it might be possible to use the induced Dirichlet for a model like this?