# Latent class ordinal regression model w/ induced Dirichlet model

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?

You’d have to rederive the induced Dirichlet and include the scale parameter in the Jacobian transformation. The derivatives will be a bit more structured but you should achieve a similar result. That said I’m not sure if it will make a difference – because the cut points can adjust to compensate for any scale of the latent affinity distribution, having varying cut points should already be giving you all of the variation you’d get from having varying scales.

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