Hierarchical graded IRT

Interfaces brms
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Hi,

I have a follow-up question on this model. At the moment the scale of theta is pinned to 1 to deal with non-identifiability. This works well, even with the soft sum-to-zero constraint, and reduce_sum helps a lot to speed this up, although I’m struggling a bit with optimizing this further via vectorization (see Using `reduce_sum` with `ordered_logistic` - #13 by saschagobel).

As a next step, I’d like to model the variance as well, so that theta is modeled as:

theta-model

The following won’t work, because scale invariance is not addressed and the model is not identified (the covariate matrix is the same for the location and scale):

theta ~ normal(X*gamma, exp(X*lambda));

In the paper mentioned above, the author suggests an additional sum-to-zero constraint on sigma, so that:

sigma-constraint

and “the geometric mean of the prior variances of the latent preferences equals one”.

I’m not sure how to implement this in Stan, though.

I tried the following but this yields a few divergent transitions, split R-hat greater than 1.05 on many parameters, and the results are clearly off.

// Just the additions

parameters {
  ...
  vector[K] lambda;
  vector<lower=0>[J] sigma;
}

model{
  ...
  log(sigma) ~ normal(X*lambda, 1); 
  mean(sigma) ~ normal(0, 0.001 * J);
  theta ~ normal(X*gamma, sigma);
  mean(theta) ~ normal(0, 0.001 * J);
  ...
}

Thanks for your help!

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