I am modeling a dataset where each of several hundred individuals have ranked the same 10 objects from best to worst. Each individual is the member of a different group. The goal of the model is to estimate the underlying valence (desirability) of each object, taken to be continuous. I am not showing the part that relates rankings to continuous values, but the hierarchical structure of the model looks like:

```
data {
int<lower=1> n_objects; // e.g. 10
int<lower=1> n_individuals; // e.g. ~300
int<lower=1, upper=n_individuals> n_groups; // e.g. 9
int<lower=1, upper=n_groups> group_id[n_individuals]; // The integer group IDs of each individual
int<lower=1, upper=n_objects> ranks[n_individuals, n_objects]; // The matrix of integer ranks
}
parameters {
vector[n_objects] mu_global;
matrix[n_groups, n_objects] mu_group;
matrix[n_individuals, n_objects] mu_ind;
real<lower=0.0001> sigma_global;
real<lower=0.0001> sigma_group;
real<lower=0.0001> sigma_ind[n_groups];
}
transformed parameters {
real mu_mean = mean(mu_global); // Used for centering
}
model {
mu_mean ~ normal(0, 0.001); // Used for centering
mu_global ~ normal(0, sigma_global);
sigma_global ~ cauchy(0, 3);
sigma_group ~ cauchy(0, 3);
sigma_ind ~ cauchy(0, 3);
for (i in 1:n_groups) {
mu_group[i] ~ normal(mu_global, sigma_group);
}
for (i in 1:n_individuals) {
mu_ind[i] ~ normal(mu_group[group_id[i]], sigma_ind[group_id[i]]);
target += some_func(mu_ind[i], ranks[i]); // A function not shown here
}
}
```

This model fits pretty well, `R_hat`

~ 1, etc., and I get reasonable looking histograms for all parameters and point estimates that make sense given the data. I know from the data that some groups have much higher inter-individual variance than others in how they rank the objects, but allowing `sigma_ind`

to vary by group as I do (`sigma_ind[n_groups]`

) makes the model much slower to fit, and in general I am surprised at how slow sampling proceeds.

Is there a better strategy for this kind of hierarchical model? I also recognize that my model may be misparameterized in other ways, so any suggestions there would also be appreciated.