Hello everyone,

This is my first post on the Stan discussion forum! A big thank you to the developers and to the community for all the excellent documentation that is out there.

I’m trying to reproduce a model from Warton et al 2015. This approach uses latent variables to approximate the correlation among many responses – in this case, between multiple species. **However**, although my model sort of runs, the answer doesn’t look at all like simulated data.

As far as I understand it, the model looks like this:

for the i sites and j species. The key part is the Uij, which are supposed to estimate the correlation among species across all sites. g() is the link function (e.g. for count data).

I wasn’t sure how much detail to include in this question, so I will include the Stan program below. I have a completely reproducible example, including R code to simulate data and fit the model, which is all in a github repository – the entire story is in this file. You can see a rendered version of this file on this website.

```
data{
int<lower=1> N;
int<lower=1> N_site_id;
int<lower=1> N_spp_id;
int obs_abd[N];
int spp_id[N];
int site_id[N];
int<lower=2> D; // number of latent dimensions
}
transformed data {
int<lower=1> M;
// We calculate the number of below-zero loadings
// the second term is the number in the "square part" below diag
// the first term is the "non square part" below that
M = D * (N_spp_id - D) + D * (D-1)/2; // number of non-zero loadings
}
parameters{
real inter;
vector[N_site_id] site_intercept;
real<lower=0> sitevar;
vector[N_spp_id] spp_intercept;
real<lower=0> sppvar;
// adding this part: latent variables? is that you?
matrix[N_site_id, D] latent_vars;
// matrix[2, N_spp_id] spp_loadings;
// blog parameters
vector[M] spp_load_L_t; // lower diagonal elements of L = spp_loadings
vector<lower=0>[D] spp_load_L_d; // diagonal elements of L = spp_loadings
// vector<lower=0>[N_spp_id] psi; // vector of variances
// hyperparameters
//real<lower=0> mu_psi;
//real<lower=0> sigma_psi;
// real mu_lt;
// real<lower=0> sigma_lt;
}
transformed parameters{
cholesky_factor_cov[N_spp_id,D] spp_loadings; //lower triangular factor loadings Matrix
{
int idx2;
idx2 = 1;
for(i in 1:N_spp_id){
for(j in (i+1):D){
spp_loadings[i,j] = 0; //constrain the upper triangular elements to zero
}
}
for (j in 1:D) {
// add the diagonal elements
spp_loadings[j,j] = spp_load_L_d[j];
for (k in (j+1):N_spp_id) {
// add the lower triangular elements
spp_loadings[k,j] = spp_load_L_t[idx2];
idx2 = idx2 + 1;
}
}
}
}
model{
vector[N] lamb;
// the hyperpriors
// mu_psi ~ cauchy(0, 1);
// sigma_psi ~ cauchy(0,1);
// mu_lt ~ cauchy(0, 1);
// sigma_lt ~ cauchy(0,1);
// the priors
// note that spp_load_L_d is constrained with lower=0
spp_load_L_d ~ cauchy(0,3);
spp_load_L_t ~ cauchy(0, 4);
// psi ~ cauchy(mu_psi,sigma_psi);
sppvar ~ cauchy( 0 , 4 );
spp_intercept ~ normal( 0 , sppvar );
sitevar ~ cauchy( 0 , 4 );
site_intercept ~ normal( 0 , sitevar );
inter ~ normal( 0 , 5 );
// latent variable priors??
for (j in 1:N_site_id){
for(k in 1:D ){
latent_vars[j, k] ~ normal(0, 1);
}
}
// the liklihood
// add the row of
for ( i in 1:N ) {
lamb[i] = inter + site_intercept[site_id[i]] + spp_intercept[spp_id[i]] + latent_vars[site_id[i],] * spp_loadings[spp_id[i],]';
}
obs_abd ~ poisson_log( lamb );
}
generated quantities{
vector[N] lamb;
matrix[N_spp_id,N_spp_id] Q; //Covariance mat
real dev;
dev = 0;
for ( i in 1:N ) {
lamb[i] = inter + site_intercept[site_id[i]] + spp_intercept[spp_id[i]] + latent_vars[site_id[i],] * spp_loadings[spp_id[i],]';
}
dev = dev + (-2)*poisson_log_lpmf( obs_abd | lamb );
//add the correlation matrix in here! Q = L*L'
Q = spp_loadings * spp_loadings'; //+ diag_matrix(psi);
// OK this gives me an error when I declare that Q needs to be positive definite, by using cov_matrix[N_spp_id]
}
```

To illustrate the problem, here is a quick plot of a DCA with the original simulated data on the left and one produced from latent variables on the right, which i *think* are supposed to resemble each other.