Hi all,

I’ve been working on developing a three-level occupancy model, that considers \psi_{ij} (the probability site j is occupied by species i) \theta_{ijk} (the probability that sample k in site j is occupied by species i), and \lambda_{ijk} (the (normalised) rate at which DNA sequences of species i are encountered in sample k at site j) - see Conditional posterior predictions for three-level occupancy model for some more details. To fit the model for multiple species, I have put independent hierarchical priors on each level of the model to estimate the population of species responses at each level.

I have six total datasets of varying sizes, and 5 out of 6 of these datasets fit well, with Rhats <1.01 and (with perhaps a sometimes slightly forgiving eye) the MCMC traceplots and other graphical diagnostics look pretty good. However, for one model, I get multimodality specifically in the site-level population parameters for the hierarchical model. Here, `sigmas_occ[1]`

is a random intercept, and the others all represent slopes to 10 other continuous environmental predictors. N_species=2063, N_sites=30, and N_replicates=300. There are no max treedepth warnings or divergences. There is no multimodality present at the higher levels.

Which seems to be due to some chains exploring completely different regions of parameter space:

I had wondered if the problem was due to parameterisation, so I tried reparameterising the model using an implementation of the rstanarm decov() prior, where there is a single variance parameter that is split into standard deviations using a simplex, but this ended up looking pretty much the same, but with divergences to boot.

I’m at a bit of a loss at what to try next - I suppose one option might be to tighten up the priors, and another would be to try another parameterisation. One thing that’s probably worth mentioning is that I use the same 11 environmental variables at all three levels of the hierarchy (though at the sample and abundance levels, there are extra predictors too) - the thinking being, for example, if a species responds negatively to increasing rainfall, then this is likely to be reflected both in the probability of finding it at a site, the probability of it being present in a sample, and the number of sequences detected - and perhaps there’s a better way of representing this dependency among the levels that might help fix this problem?

Any insight much appreciated!

Model code is below:

```
functions {
real sample_detection_lpmf(int Y, real lambda, real theta, real phi){
return bernoulli_logit_lpmf(1 | theta) +
neg_binomial_2_log_lpmf(Y | lambda, phi);
}
real sample_nondetection_lpmf(int Y, real lambda, real theta, real phi) {
return log_sum_exp(
sample_detection_lpmf(Y | lambda, theta, phi), bernoulli_logit_lpmf(0 | theta));
}
real sample_occupancy_sum_lpmf(array[] int Y, vector lambda, vector theta, real phi) {
int K = size(Y);
real temp_target = 0;
for(k in 1:K) {
if(Y[k] > 0) {
temp_target += sample_detection_lpmf(Y[k] | lambda[k], theta[k], phi);
} else {
temp_target += sample_nondetection_lpmf(0 | lambda[k], theta[k], phi);
}
}
return(temp_target);
}
real partial_sum_lpmf(array[,] int Y,
int start, int end,
array[,] int Q,
int N_sites,
int N_replicates,
matrix X_reads,
matrix X_rep,
matrix X_occ,
vector b_reads,
vector b_rep,
vector b_occ,
matrix b_reads_species,
matrix b_rep_species,
matrix b_occ_species,
vector phi,
vector SF,
array[] int rep_group_sizes) {
int N = size(Y);
real temp_target = 0;
array[N, N_sites] int Q_local = Q[start:end,];
vector[N] phi_local = phi[start:end];
matrix[N_replicates, N] lambda = rep_matrix(SF, N) +
rep_matrix(X_reads * b_reads, N) +
(X_reads * b_reads_species[,start:end]);
matrix[N_replicates, N] theta = rep_matrix(X_rep * b_rep, N) + X_rep * b_rep_species[,start:end];
matrix[N_sites, N] psi = rep_matrix(X_occ * b_occ, N) + X_occ * b_occ_species[,start:end];
for(i in 1:N) {
int loc_start=1; //replicates are ordered by site - starting at replicate 1
for(j in 1:N_sites) {
int loc_end = loc_start + rep_group_sizes[j] - 1; // add number of reps for site j and subtract 1
if(Q_local[i, j] == 1) {
temp_target += bernoulli_logit_lpmf(1 | psi[j, i]) +
sample_occupancy_sum_lpmf(Y[i,loc_start:loc_end] | lambda[loc_start:loc_end, i], theta[loc_start:loc_end, i], phi_local[i]);
}
if(Q_local[i, j] == 0) {
temp_target += log_sum_exp(bernoulli_logit_lpmf(1 | psi[j, i]) +
sample_occupancy_sum_lpmf(Y[i, loc_start:loc_end] | lambda[loc_start:loc_end, i], theta[loc_start:loc_end, i], phi_local[i]),
bernoulli_logit_lpmf(0 | psi[j, i]));
}
loc_start += rep_group_sizes[j]; // update start point for next site
}
}
return temp_target;
}
}
data {
int N_species; // Number of species
int N_sites; // Number of sites
int N_replicates; // Number of replicates
array[N_species, N_replicates] int Y; // Raw data [OTU][Replicate]
array[N_species, N_sites] int Q; // Site occupation factor
// Different number of replicates per site, so the input replicates are passed in site
// order - then we can access them per site by looping over the group sizes
array[N_sites] int rep_group_sizes;
// Site occupancy predictors
int N_pred_occ; // Number of predictors
matrix[N_sites, N_pred_occ] X_occ; // Predictor matrix for occupancy
// Replicate occupancy predictors
int N_pred_rep;
matrix[N_replicates, N_pred_rep] X_rep;
// Read abundance predictors
int N_pred_reads;
matrix[N_replicates, N_pred_reads] X_reads;
// Parallelisation options
int grainsize;
}
parameters {
// Site occupancy parameters
vector[N_pred_occ] b_occ;
cholesky_factor_corr[N_pred_occ] L_Rho_occ;
vector<lower=0>[N_pred_occ] sigmas_occ;
matrix[N_pred_occ, N_species] z_occ;
// Sample occupancy parameters
vector[N_pred_rep] b_rep;
cholesky_factor_corr[N_pred_rep] L_Rho_rep;
vector<lower=0>[N_pred_rep] sigmas_rep;
matrix[N_pred_rep, N_species] z_rep;
// Size factors/row effects
vector<lower=0>[N_replicates] SF;
// Read abundance parameters
vector[N_pred_reads] b_reads;
cholesky_factor_corr[N_pred_reads] L_Rho_reads;
vector<lower=0>[N_pred_reads] sigmas_reads;
matrix[N_pred_reads, N_species] z_reads;
// Read dispersion
vector[N_species] z_phi;
real<lower=0> sigma_phi;
real a_phi;
}
transformed parameters {
// Site occupancy parameters
matrix[N_pred_occ, N_species] b_occ_species;
b_occ_species = (diag_pre_multiply(sigmas_occ, L_Rho_occ) * z_occ);
// Sample occupancy parameters
matrix[N_pred_rep, N_species] b_rep_species;
b_rep_species = (diag_pre_multiply(sigmas_rep, L_Rho_rep) * z_rep);
// Read abundance parametters
matrix[N_pred_reads, N_species] b_reads_species;
b_reads_species = (diag_pre_multiply(sigmas_reads, L_Rho_reads) * z_reads);
// Read dispersion
vector<lower=0>[N_species] phi;
phi = exp((a_phi + z_phi) * sigma_phi);
}
model {
// Priors
// Site occupancy
target += std_normal_lpdf(b_occ);
target += lkj_corr_cholesky_lpdf(L_Rho_occ | 3);
target += exponential_lpdf(sigmas_occ | 1);
target += std_normal_lpdf(to_vector(z_occ));
// Sample occupancy
target += std_normal_lpdf(b_rep);
target += lkj_corr_cholesky_lpdf(L_Rho_rep | 3);
target += exponential_lpdf(sigmas_rep | 1);
target += std_normal_lpdf(to_vector(z_rep));
// Size factors
target += lognormal_lpdf(SF | 0, 1);
// Read abundance
target += normal_lpdf(b_reads | 0, 3);
target += lkj_corr_cholesky_lpdf(L_Rho_reads | 3);
target += exponential_lpdf(sigmas_reads | 1);
target += std_normal_lpdf(to_vector(z_reads));
// Read dispersion
target += std_normal_lpdf(a_phi);
target += std_normal_lpdf(z_phi);
target += exponential_lpdf(sigma_phi | 1);
//array[,,] int Y, int grainsize, array[,] int Q, int N_sites, int N_replicates, matrix X_reads,
//matrix X_rep, matrix X_occ, vector b_reads, vector b_rep, vector b_occ, matrix b_reads_species,
//matrix b_rep_species, matrix b_occ_species, vector phi, vector SF, array[] int rep_group_sizes
target += reduce_sum(partial_sum_lpmf, Y, grainsize, Q, N_sites, N_replicates, X_reads, X_rep, X_occ, b_reads,
b_rep, b_occ, b_reads_species, b_rep_species, b_occ_species, phi, SF, rep_group_sizes);
}
```