## Background

I’m reproducing the code from https://onlinelibrary.wiley.com/doi/10.1002/pst.1730 in Stan (originally written with BUGS).

The basic data setup is that there are J = 10 clinical trials of varying sample size, each with an unknown log-odds \theta_j of success. We observe r_j successes out of n_j total patients in each trial.

The model is a hierarchical mixture model where with probability p_\text{exch}, \theta_j is exchangeable with the other strata (partial pooling) and with probability 1 - p_\text{exch}, \theta_j is non-exchangeable with the other strata (no pooling).

By setting p_\text{exch} to 0 for each strata we obtain a completely stratified analysis, the exchangeable/non-exchangeable (EXNEX) mixture with 0.5, and the standard hierarchical model partial pooling with when setting p_\text{exch} to 1.

I am able to reproduce the results of the original paper with my below code. However, I am running into some convergence issues. The model runs fine with p_\text{exch} = 0 and p_\text{exch} = 0.5 but the difficulties arise when p_\text{exch} is close to 1. I rewrote the model for the special case when p_\text{exch} = 1 using the non-centered parameterization (code also included below), and the model ran much better.

My question is, using the mixture model formulation shown in the code below, how can I implement the non-centered parameterization? Any guidance would be very helpful.

## EXNEX Stan Code

```
data {
int<lower=1> J; // Number of trial strata
array[J] int n; // Number of subjects per strata
array[J] int r; // Number of successes per strata
array[J] real<lower=0, upper=1> p_exch; // Mixture probability of exchangeability for each strata
}
parameters {
real mu; // Mean of hierarchical log-odds distribution
real<lower=0> tau; // SD of hierarchical log-odds distribution
vector[J] theta; // Log-odds for each strata
}
model {
// Priors
mu ~ normal(-1.73, 2.616); // Centered at logit(0.15)
tau ~ normal(0, 1); // Half-normal prior because of the constraint
for (j in 1:J) { // Loop over the trial strata
target += log_mix(p_exch[j], // Strata-specific mixing parameter
normal_lpdf(theta[j] | mu, tau), // EX
normal_lpdf(theta[j] | -1.734, 2.801) // NEX
);
r[j] ~ binomial_logit(n[j], theta[j]); // Binomial likelihood with built-in expit transform
}
}
```

## Special case for p_exch = 1 with non-centered parameterization

```
data {
int<lower=1> J; // Number of trial strata
array[J] int n; // Number of subjects per strata
array[J] int r; // Number of successes per strata
}
parameters {
real mu; // Mean of hierarchical log-odds distribution
real<lower=0> tau; // SD of hierarchical log-odds distribution
vector[J] theta_std; // Log-odds of response for each strata (standardized for non-centered parameterization)
}
model {
for (j in 1:J) { // Loop over the trial strata
theta_std[j] ~ normal(mu, 1); // Hierarchical structure + use non-centered parameterization
r[j] ~ binomial_logit(n[j], theta_std[j]*tau); // Binomial likelihood with built-in expit transform
}
// Priors
mu ~ normal(-1.73, 2.616); // Centered at logit(0.15)
tau ~ normal(0, 1); // Half-normal prior because of the constraint
}
```