@mike-lawrence Thank you for the response!

I include the full model code below. At the most basic, I am modelling cross-sectional human growth where x is a continuous `Real`

that equates to chronological age (from ~birth to 20 years of age in my data) and Y is a large response vector of length k. In other words, I have measured k continuous growth variables (e.g., size, breadth, etc.) I have parameterized the mean \mu by an offset power law and the noise \sigma that scales linearly demonstrating heteroskedasticity as chronological age increases. In the code below, I assume Y is distributed as a multivariate normal with a mean per response k and decomposed \Sigma to disentangle the noise from the correlation structure. A few notes, I have also fit this model with a Multivariate Studentâ€™s T to compare against the MVN model. Further, I am in the process of trying to fit the model when the data is mixed. That is, the response vector k contains both continuous and ordinal responses variables. I am trying to do this using the GHK algorithm. I have been able to fit continuous and ordinal separately, but have yet to get them to work togetherâ€¦ I have posted other times through the discourse here and here for further information about the model.

In terms of the covariance structure and/or the correlation structure, my ultimate goal here is less concerned about the parameters associated with the mean and noise. Instead, I am most concerned with the nature of the covariance structure between growth responses and how they change with age (x). As it is currently written, I have let the noise vary as a function of x, but the correlations between growth traits are assumed to remain unchanged - there is just a single correlation term for each response variable regardless of age. My domain expertise suggests \rho like \sigma, may also vary as an individual ages.

I hope all of this makes sense!

```
data{
int N; // # of individuals
int K; // # of responses
vector[K] y[N]; // vector of growth responses per individual
real X[N]; //age
}
parameters{
vector<lower=0>[K] a; // multiplicative constant
vector<lower=0>[K] r; // scaling parameter
vector[K] b; // offset
vector<lower=0>[K] s_scale; //constant noise
vector<lower=0>[K] kappa; // slope/gradient of linear noise function
cholesky_factor_corr[K] L_Omega; // cholesky factor the corr matrix for efficiency
}
transformed parameters{
vector[K] mu[N]; //mean array of size N containing vectors of length K
vector[K] s[N]; //sd array of size N containing vectors of length K
matrix[K,K] Sigma[N]; // cov array of size N with K x K cov matrices
for(i in 1:N){
for(k in 1:K){
mu[i,k] = a[k]*X[i]^(r[k])+b[k]; // mean function
s[i,k] = s_scale[k]*(1 + kappa[k]*X[i]); // sd function
}
}
for(i in 1:N){
Sigma[i] = diag_pre_multiply(s[i],L_Omega); // combining the cholesky corr matrix with the noise
}
}
model{
//priors
a ~ normal(0,10); // half-normal
r ~ normal(0,1); //half-normal
b ~ normal(0,10);
kappa ~ normal(0,1); //half-normal
s_scale ~ cauchy(0,5); //half-cauchy
L_Omega ~ lkj_corr_cholesky(0.5);
for(i in 1:N){
y[i] ~ multi_normal_cholesky(mu[i], Sigma[i]); //likelihood
}
}
generated quantities{
vector[K] Y_mean[N]; //posterior mean
vector[K] Y_pred[N]; //posterior predictive check
corr_matrix[K] Omega; //re-factor corr matrix
cov_matrix[K] Sigma_new[N];//put cov matrix back together
real<lower=0> a_prior = normal_rng(0,10); // a prior
real<lower=0> r_prior = normal_rng(0,1); // r prior
real b_prior = normal_rng(0,10); // b prior
real<lower=0> s_const_prior = cauchy_rng(0,5); // constant s prior
real<lower=0> kappa_prior = normal_rng(0,1); // kappa prior
real s_prior_lin[N] = s_const_prior*(1+kappa_prior*X[N]); //heterskedatsic w/ prior
vector[K] log_lik[N]; // log-likelihood for model comparison
Omega = multiply_lower_tri_self_transpose(L_Omega); //corr matrix
for(i in 1:N){
Sigma_new[i] = quad_form_diag(Omega, s[i]); //posterior cov matrix
}
for(i in 1:N){
for(k in 1:K){
Y_mean[i,k] = a[k]*X[i]^r[k] +b[k]; //posterior mean
}
Y_pred[i] = multi_normal_rng(Y_mean[i],Sigma_new[i]); //posterior predictive
}
real y_sim[N] = normal_rng(a_prior*X^(r_prior) + b_prior,s_prior_lin); // prior pred distribution
for(n in 1:N){
log_lik[n] = multi_normal_cholesky_lpdf(y[n]|mu[n],Sigma[n]); // log lik of each obs
}
}
```