Hello,

I have data with a three-level hierarchy, i.e. `n[j,k]`

individuals nested within J populations, nested within K=2 super-populations (let us call them `A`

and `B`

). Within any population j, the two super-populations may be more or less similar, but I want to estimate the average heterogeneity of the super-populations, i.e. averaged over all J populations. Since I am not specifically interested in the population-specific heterogeneity, I am wondering whether it is required to explicitely estimate it in the first place. As an example:

The standard way (I guess):

```
model {
for(j in 1:J){
for(k in 1:K){
y[j,k] ~ normal(mu[j,k], sigma[j,k]); // individual level
}
mu[j,1:K] ~ normal(nu[j], tau[j]); // population-specific heterogeneity expressed by tau[j]
tau[j] ~ lognormal(tau_mu, tau_sigma); // tau_mu represents average heterogeneity of the super-populations
}
tau_mu ~ normal(prior_tau_mu, prior_tau_sigma); // prior for tau_mu
tau_sigma ~ gamma(prior_shape, prior_rate); // prior for tau_sigma
}
```

The “short-cut”:

```
model {
for(j in 1:J){
for(k in 1:K){
y[j,k] ~ normal(mu[j,k], sigma[j,k]); // individuals
}
mu[j,1:K] ~ normal(nu[j], tau); // average heterogeneity expressed by tau
}
tau ~ gamma(shape, rate); // prior for tau
}
```

So my questions are: What is the difference between the following two models (except that the first model explicitely estimates the population-specific `tau`

)? And is the “short-cut” or any of the two models valid?