# Apparently equivalent noncentered parameterization, different outcomes

**URL:** <https://discourse.mc-stan.org/t/apparently-equivalent-noncentered-parameterization-different-outcomes/36157>\
**Category:** Modeling\
**Tags:** rstan, fitting-issues\
**Created:** [August 7, 2024, 7:30pm UTC](https://discourse.mc-stan.org/t/apparently-equivalent-noncentered-parameterization-different-outcomes/36157 "2024-08-07T19:30:21Z")\
**Posts on this page:** 3\
**Page:** 1

<div class="post-metadata">

**Author:** ![ashmadoo](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/ashmadoo/32/18419_2.png) [@ashmadoo](https://discourse.mc-stan.org/u/ashmadoo)\
**Post date:** [August 7, 2024, 7:30pm UTC](https://discourse.mc-stan.org/t/apparently-equivalent-noncentered-parameterization-different-outcomes/36157/1 "2024-08-07T19:30:21Z")

</div>

I’m having a funny issue with a hierarchical model I’m fitting in Rstan. It runs fine, and the posterior predictive checks all seem to check out. The model is a negative binomial regression with three crossed random effects. I’ve used noncentered parameterization to avoid the funnel issue. However, the model only runs successfully without divergent transitions if I specify redundant mean values for each of the random intercepts.

I think that the more “correct” approach is supposed to be something like this:

```stan
model {

for(a in 1:AA){
a1[a] = b0 + sigma_a * raw_a[a];
}

for(b in 1:BB){
b2[b] = sigma_b * raw_b[b];
}

for(c in 1:CC){
c2[c] = sigma_c * raw_c[c];
}

for(i in 1:N){
log_y_hat = a1[AA[i]] + b2[BB[i]] + c2[CC[i]] + coef * x[i];
} 

y ~ neg_binomial_2_log(log_y_hat, phi);

b0 ~ normal(0,1);
raw_a ~ normal(0,1);
raw_b ~ normal(0,1);
raw_c ~ normal(0,1);

sigma_a~exponential(1);
sigma_b~exponential(1);
sigma_c~exponential(1);

phi ~ gamma(1,2);
coef ~ lognormal(-1, 1);

}

```

But the model will only fit without divergent transitions if I include a redundant mean value for each of the random intercepts (a\_mu, b\_mu, c\_mu)

```stan
model {

for(a in 1:AA){
a1[a] = a_mu + sigma_a * raw_a[a];
}

for(b in 1:BB){
b2[b] = b_mu + sigma_b * raw_b[b];
}

for(c in 1:CC){
c2[c] = c_mu + sigma_c * raw_c[c];
}

for(i in 1:N){
log_y_hat = b0 + a1[AA[i]] + b2[BB[i]] + c2[CC[i]] + coef * x[i];
} 

y ~ neg_binomial_2_log(log_y_hat, phi);

b0 ~ normal(0,1);
raw_a ~ normal(0,1);
raw_b ~ normal(0,1);
raw_c ~ normal(0,1);

a_mu ~ normal(0,1);
b_mu~normal(0,1);
c_mu~normal(0,1);

sigma_a~exponential(1);
sigma_b~exponential(1);
sigma_c~exponential(1);

phi ~ gamma(1,2);
coef ~ lognormal(-1, 1);

}

```

Have any of you run into this before? Or is there something I’m missing? I’d appreciate your thoughts!

---

<div class="post-metadata">

**Author:** ![Corey.Plate](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/corey.plate/32/20600_2.png) [@Corey.Plate](https://discourse.mc-stan.org/u/Corey.Plate)\
**Post date:** [August 7, 2024, 8:06pm UTC](https://discourse.mc-stan.org/t/apparently-equivalent-noncentered-parameterization-different-outcomes/36157/2 "2024-08-07T20:06:36Z")

</div>

Why do you feel that these parameters are redundant? In one case, you have a gaussian multiplied by an exponential, and, in the other, a mixture of a gaussian and a gaussian mutliplied by an exponential. Their geometries are not the same:

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---

<div class="post-metadata">

**Author:** ![ashmadoo](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/ashmadoo/32/18419_2.png) [@ashmadoo](https://discourse.mc-stan.org/u/ashmadoo)\
**Post date:** [August 8, 2024, 12:50pm UTC](https://discourse.mc-stan.org/t/apparently-equivalent-noncentered-parameterization-different-outcomes/36157/3 "2024-08-08T12:50:32Z")

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Hi Corey,  
Thanks, I see what you mean. I was thinking in terms of the intercepts’ role in the model rather than their geometries. Since one intercept is already specified (b0), I thought that the random intercepts would only need to be included as some deviation from that mean value. I don’t think it’s a problem to have those three extra intercept terms, but I’m hoping to understand why they are necessary for the model fit. (Right now I think my response to any reviewer questions about it would be kind of lame.)