# The issue of adding random slopes in non-central parameterization

**URL:** <https://discourse.mc-stan.org/t/the-issue-of-adding-random-slopes-in-non-central-parameterization/34034>\
**Category:** Modeling\
**Tags:** rstan, specification\
**Created:** [February 5, 2024, 8:38am UTC](https://discourse.mc-stan.org/t/the-issue-of-adding-random-slopes-in-non-central-parameterization/34034 "2024-02-05T08:38:10Z")\
**Posts on this page:** 4\
**Page:** 1

<div class="post-metadata">

**Author:** ![zrn](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/zrn/32/17777_2.png) [@zrn](https://discourse.mc-stan.org/u/zrn)\
**Post date:** [February 5, 2024, 8:38am UTC](https://discourse.mc-stan.org/t/the-issue-of-adding-random-slopes-in-non-central-parameterization/34034/1 "2024-02-05T08:38:10Z")

</div>

Hello everyone! I’ve encountered some issues while using rstan. My model is an intermediate model with two crossed random factors. Upon recommendation, I employed the non-central parameterization method to improve the fitting performance and avoid warnings. Initially, I constructed a model with only random intercepts (code provided below), and at this point, all coefficients appeared normal. However, when I added random slopes to this model, none of the coefficients for fixed effects were significant. I’m wondering why this is happening and how I can improve it. Any help would be greatly appreciated.

> randomalpha="data{  
> int\<lower=1\> N;  
> vector[N] c1;  
> vector[N] c2;  
> vector[N] aq;  
> vector[N] cr;  
> vector[N] adt;  
> int\<lower=1\> J;  
> int\<lower=1, upper=J\> Group[N];  
> int\<lower=1\> G;  
> int\<lower=1, upper=G\> Group2[N];  
> }  
> parameters{  
> real alphaaq;  
> real beta1aq;  
> real beta2aq;  
> real\<lower=0\> sigma\_alphaaq;  
> real\<lower=0\> sigma\_beta1aq;  
> real\<lower=0\> sigma\_beta2aq;  
> real\<lower=0\> sigmaaq;  
> real alphacr;  
> real beta1cr;  
> real beta2cr;  
> real\<lower=0\> sigma\_alphacr;  
> real\<lower=0\> sigma\_beta1cr;  
> real\<lower=0\> sigma\_beta2cr;  
> real\<lower=0\> sigmacr;  
> real alphaadt;  
> real beta1adt;  
> real beta2adt;  
> real beta3adt;  
> real beta4adt;  
> real\<lower=0\> sigma\_alphaadt;  
> real\<lower=0\> sigma\_beta1adt;  
> real\<lower=0\> sigma\_beta2adt;  
> real\<lower=0\> sigma\_beta3adt;  
> real\<lower=0\> sigma\_beta4adt;  
> real\<lower=0\> sigma\_adt;  
> cholesky\_factor\_corr[3] Omega\_chol1;  
> cholesky\_factor\_corr[3] Omega\_chol2;  
> vector\<lower=0\>[3] sigma\_ranef1;  
> vector\<lower=0\>[3] sigma\_ranef2;  
> matrix[J,3] gamma1\_raw; // random effects for id  
> matrix[G,3] gamma2\_raw; // random effects for topic  
> vector[3] mu1;  
> vector[3] mu2;  
> }  
> transformed parameters{  
> vector[J] gamma1alphaaq;  
> vector[J] gamma1alphacr;  
> vector[J] gamma1alpha;  
> vector[G] gamma2alphaaq;  
> vector[G] gamma2alphacr;  
> vector[G] gamma2alpha;  
> vector[N] mu\_adt;  
> vector[N] mu\_aq;  
> vector[N] mu\_cr;  
> matrix[J,3] gamma1 = rep\_matrix(mu1’, J) + gamma1\_raw \* diag\_pre\_multiply(sigma\_ranef1, Omega\_chol1)‘;  
> matrix[G,3] gamma2 = rep\_matrix(mu2’, G) + gamma2\_raw \* diag\_pre\_multiply(sigma\_ranef2, Omega\_chol2)';  
> matrix[3,3] DC1 = diag\_pre\_multiply(sigma\_ranef1, Omega\_chol1);  
> matrix[3,3] DC2 = diag\_pre\_multiply(sigma\_ranef2, Omega\_chol2);  
> for (j in 1:J){  
> gamma1alphaaq[j] = gamma1[j,1];  
> gamma1alphacr[j] = gamma1[j,2];  
> gamma1alpha[j]= gamma1[j,3];  
> }  
> for (g in 1:G) {  
> gamma2alphaaq[g] = gamma2[g,1];  
> gamma2alphacr[g] = gamma2[g,2];  
> gamma2alpha[g]= gamma2[g,3];  
> }  
> for (n in 1:N) {  
> mu\_aq[n] = alphaaq + gamma1alphaaq[Group[n]] + gamma2alphaaq[Group2[n]]+  
> (beta1aq) \* c1[n] +  
> (beta2aq) \* c2[n];  
> mu\_cr[n] = alphacr + gamma1alphacr[Group[n]] + gamma2alphacr[Group2[n]]+  
> (beta1cr) \* c1[n] +  
> (beta2cr) \* c2[n];  
> mu\_adt[n] = alphaadt + gamma1alpha[Group[n]] + gamma2alpha[Group2[n]]+  
> (beta1adt) \* c1[n] +  
> (beta2adt) \* c2[n] +  
> (beta3adt) \* aq[n] +  
> (beta4adt) \* cr[n];  
> }  
> }model{  
> sigma\_alphaaq ~ student\_t(5, 0, 2);  
> sigma\_beta1aq ~ student\_t(5, 0, 2);  
> sigma\_beta2aq ~ student\_t(5, 0, 2);  
> alphaaq ~ normal(0, sigma\_alphaaq);  
> beta1aq ~ normal(0, sigma\_beta1aq);  
> beta2aq ~ normal(0, sigma\_beta2aq);  
> sigmaaq ~ student\_t(5, 0, 2);  
> sigma\_alphacr ~ student\_t(5, 0, 2);  
> sigma\_beta1cr ~ student\_t(5, 0, 2);  
> sigma\_beta2cr ~ student\_t(5, 0, 2);  
> alphacr ~ normal(0, sigma\_alphacr);  
> beta1cr ~ normal(0, sigma\_beta1cr);  
> beta2cr ~ normal(0, sigma\_beta2cr);  
> to\_vector(gamma1\_raw) ~ normal(0, 1);  
> to\_vector(gamma2\_raw) ~ normal(0, 1);  
> sigmacr ~ student\_t(5, 0, 2);  
> sigma\_alphaadt ~ student\_t(5, 0, 2);  
> sigma\_beta1adt ~ student\_t(5, 0, 2);  
> sigma\_beta2adt ~ student\_t(5, 0, 2);  
> sigma\_beta3adt ~ student\_t(5, 0, 2);  
> sigma\_beta4adt ~ student\_t(5, 0, 2);  
> alphaadt ~ normal(0, sigma\_alphaadt);  
> beta1adt ~ normal(0, sigma\_beta1adt);  
> beta2adt ~ normal(0, sigma\_beta2adt);  
> beta3adt ~ normal(0, sigma\_beta3adt);  
> beta4adt ~ normal(0, sigma\_beta4adt);  
> sigma\_adt ~ student\_t(5, 0, 2);  
> sigma\_ranef1 ~ student\_t(5, 0, 2);  
> sigma\_ranef2 ~ student\_t(5, 0, 2);  
> Omega\_chol1 ~ lkj\_corr\_cholesky(2.0);  
> Omega\_chol2 ~ lkj\_corr\_cholesky(2.0);  
> aq ~ normal(mu\_aq, sigmaaq);  
> cr ~ normal(mu\_cr, sigmacr);  
> adt ~ normal(mu\_adt, sigma\_adt);  
> mu1 ~ normal(0, 2);  
> mu2 ~ normal(0, 2);  
> // for (j in 1:J)  
> // gamma1[j] ~ multi\_normal\_cholesky(mu1, DC1);  
> // for (g in 1:G)  
> // gamma2[g] ~ multi\_normal\_cholesky(mu2, DC2);  
> }  
> generated quantities{  
> real naiveIndEffect1;  
> real naiveIndEffect2;  
> real naiveIndEffect3;  
> real naiveIndEffect4;  
> real totalEffect;  
> naiveIndEffect1=beta1aq_beta3adt;  
> naiveIndEffect2=beta1cr_beta4adt;  
> naiveIndEffect3=beta2aq_beta3adt;  
> naiveIndEffect4=beta2cr_beta4adt;  
> totalEffect=naiveIndEffect1+naiveIndEffect2+naiveIndEffect3+naiveIndEffect4+beta1adt  
> +beta2adt;  
> }  
> "

randombeta model:

> randombeta="data{  
> int\<lower=1\> N;  
> vector[N] c1;  
> vector[N] c2;  
> vector[N] aq;  
> vector[N] cr;  
> vector[N] adt;  
> int\<lower=1\> J;  
> int\<lower=1, upper=J\> Group[N];  
> int\<lower=1\> G;  
> int\<lower=1, upper=G\> Group2[N];  
> }  
> parameters{  
> real alphaaq;  
> real beta1aq;  
> real beta2aq;  
> real\<lower=0\> sigma\_alphaaq;  
> real\<lower=0\> sigma\_beta1aq;  
> real\<lower=0\> sigma\_beta2aq;  
> real\<lower=0\> sigmaaq;  
> real alphacr;  
> real beta1cr;  
> real beta2cr;  
> real\<lower=0\> sigma\_alphacr;  
> real\<lower=0\> sigma\_beta1cr;  
> real\<lower=0\> sigma\_beta2cr;  
> real\<lower=0\> sigmacr;  
> real alphaadt;  
> real beta1adt;  
> real beta2adt;  
> real beta3adt;  
> real beta4adt;  
> real\<lower=0\> sigma\_alphaadt;  
> real\<lower=0\> sigma\_beta1adt;  
> real\<lower=0\> sigma\_beta2adt;  
> real\<lower=0\> sigma\_beta3adt;  
> real\<lower=0\> sigma\_beta4adt;  
> real\<lower=0\> sigma\_adt;  
> cholesky\_factor\_corr[11] Omega\_chol1;  
> cholesky\_factor\_corr[11] Omega\_chol2;  
> vector\<lower=0\>[11] sigma\_ranef1;  
> vector\<lower=0\>[11] sigma\_ranef2;  
> matrix[J,11] gamma1\_raw;  
> matrix[G,11] gamma2\_raw;  
> vector[11] mu1;  
> vector[11] mu2;  
> }  
> transformed parameters{  
> vector[J] gamma1alphaaq;  
> vector[J] gamma1alphacr;  
> vector[J] gamma1alpha;  
> vector[J] gamma1beta1aq;  
> vector[J] gamma1beta2aq;  
> vector[J] gamma1beta1cr;  
> vector[J] gamma1beta2cr;  
> vector[J] gamma1beta1adt;  
> vector[J] gamma1beta2adt;  
> vector[J] gamma1beta3adt;  
> vector[J] gamma1beta4adt;  
> vector[G] gamma2alphaaq;  
> vector[G] gamma2alphacr;  
> vector[G] gamma2alpha;  
> vector[G] gamma2beta1aq;  
> vector[G] gamma2beta2aq;  
> vector[G] gamma2beta1cr;  
> vector[G] gamma2beta2cr;  
> vector[G] gamma2beta1adt;  
> vector[G] gamma2beta2adt;  
> vector[G] gamma2beta3adt;  
> vector[G] gamma2beta4adt;  
> vector[N] mu\_adt;  
> vector[N] mu\_aq;  
> vector[N] mu\_cr;  
> matrix[J,11] gamma1 = rep\_matrix(mu1’, J) + gamma1\_raw \* diag\_pre\_multiply(sigma\_ranef1, Omega\_chol1)‘; // random effects for id  
> matrix[G,11] gamma2 = rep\_matrix(mu2’, G) + gamma2\_raw \* diag\_pre\_multiply(sigma\_ranef2, Omega\_chol2)';  
> matrix[11,11] DC1 = diag\_pre\_multiply(sigma\_ranef1, Omega\_chol1);  
> matrix[11,11] DC2 = diag\_pre\_multiply(sigma\_ranef2, Omega\_chol2);  
> for (j in 1:J){  
> gamma1alphaaq[j] = gamma1[j,1];  
> gamma1alphacr[j] = gamma1[j,2];  
> gamma1alpha[j]= gamma1[j,3];  
> gamma1beta1aq[j]= gamma1[j,4];  
> gamma1beta1cr[j]= gamma1[j,5];  
> gamma1beta1adt[j]= gamma1[j,6];  
> gamma1beta2aq[j]= gamma1[j,7];  
> gamma1beta2cr[j]= gamma1[j,8];  
> gamma1beta2adt[j]= gamma1[j,9];  
> gamma1beta3adt[j]= gamma1[j,10];  
> gamma1beta4adt[j]= gamma1[j,11];  
> }  
> for (g in 1:G) {  
> gamma2alphaaq[g] = gamma2[g,1];  
> gamma2alphacr[g] = gamma2[g,2];  
> gamma2alpha[g]= gamma2[g,3];  
> gamma2beta1aq[g]= gamma2[g,4];  
> gamma2beta1cr[g]= gamma2[g,5];  
> gamma2beta1adt[g]= gamma2[g,6];  
> gamma2beta2aq[g]= gamma2[g,7];  
> gamma2beta2cr[g]= gamma2[g,8];  
> gamma2beta2adt[g]= gamma2[g,9];  
> gamma2beta3adt[g]= gamma2[g,10];  
> gamma2beta4adt[g]= gamma2[g,11];  
> }  
> for (n in 1:N) {  
> mu\_aq[n] = alphaaq + gamma1alphaaq[Group[n]] + gamma2alphaaq[Group2[n]]+  
> (beta1aq+gamma1beta1aq[Group[n]]+gamma2beta1aq[Group2[n]]) \* c1[n] +  
> (beta2aq+gamma1beta2aq[Group[n]]+gamma2beta2aq[Group2[n]]) \* c2[n];  
> mu\_cr[n] = alphacr + gamma1alphacr[Group[n]] + gamma2alphacr[Group2[n]]+  
> (beta1cr+gamma1beta1cr[Group[n]]+gamma2beta1cr[Group2[n]]) \* c1[n] +  
> (beta2cr+gamma1beta2cr[Group[n]]+gamma2beta2cr[Group2[n]]) \* c2[n];  
> mu\_adt[n] = alphaadt + gamma1alpha[Group[n]] + gamma2alpha[Group2[n]]+  
> (beta1adt+gamma1beta1adt[Group[n]]+gamma2beta1adt[Group2[n]]) \* c1[n] +  
> (beta2adt+gamma1beta2adt[Group[n]]+gamma2beta2adt[Group2[n]]) \* c2[n] +  
> (beta3adt+gamma1beta3adt[Group[n]]+gamma2beta4adt[Group2[n]]) \* aq[n] +  
> (beta4adt+gamma1beta4adt[Group[n]]+gamma2beta4adt[Group2[n]]) \* cr[n];  
> }  
> }  
> model{  
> sigma\_alphaaq ~ student\_t(5, 0, 2);  
> sigma\_beta1aq ~ student\_t(5, 0, 2);  
> sigma\_beta2aq ~ student\_t(5, 0, 2);  
> alphaaq ~ normal(0, sigma\_alphaaq);  
> beta1aq ~ normal(0, sigma\_beta1aq);  
> beta2aq ~ normal(0, sigma\_beta2aq);  
> sigmaaq ~ student\_t(5, 0, 2);  
> sigma\_alphacr ~ student\_t(5, 0, 2);  
> sigma\_beta1cr ~ student\_t(5, 0, 2);  
> sigma\_beta2cr ~ student\_t(5, 0, 2);  
> alphacr ~ normal(0, sigma\_alphacr);  
> beta1cr ~ normal(0, sigma\_beta1cr);  
> beta2cr ~ normal(0, sigma\_beta2cr);  
> to\_vector(gamma1\_raw) ~ normal(0, 1);  
> to\_vector(gamma2\_raw) ~ normal(0, 1);  
> sigmacr ~ student\_t(5, 0, 2);  
> sigma\_alphaadt ~ student\_t(5, 0, 2);  
> sigma\_beta1adt ~ student\_t(5, 0, 2);  
> sigma\_beta2adt ~ student\_t(5, 0, 2);  
> sigma\_beta3adt ~ student\_t(5, 0, 2);  
> sigma\_beta4adt ~ student\_t(5, 0, 2);  
> alphaadt ~ normal(0, sigma\_alphaadt);  
> beta1adt ~ normal(0, sigma\_beta1adt);  
> beta2adt ~ normal(0, sigma\_beta2adt);  
> beta3adt ~ normal(0, sigma\_beta3adt);  
> beta4adt ~ normal(0, sigma\_beta4adt);  
> sigma\_adt ~ student\_t(5, 0, 2);  
> sigma\_ranef1 ~ student\_t(5, 0, 2);  
> sigma\_ranef2 ~ student\_t(5, 0, 2);  
> Omega\_chol1 ~ lkj\_corr\_cholesky(2.0);  
> Omega\_chol2 ~ lkj\_corr\_cholesky(2.0);  
> aq ~ normal(mu\_aq, sigmaaq);  
> cr ~ normal(mu\_cr, sigmacr);  
> adt ~ normal(mu\_adt, sigma\_adt);  
> mu1 ~ normal(0, 2);  
> mu2 ~ normal(0, 2);  
> // for (j in 1:J)  
> // gamma1[j] ~ multi\_normal\_cholesky(mu1, DC1);  
> // for (g in 1:G)  
> // gamma2[g] ~ multi\_normal\_cholesky(mu2, DC2);  
> }  
> generated quantities{  
> real naiveIndEffect1;  
> real naiveIndEffect2;  
> real naiveIndEffect3;  
> real naiveIndEffect4;  
> real avgIndEffect1\_1;  
> real avgIndEffect1\_2;  
> real avgIndEffect2\_1;  
> real avgIndEffect2\_2;  
> real avgIndEffect3\_1;  
> real avgIndEffect3\_2;  
> real avgIndEffect4\_1;  
> real avgIndEffect4\_2;  
> real totalEffect;  
> naiveIndEffect1=beta1aq \* beta3adt;  
> avgIndEffect1\_1=beta1aq \* beta3adt+DC1[4,10];  
> avgIndEffect1\_2=beta1aq \* beta3adt+DC2[4,10];  
> naiveIndEffect2=beta1cr \* beta4adt;  
> avgIndEffect2\_1=beta1cr \* beta4adt+DC1[5,11];  
> avgIndEffect2\_2=beta1cr \* beta4adt+DC2[5,11];  
> naiveIndEffect3=beta2aq \* beta3adt;  
> avgIndEffect3\_1=beta2aq \* beta3adt+DC1[7,10];  
> avgIndEffect3\_2=beta2aq \* beta3adt+DC2[7,10];  
> naiveIndEffect4=beta2cr \* beta4adt;  
> avgIndEffect4\_1=beta2cr \* beta4adt+DC1[8,11];  
> avgIndEffect4\_2=beta2cr \* beta4adt+DC2[8,11];  
> totalEffect=naiveIndEffect1+naiveIndEffect2+naiveIndEffect3+naiveIndEffect4+beta1adt  
> +beta2adt;  
> }  
> "

---

<div class="post-metadata">

**Author:** ![Bob\_Carpenter](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/bob_carpenter/32/9230_2.png) [@Bob\_Carpenter](https://discourse.mc-stan.org/u/Bob_Carpenter)\
**Post date:** [February 12, 2024, 11:23pm UTC](https://discourse.mc-stan.org/t/the-issue-of-adding-random-slopes-in-non-central-parameterization/34034/3 "2024-02-12T23:23:05Z")

</div>

Hi, @zrn. Did you sort out your issue? It looks like you marked your reply as an answer and then deleted it.

If by significant, you mean having a posterior interval bounded away from zero, then the question is why you would expect the coefficients to be significant in this way? My suggestion would be to simulate data and see if your model recovers it.

I’d also strongly recommend putting your Stan programs in separate files with indentation. That way, line number based errors are easy to resolve.

You can define `mu_aq` in a one-linear using vectorization.

The `beta1adt` etc. have centered parameterizations which may prove difficult to fit. You can switch to non-centered by changing the declaration to:

```stan
real<multiplier=sigma_beta1adt> beta1adt;

```

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<div class="post-metadata">

**Author:** ![zrn](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/zrn/32/17777_2.png) [@zrn](https://discourse.mc-stan.org/u/zrn)\
**Post date:** [February 13, 2024, 5:41am UTC](https://discourse.mc-stan.org/t/the-issue-of-adding-random-slopes-in-non-central-parameterization/34034/4 "2024-02-13T05:41:13Z")

</div>

@ [Bob\_Carpenter](https://discourse.mc-stan.org/u/Bob_Carpenter) Unfortunately, the problem has not been solved. Thank you for your reply! I put the code for the centered parameterized and non-centered parameterized versions in [Why The effects unexpectedly became smaller after employing non-central parameterization？](https://discourse.mc-stan.org/t/why-the-effects-unexpectedly-became-smaller-after-employing-non-central-parameterization/34082). I am puzzled why the significance of the effect is so different in the two versions of the model.  
I really need help：（

---

<div class="post-metadata">

**Author:** ![zrn](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/zrn/32/17777_2.png) [@zrn](https://discourse.mc-stan.org/u/zrn)\
**Post date:** [February 13, 2024, 5:29pm UTC](https://discourse.mc-stan.org/t/the-issue-of-adding-random-slopes-in-non-central-parameterization/34034/5 "2024-02-13T17:29:00Z")

</div>

I removed

> vector[11] mu1;  
> vector[11] mu2;  
> mu1 ~ normal(0, 2);  
> mu2 ~ normal(0, 2);

And changed

> matrix[J,11] gamma1 = rep\_matrix(mu1’, J) + gamma1\_raw \* diag\_pre\_multiply(sigma\_ranef1, Omega\_chol1) ; // random effects for id  
> matrix[G,11] gamma2 = rep\_matrix(mu2’, G) + gamma2\_raw \* diag\_pre\_multiply(sigma\_ranef2, Omega\_chol2)';

to

> matrix[J, 11] gamma1 = gamma1\_raw \* diag\_pre\_multiply(sigma\_ranef1, Omega\_chol1)‘;  
> matrix[G, 11] gamma2 = gamma2\_raw \* diag\_pre\_multiply(sigma\_ranef2, Omega\_chol2)’;

And I obtained mostly significant effects (similar to the case with only random intercept version). Does this imply that mu has an impact on gamma and I should delete it?  
result with mu：

 ![result](https://canada1.discourse-cdn.com/flex030/uploads/mc_stan/original/3X/c/9/c92563b285670657e0cfd4c88570719ef15359ef.png)  
result without mu：  
 ![WechatIMG571](https://canada1.discourse-cdn.com/flex030/uploads/mc_stan/original/3X/0/c/0c8f6e23629c46ca3ee0f79433fa2a2417c70a9d.png)