# Reproducing brms results in rstan yields different estimates

• Operating System: OSX Catalina
• brms Version: 2.13

I’m trying to understand how brms implements hierarchical models by writing similar models in Stan. From the documentation, `prior_summary`, and the generated stan code from brms, I understand that the following brms model…

``````library(brms)
library(tidyverse)
baseline_prior<-set_prior('normal(0.61, 0.12)', class = 'Intercept')
variant_prior<-set_prior('normal(0,1)', class = 'b', coef = 'variantB')

brms_model = brm(y|trials(n) ~ variant + (1|member_id),
data = ab_data,
prior = baseline_prior + variant_prior,
family  = binomial())
``````

should be equivalent to the following Stan model

``````stan_code="
data{
int N;
int n[N];
int y[N];
vector[N] variant;
}
parameters{
real Intercept;
real variantB;
vector[N] z;
real<lower=0> sigma;
}
model{
Intercept~normal(0.61, 0.12);
variantB~normal(0,1);
z~normal(0,1);
sigma~student_t(3,0,2.5);
y~binomial_logit(n,Intercept + z*sigma + variantB*variant);
}
"
``````

However, running these two models yields different results for the coefficient of `variant` (code to run models found here

``````# My Stan Code

data = list(N = nrow(ab_data),
y = ab_data\$y,
n = ab_data\$n,
variant = if_else(ab_data\$variant=='A',0,1))

model = stan_model(model_code = stan_code)

fit=sampling(model, data = data, pars = c('Intercept','variantB'))

mean se_mean    sd     2.5%      25%      50%      75%    97.5% n_eff Rhat
Intercept     0.75    0.00  0.08     0.60     0.70     0.75     0.80     0.90  2376    1
variantB      0.24    0.00  0.13    -0.02     0.15     0.24     0.32     0.50  2124    1
lp__      -1735.62    1.97 46.89 -1826.75 -1767.32 -1736.25 -1704.76 -1643.39   567    1

# brms model

brms_model\$fit

mean se_mean    sd     2.5%      25%      50%      75%    97.5% n_eff Rhat
b_Intercept                       0.77    0.00  0.09     0.59     0.70     0.77     0.83     0.96  2131 1.00
b_variantB                        0.13    0.00  0.14    -0.14     0.04     0.13     0.23     0.41  2386 1.00
``````

It is entirely possible that I have a) misunderstood the documentation for brms default priors/`prior_summary`, and/or b) have not properly coded up the model I want in Stan. Which of these happens to be the case?

I think this is because brms mean-centers the input data prior to analysis, and so the prior on the intercept is then a prior on the centered intercept. In contrast, your Stan code is then a prior on the non-centered intercept, so you’re implying two different models.

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Just to add - you can use `make_standata` from `brms` to see the data that is actually passed to Stan (and hunt for differences to what you pass).

But yes, this is a good approach to get better understanding of what is going on under the hood, best of luck!

Brms implements mean centering in the Stan model.

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So the binary indicator is mean centered as well?

My understanding is that brms uses model.matrix and other functions to generate the design matrix for fixed effects X.
This design matrix is part of the standata list submitted to the Stan model. In the Stan model each column in X will be mean centered.

That should be easy to see in the transformed data block in the Stan model. I recommend that you check a Stan model to verify that my recollection is correct.

After inspecting the Stan code, it does seem that brms centers the covariates.

Yes, that’s what I thought. If you want to compare brms results with e.g. rstanarm or other approaches, you can always center your covariates in the data that is submitted to brms and e.g. rstanarm. brms will still center, but it won’t have an effect.

It’s a bit harder with ordinal or nominal covariates: Then you’d have to first generate a design matrix with model.matrix,.then center all columns, and finally use the centered design matrix as data.

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