I want to make a bayesian multilevel for data with a panel structure. Therefore, I use the brms package. I started by making a standard multilevel model where only the intercept is group dependent and just one explanatory variable using the following command:

Everything works perfectly, but I have a question regarding the interpretation of the results. When I look at the results using the summary() function, I only see the standard deviation of the group-level intercept. I expected to also see an estimate for the mean of this group-level intercept, but I don’t know why this is not displayed. Can someone explain how to interpret this result and why I don’t get to see the mean estimate? I am completely new to stan and brms, so please ask for further clarification if my question is not clear. Many thanks in advance!

Ah oke, so the estimate of the intercept under “Population-level effects” is the estimate of the mean of the posterior distribution of the intercept and under “Group-Level effects” you can find the estimated standard deviation of this distribution?

Oke thank for you answers! So, if I am correct with my example you estimate \alpha_{i} \sim N(\gamma, \sigma), where the posterior distribution of \gamma is summarized in “Population-level effects” and the posterior distribution for \sigma is summarized in “Group-Level effects”? Here the normal distribution is just an example

I still have on more question about the interpretation of the results. When I look at the sampling results using as.data.frame(result_brms) I get, among others, columns for b_intercept, sd_ID_intercept and many columns with draws for each ID-specific \alpha_{i}. However, I do not know how to get the b_intercept and sd_ID_intercept from the individual-specific draws of the intercept. Could you maybe explain that? I would think that the b_intercept and sd_ID_intercept are the rowwise mean and standard deviation of the individual-specific intercepts, but that is not true.

The individual alphas are centered around zero, not around b_Intercept. To extract the alphas + b_Intercept use the coef() method. The SD of the alphas will be close to sd_ID_intercept but not identical due to sampling error.

Ah great thanks! But why there are separate b_Intercept and \alpha_{i} draws? And why is the mean of the individual-specific a combination of the two and the standard deviation just the sd of the \alpha_{i}'s?

Ah oke, that’s clear. My final question is how to get from the b_Intercept and \alpha_{i}'s to the real estimated individual-specific intercepts when using the coef() function?

Yes, I know, but do you maybe know the math behind it? Do you maybe also know a scientific paper where the non-centered parameterization is based on in the brms package?

It’s simply that the “random effects” (denoted u in the paper) are centered around zero and that the corresponding mean parameters are part of the regression coefficients vector \beta. This works because

Ah thanks! That clarifies a lot! Incredibly useful to have a discussion forum like this!

Am I right that they talk about the non-centered parameterization in this paper: Betancourt, M., & Girolami, M. (2015). Hamiltonian Monte Carlo for hierarchical models. Current trends in Bayesian methodology with applications , 79 , 30.