Fitting Bayesian Linear Mixed Models for continuous and binary data using Stan: A quick tutorial

I want to give a quick tutorial on fitting Linear Mixed Models (hierarchical models) with a full variance-covariance matrix for random effects (what Barr et al 2013 call a maximal model) using Stan.

For a longer version of this tutorial, see: Sorensen, Hohenstein, Vasishth, 2016.

Prerequisites: You need to have R and preferably RStudio installed; RStudio is optional. You need to have rstan installed. See here. I am also assuming you have fit lmer models like these before:

lmer(log(rt) ~ 1+RCType+dist+int+(1+RCType+dist+int|subj) + (1+RCType+dist+int|item), dat)

If you don't know what the above code means, first read chapter 4 of my lecture notes.

The code and data format needed to fit LMMs in Stan

The data

I assume you have a 2x2 repeated measures design with some continuous measure like reading time (rt) data and want to do a main effects and interaction contrast coding. Let's say your main effects are RCType and dist, and the interaction is coded as int. All these contrast codings are ±1. If you don't know what contrast coding is, see these notes and read section 4.3 (although it's best to read the whole chapter). I am using an excerpt of an example data-set from Husain et al. 2014.

"subj" "item" "rt""RCType" "dist" "int"
1       14    438  -1        -1      1
1       16    531   1        -1     -1
1       15    422   1         1      1
1       18   1000  -1        -1      1
...

Assume that these data are stored in R as a data-frame with name rDat.

The Stan code

Copy the following Stan code into a text file and save it as the file matrixModel.stan. For continuous data like reading times or EEG, you never need to touch this file again. You will only ever specify the design matrix X and the structure of the data. The rest is all taken care of.

data {
  int<lower=0> N;               //no trials
  int<lower=1> P;               //no fixefs
  int<lower=0> J;               //no subjects
  int<lower=1> n_u;             //no subj ranefs
  int<lower=0> K;               //no items
  int<lower=1> n_w;             //no item ranefs
  int<lower=1,upper=j> subj[N]; //subject indicator
  int<lower=1,upper=k> item[N]; //item indicator
  row_vector[P] X[N];           //fixef design matrix
  row_vector[n_u] Z_u[N];       //subj ranef design matrix
  row_vector[n_w] Z_w[N];       //item ranef design matrix
  vector[N] rt;                 //reading time
}

parameters {
  vector[P] beta;               //fixef coefs
  cholesky_factor_corr[n_u] L_u;  //cholesky factor of subj ranef corr matrix
  cholesky_factor_corr[n_w] L_w;  //cholesky factor of item ranef corr matrix
  vector<lower=0>[n_u] sigma_u; //subj ranef std
  vector<lower=0>[n_w] sigma_w; //item ranef std
  real<lower=0> sigma_e;        //residual std
  vector[n_u] z_u[J];           //spherical subj ranef
  vector[n_w] z_w[K];           //spherical item ranef
}

transformed parameters {
  vector[n_u] u[J];             //subj ranefs
  vector[n_w] w[K];             //item ranefs
  {
    matrix[n_u,n_u] Sigma_u;    //subj ranef cov matrix
    matrix[n_w,n_w] Sigma_w;    //item ranef cov matrix
    Sigma_u = diag_pre_multiply(sigma_u,L_u);
    Sigma_w = diag_pre_multiply(sigma_w,L_w);
    for(j in 1:J)
      u[j] = Sigma_u * z_u[j];
    for(k in 1:K)
      w[k] = Sigma_w * z_w[k];
  }
}

model {
  //priors
  L_u ~ lkj_corr_cholesky(2.0);
  L_w ~ lkj_corr_cholesky(2.0);
  for (j in 1:J)
    z_u[j] ~ normal(0,1);
  for (k in 1:K)
    z_w[k] ~ normal(0,1);
  //likelihood
  for (i in 1:N)
    rt[i] ~ lognormal(X[i] * beta + Z_u[i] * u[subj[i]] + Z_w[i] * w[item[i]], sigma_e);
}

Define the design matrix

Since we want to test the main effects coded as the columns RCType, dist, and int, our design matrix will look like this:

# Make design matrix
X <- unname(model.matrix(~ 1 + RCType + dist + int, rDat))
attr(X, "assign") <- NULL

Prepare data for Stan

Stan expects the data in a list form, not as a data frame (unlike lmer). So we set it up as follows:

# Make Stan data
stanDat <- list(N = nrow(X),
P = ncol(X),
n_u = ncol(X),
n_w = ncol(X),
X = X,
Z_u = X,
Z_w = X,
J = nlevels(rDat$subj),
K = nlevels(rDat$item),
rt = rDat$rt,
subj = as.integer(rDat$subj),
item = as.integer(rDat$item))

Load library rstan and fit Stan model

library(rstan)
rstan_options(auto_write = TRUE)
options(mc.cores = parallel::detectCores())

# Fit the model
matrixFit <- stan(file = "matrixModel.stan", data = stanDat,
iter = 2000, chains = 4)

Examine posteriors

print(matrixFit)

This print output is overly verbose. I wrote a simple function to get the essential information quickly.

stan_results<-function(m,params=paramnames){
  m_extr<-extract(m,pars=paramnames)
  par_names<-names(m_extr)
  means<-lapply(m_extr,mean)
  quantiles<-lapply(m_extr,
                    function(x)quantile(x,probs=c(0.025,0.975)))
  means<-data.frame(means)
  quants<-data.frame(quantiles)
  summry<-t(rbind(means,quants))
  colnames(summry)<-c("mean","lower","upper")
  summry
}

For example, if I want to see only the posteriors of the four beta parameters, I can write:

stan_results(matrixFit, params=c("beta[1]","beta[2]","beta[3]","beta[4]"))

For more details, such as interpreting the results and computing things like Bayes Factors, seeNicenboim and Vasishth 2016.

FAQ: What if I don't want to fit a lognormal?

In the Stan code above, I assume a lognormal function for the reading times:

 rt[i] ~ lognormal(X[i] * beta + Z_u[i] * u[subj[i]] + Z_w[i] * w[item[i]], sigma_e);

If this upsets you deeply and you want to use a normal distribution (and in fact, for EEG data this makes sense), go right ahead and change the lognormal to normal:

 rt[i] ~ normal(X[i] * beta + Z_u[i] * u[subj[i]] + Z_w[i] * w[item[i]], sigma_e);

FAQ: What if I my dependent measure is binary (0,1) responses?

Use this Stan code instead of the one shown above. Here, I assume that you have a column called response in the data, which has 0,1 values. These are the trial level binary responses.

data {
  int<lower=0> N;               //no trials
  int<lower=1> P;               //no fixefs
  int<lower=0> J;               //no subjects
  int<lower=1> n_u;             //no subj ranefs
  int<lower=0> K;               //no items
  int<lower=1> n_w;             //no item ranefs
  int<lower=1,upper=j> subj[N]; //subject indicator
  int<lower=1,upper=k> item[N]; //item indicator
  row_vector[P] X[N];           //fixef design matrix
  row_vector[n_u] Z_u[N];       //subj ranef design matrix
  row_vector[n_w] Z_w[N];       //item ranef design matrix
  int response[N];                 //response
}

parameters {
  vector[P] beta;               //fixef coefs
  cholesky_factor_corr[n_u] L_u;  //cholesky factor of subj ranef corr matrix
  cholesky_factor_corr[n_w] L_w;  //cholesky factor of item ranef corr matrix
  vector<lower=0>[n_u] sigma_u; //subj ranef std
  vector<lower=0>[n_w] sigma_w; //item ranef std
  vector[n_u] z_u[J];           //spherical subj ranef
  vector[n_w] z_w[K];           //spherical item ranef
}

transformed parameters {
  vector[n_u] u[J];             //subj ranefs
  vector[n_w] w[K];             //item ranefs
  {
    matrix[n_u,n_u] Sigma_u;    //subj ranef cov matrix
    matrix[n_w,n_w] Sigma_w;    //item ranef cov matrix
    Sigma_u = diag_pre_multiply(sigma_u,L_u);
    Sigma_w = diag_pre_multiply(sigma_w,L_w);
    for(j in 1:J)
      u[j] = Sigma_u * z_u[j];
    for(k in 1:K)
      w[k] = Sigma_w * z_w[k];
  }
}

model {
  //priors
  beta ~ cauchy(0,2.5);
  sigma_u ~ cauchy(0,2.5);
  sigma_w ~ cauchy(0,2.5);
  L_u ~ lkj_corr_cholesky(2.0);
  L_w ~ lkj_corr_cholesky(2.0);
  for (j in 1:J)
    z_u[j] ~ normal(0,1);
  for (k in 1:K)
    z_w[k] ~ normal(0,1);
  //likelihood
  for (i in 1:N)
    response[i] ~ bernoulli_logit(X[i] * beta + Z_u[i] * u[subj[i]] + Z_w[i] * w[item[i]]);
}

For reproducible example code

See here.