## Monday, March 27, 2017

### 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 $\pm 1$. 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 N;               //no trials
int P;               //no fixefs
int J;               //no subjects
int n_u;             //no subj ranefs
int K;               //no items
int n_w;             //no item ranefs
int subj[N]; //subject indicator
int 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
}

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[n_u] sigma_u; //subj ranef std
vector[n_w] sigma_w; //item ranef std
real 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
beta ~ cauchy(0,2.5);
sigma_e ~ 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)
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","beta","beta","beta"))

For more details, such as interpreting the results and computing things like Bayes Factors, see Nicenboim 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 N;               //no trials
int P;               //no fixefs
int J;               //no subjects
int n_u;             //no subj ranefs
int K;               //no items
int n_w;             //no item ranefs
int subj[N]; //subject indicator
int 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[n_u] sigma_u; //subj ranef std
vector[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]]);
}


See here.