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 { intN; //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 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 [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 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, 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 { intN; //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]]); }