Convergence Error (Multilevel Model)

Modeling
1

Here is mu multilevel model:

data {
  int<lower=1> N;            //number of observations
  real M[N];                //reaction times
  real X[N];   //predictor (days of sleep deprivation)

  // grouping factor
  int<lower=1> J;                   //number of subjects
  int<lower=1,upper=J> Subject[N];  //subject id
}

parameters {
  vector[2] beta;                   // fixed-effects parameters
  real<lower=0> sigma_e;            // residual std
  vector<lower=0>[2] sigma_u;       // random effects standard deviations

  //declare L_u to be the Choleski factor of a 2x2 correlation matrix
  cholesky_factor_corr[2] L_u;

  matrix[2,J] z_u;                  // random effect matrix
}

transformed parameters {
  // this transform random effects so that they have the correlation
  // matrix specified by the correlation matrix above
   matrix[2,J] u;
   u = diag_pre_multiply(sigma_u, L_u) * z_u;

}

model {
  real mu; // conditional mean of the dependent variable

  //priors
  L_u ~ lkj_corr_cholesky(2); // LKJ prior for the correlation matrix
  to_vector(z_u) ~ normal(0,5);
  sigma_e ~ normal(0, 5);       // prior for residual standard deviation
  beta[1] ~ normal(0, 0.5);   // prior for fixed-effect intercept
  beta[2] ~ normal(0, 2);     // prior for fixed-effect slope

  //likelihood
  for (i in 1:N){
    mu = beta[1] + u[1,Subject[i]] + beta[2]*X[i] + u[2,Subject[i]]*X[i];
    M[i] ~ normal(mu, sigma_e);
  }
}

This is my stan model. However, when I try to run it using my data, it does not work.
Warning messages:
1: The largest R-hat is NA, indicating chains have not mixed.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#r-hat 
2: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-ess 
3: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-ess [sim.csv|attachment]

Here the 'data' in  R:
data=list(N = nrow(MyData),
          X = MyData$x,
          M = MyData$m,
          J = length(unique(MyData$id)),
          Subject = as.numeric(factor(MyData$id, 
          labels = 1:length(unique(MyData$id)))))

Here is the model in R:
fit = stan(file = 'mediationHM.stan', data=data, iter=4000, chains =4).

What am I doing wrong?
(upload://nMC1lZfQXULLasNi8i4Js20MEl4.csv) (49.9 KB)
2

The issue is that you haven’t quite got the non-centered parameterisation set up correctly.

Instead of giving z_u a normal(0,5) prior, you need to give it a normal(0,1) prior:

to_vector(z_u) ~ std_normal();
3

Thanks, however, it still does not work. The problem seems to stem from the Cholesky Factorization as only the L_um and L_uy parameters give to give Rhat values of NA.

4

Can you try putting a prior on sigma_u and seeing if that changes anything?

5

Nope. Bu the problem really seems to come from L_u. Maybe I should not use the Cholesky factorization?!