Please share your Stan program and accompanying data if possible.

Hi everyone,

I’m new to working with stan and I am trying to fit this multivariate model. I saw one article online on how to analyse “Bayesian Varying Effects Models in R and Stan”. Bayesian Varying Effects Models in R and Stan - Will Hipson.

The data was simulated this way:

```
theme_set(theme_bw(base_size = 14)) # setting ggplot2 theme
library(mvtnorm)
set.seed(12407) # for reproducibility
# unstandardized means and sds
real_mean_y <- 4.25
real_mean_x <- 3.25
real_sd_y <- 0.45
real_sd_x <- 0.54
N <- 30 # number of people
n_days <- 7 # number of days
total_obs <- N * n_days
sigma <- 1 # population sd
beta <- c(0, 0.15) # average intercept and slope
sigma_p <- c(1, 1) # intercept and slope sds
rho <- -0.36 # covariance between intercepts and slopes
cov_mat <- matrix(c(sigma_p[1]^2, sigma_p[1] * sigma_p[2] * rho, sigma_p[1] * sigma_p[2] * rho, sigma_p[2]^2), nrow = 2)
beta_p <- rmvnorm(N, mean = beta, sigma = cov_mat) # participant intercepts and slopes
x <- matrix(c(rep(1, N * n_days), rnorm(N * n_days, 0, 1)), ncol = 2) # model matrix
pid <- rep(1:N, each = n_days) # participant id
sim_dat <- map_dfr(.x = 1:(N * n_days), ~data.frame(
mu = x[.x, 1] * beta_p[pid[.x], 1] + x[.x, 2] * beta_p[pid[.x], 2],
pid = pid[.x],
x = x[.x, 2]
))
sim_dat$y <- rnorm(210, sim_dat$mu, sigma) # creating observed y from mu and sigma
dat <- sim_dat %>%
select(-mu) %>% # removing mu
mutate(y = real_mean_y + (y * real_sd_y), # unstandardize
x = real_mean_x + (x * real_sd_x)) # unstandardize
stan_dat3 <- list(
N_obs = nrow(dat),
N_pts = max(as.numeric(dat$pid)),
K = 2, # intercept + slope
pid = as.numeric(dat$pid),
x = matrix(c(rep(1, nrow(dat)), (dat$x - mean(dat$x)) / sd(dat$x)), ncol = 2), # z-score for x
y = (dat$y - mean(dat$y)) / sd(dat$y) # z-score for y
)
```

and here is the stan model saved as mod_c.stan:

```
data {
int N_obs; // number of observations
int N_pts; // number of participants
int K; // number of predictors + intercept
int pid[N_obs]; // participant id vector
matrix[N_obs, K] x; // matrix of predictors
real y[N_obs]; // y vector
}
parameters {
matrix[K, N_pts] z_p; // matrix of intercepts and slope
vector<lower=0>[K] sigma_p; // sd for intercept and slope
vector[K] beta; // intercept and slope hyper-priors
cholesky_factor_corr[K] L_p; // Cholesky correlation matrix
real<lower=0> sigma; // population sigma
}
transformed parameters {
matrix[K, N_pts] z; // non-centered version of beta_p
z = diag_pre_multiply(sigma_p, L_p) * z_p;
}
model {
vector[N_obs] mu;
// priors
beta ~ normal(0, 1);
sigma ~ exponential(1);
sigma_p ~ exponential(1);
L_p ~ lkj_corr_cholesky(2);
to_vector(z_p) ~ normal(0, 1);
// likelihood
for(i in 1:N_obs) {
mu[i] = beta[1] + z[1, pid[i]] + (beta[2] + z[2, pid[i]]) * x[i, 2];
}
y ~ normal(mu, sigma);
}
generated quantities {
matrix[2, 2] Omega;
Omega = multiply_lower_tri_self_transpose(L_p);
}
```

I used this to run the model

```
stan_fit3 <- stan(file = "mod_c.stan",
data = stan_dat3,
chains = 4, cores = 4)
stan_fit3
```

and I got the same result.

Now here is my application: I divided the dataset into two parts (105 each)[this was done by plotting the dataset and using the plot to divide] and I used this model to run the first part (first data) (where my pid = rep(1:N/2, each = n_days=7) (I used N/2 =15 since I divided the data into two) ) and also run the second part. In the model, K=2, N_obs = nrow(dat)=105, N_pts = max(as.numeric(dat$pid))=15.

For the first part, I got my random effect correlation to be

\bigg(\begin{matrix}
1 & -0.40\\
-0.40& 1
\end{matrix}\bigg)

For the second part also, I got my random effect correlation to be

\bigg(\begin{matrix}
1 & -0.36\\
-0.36& 1
\end{matrix}\bigg)

Now using the result from the two dataset, I simulated another dataset by using correlation 0.6 and -0.6 to join the random effect correlation from the two dataset together and this gives a 4 by 4 matrix: Like this

\begin{pmatrix}
1 & 0.6&-0.40&-0.6\\
0.6& 1&-0.6&-0.36\\
-0.4& -0.6&1&0.6\\
-0.6& -0.36&0.6&1
\end{pmatrix}

In this case, my random effect follows MVNormal(c(\beta_0[1],\beta_0[2],\beta_1[1],\beta_1[2]), \Sigma), where \beta_0[1] and \beta_1[1] are from data 1, \beta_0[2] and \beta_1[2] are from data 2 and \Sigma is the sigma of the random effect after applying the correlation above.

Here is my stan code using the simulated dataset:

```
data {
int N_obs; // number of observations
int N_pts; // number of participants
int<lower=1>J;
int<lower=1,upper=J>dam[N_obs];
int K; // number of predictors + intercept
int pid[N_obs]; // participant id vector
matrix[N_obs, K] x; // matrix of predictors
real y[N_obs]; // y vector
}
parameters {
matrix[K, N_pts] z_p; // matrix of intercepts and slope
vector<lower=0>[K] sigma_p; // sd for intercept and slope
vector[K] mu1; // intercept and slope hyper-priors
cholesky_factor_corr[K] L_p; // Cholesky correlation matrix
vector[2] sigma; // population sigma
}
transformed parameters {
matrix[K, N_pts] z; // non-centered version of beta_p
z = diag_pre_multiply(sigma_p, L_p) * z_p;
}
model {
vector[2] beta1;
vector[2] beta2;
vector[N_obs] mu;
// priors
mu1~ normal(0, 1);
sigma ~ exponential(1);
sigma_p ~ exponential(1);
L_p ~ lkj_corr_cholesky(2);
to_vector(z_p) ~ normal(0, 1);
// likelihood
for(i in 1:N_obs) {
beta1[1] = mu1[1]+z[1,pid[i]];
beta1[2] = mu1[2]+z[2,pid[i]];
beta2[1] =mu1[3]+z[3,pid[i]];
beta2[2]=mu2[4]+z[4,pid[i]];
mu[i] = beta1[dam[i]]+ (beta2[dam[i]]) * x[i, 2];
}
y[i] ~ normal(mu, sigma[dam[i]]);
}
generated quantities {
matrix[4,4] Omega;
Omega = multiply_lower_tri_self_transpose(L_p);
}
```

where my Dam is a factor of (1,1,1,…,2,2,2) for “1” represent data 1 and “2” represent data 2, dam=as.integer(Dam) and J=2. Also pid = rep(1:N"=30", each =7) , K=4, N_obs = nrow(dat)=210, N_pts = max(as.numeric(dat$pid))=30.

Using my simulated dataset, I expect my code to capture the 0.6 and -0.6 correlation but I got 0.002 and -0.04 instead. Is there anything I’m not doing right? Please help.

Thank you