I asked a question the other day but included the wrong information. Below is my stan code, sorry about all the comments but those are things i have tried. I cannot seem to disentangle the effects of epsilon and beta_coefficients. For a description of what the parameters are, see the data block comments. Would anyone have any suggestions? This particular model assumes that epsilon is essentially the error of the item type effect, and thus each item is centered around its respective item type effect.

There is also another version where epsilon is set as \epsilon ~ N(0, \sigma_e^2 ) as to treat epsilon as more of random item effect (most likely due to something like the actual content of the item). Here are the descriptions of the differences if these are of interest:

When \epsilon_i\ ~ \ N(0,{\ \sigma}_e^2) , this stresses a random error interpretation of the random item effect. Items have a structural part \sum_{k\ =\ 0}^{K}\beta_kX_{ik}, and an item-specific deviation part, \epsilon_i. {\ \sigma}_e^2 refers to the residual variance in the regression of the \beta_i on the item predictors X_{ik}. The smaller the residual variance in comparison with the total variance of the \beta_i, the better explanatory power of the item predictors. Simply put, if the residual variance is large, that indicates the item predictor effects do not do a good job at explaining the item difficulty. If the residual variance was small, then the error would be close to 0, indicating that the added effect of the error is inconsequential.

The other interpretation is that the item parameters \beta_i are randomly sampled. We can see that in \beta_i\ ~\ \ N(\beta_i\prime,\ \ \sigma_e^2) in which our individual difficulty parameters are randomly sampled. This means that each item population is characterized by an expected difficulty (being centered at \beta_i\prime), and a within-population variance \sigma_e^2.

I have tried both of these approaches but have the same indeterminacy problems in each.

```
data {
int<lower=0> n_examinee; // Number of examinees
int<lower=0> n_item; // Number of items
int<lower=0> Y[n_examinee, n_item]; //The data matrix (count)
int<lower=1> K; // // number of item types, could be 3 or 4.
matrix[n_item, K] X; //dummy matrix of covariates, includes a column of 1s for intercept that replaces 1 of the covariates
}
parameters {
// real<lower=0> sd_theta;
// vector[n_examinee] theta_raw;
vector[K] beta_coefficients;
vector[n_item] epsilon;
vector[n_examinee] theta;
// real<lower=0> sigma2theta;
// real<lower=0> sigma2ep;
// vector[n_item] epsilon_raw;
// real<lower=0> sd_epsilon;
}
transformed parameters {
// vector[n_examinee] theta;
// theta = theta_raw * sd_theta;
// vector[n_item] epsilon;
// epsilon = epsilon_raw * sd_epsilon;
}
model {
theta ~ normal(0,1)
beta_coefficients ~ normal(0,1);
// theta_raw ~ normal(0,1);
// sd_theta ~ lognormal(0,1);
// epsilon_raw ~ normal(0,1);
// sd_epsilon ~ lognormal(0,1);
// sigma2theta ~ scaled_inv_chi_square(1, 1);
// sigma2ep ~ scaled_inv_chi_square(1, 1);
//epsilon ~ normal(0, 1);
matrix [n_examinee, n_item] lambdas;
for (i in 1:n_item) {
epsilon ~ normal((dot_product(X[i], beta_coefficients)), 10);
lambdas[,i] = exp(theta - (dot_product(X[i], beta_coefficients)) + epsilon[i]);
target += poisson_lpmf(Y[,i] | lambdas[,i]);
}
}
```