I am looking for advice on how to specify prior distributions in a two-parameter logistic item response theory model with predictor variables. In setting up my models, I follow @paul.buerkner’s Bayesian Item Response Modeling in R with brms and Stan and constrain the respondent latent variable (“ability”) \theta_j to follow a standard normal distribution.

Once I add respondent-level predictor variables, this strategy now constrains the *residual* standard deviation of the respondent latent variable to be 1. Instead, I would like the *total* variance of the respondent component to be 1 so that the item characteristics (\alpha_i and \beta_i) are comparable in magnitude across the models with and without predictors.

So far, I have found a way to do this by z-standardizing \theta_j in the `transformed parameters`

block.

```
data {
int<lower = 1> N; // # of responses
int<lower = 1> M; // # of predictors
int<lower = 1> I; // # of actions
int<lower = 1> J; // # of participants
array[N] int<lower = 1, upper = I> ii;
array[N] int<lower = 1, upper = J> jj;
array[N] int<lower = 0, upper = 1> y;
matrix[J, M] X; // matrix of (z-standardized) predictors
}
parameters {
vector[J] z_J;
matrix[2, I] z_I;
vector<lower = 0>[2] sigma_I;
cholesky_factor_corr[2] L_I;
real mu_alpha;
real mu_beta;
vector[M] b_x;
}
transformed parameters {
matrix[I, 2] r_I = transpose(diag_pre_multiply(sigma_I, L_I) * z_I);
vector<lower = 0>[I] alpha = exp(mu_alpha + r_I[, 1]);
vector[I] beta = mu_beta + r_I[, 2];
real sigma_J = 1.0;
vector[J] theta = X * b_x + sigma_J * z_J;
vector[J] theta_z = (theta - mean(theta))/sd(theta);
}
model {
z_J ~ std_normal();
to_vector(z_I) ~ std_normal();
to_vector(sigma_I) ~ student_t(3, 0, 2.5);
L_I ~ lkj_corr_cholesky(2);
mu_beta ~ student_t(3, 0, 2.5);
mu_alpha ~ std_normal();
to_vector(b_x) ~ student_t(3, 0, 2.5);
y ~ bernoulli_logit(alpha[ii] .* (theta_z[jj] + beta[ii]));
}
generated quantities {
corr_matrix[2] R_I = multiply_lower_tri_self_transpose(L_I);
}
```

- Is there a more elegant way of doing this that allows me to directly assign priors to the regression coefficients so that \theta_j = \text{X} b + b_j \sim \mathcal{N}(0, 1)?
^{[1]} - Any ideas on how to implement a regularized horseshoe prior or similar regularizing prior for the regression coefficients?
- Is there a way to implement this model in
`brms`

?

Thank you in advance!

Here is an almost equivalent model specified in `brms`

.

```
brm(
formula = bf(
y ~ exp(logalpha) * eta,
eta ~ 1 + x1 + ... + (1|i|ii) + (1|jj),
logalpha ~ 1 + (1|i|ii),
nl = TRUE
),
family = brmsfamily(family = "bernoulli", link = "logit"),
prior = c(
prior(student_t(3, 0, 2.5), class = b, nlpar = eta),
prior(lkj_corr_cholesky(2), class = L),
prior(constant(1), class = sd, group = jj, nlpar = eta),
prior(student_t(3, 0, 2.5), class = sd, group = ii, nlpar = eta),
prior(normal(0, 1), class = b, nlpar = logalpha),
prior(student_t(3, 0, 2.5), class = sd, group = ii, nlpar = logalpha)
),
data = d_sim
)
```

In other words, \mathrm{Var}[\text{X} b] + \sigma_j^2 = 1 if I am not mistaken. ↩︎