Bayesian Hierarchical Model AI Prompt

Specify and interpret a Bayesian hierarchical (multilevel) model for this data.

Data structure: {{data_structure}} (units nested in groups: students in schools, measurements in subjects)
Outcome: {{outcome}}
Level-1 predictors: {{l1_predictors}} (measured at the unit level)
Level-2 predictors: {{l2_predictors}} (measured at the group level)
Group variable: {{group_variable}}
Number of groups: {{n_groups}}

1. Why hierarchical models:
   - Observations within groups are not independent (ICC > 0)
   - Using a single-level model underestimates standard errors (overestimates precision)
   - Hierarchical models borrow strength across groups (partial pooling):
     - Complete pooling: ignores group membership (too simple)
     - No pooling: estimates each group separately (too noisy for small groups)
     - Partial pooling: shrinks group estimates toward the overall mean (optimal)

2. Model specification:
   Level 1 (within groups):
   y_ij = alpha_j + beta_j * x_ij + epsilon_ij
   epsilon_ij ~ Normal(0, sigma^2)

   Level 2 (between groups):
   alpha_j ~ Normal(mu_alpha, tau_alpha^2)  [random intercepts]
   beta_j ~ Normal(mu_beta, tau_beta^2)    [random slopes, if specified]

   Hyperpriors (Bayesian):
   mu_alpha ~ Normal(0, 10)
   tau_alpha ~ HalfNormal(0, 1)  [must be positive; HalfNormal is a good weakly informative prior]

3. Intraclass correlation coefficient (ICC):
   ICC = tau^2 / (tau^2 + sigma^2)
   - ICC = 0: no clustering; single-level model is fine
   - ICC = 0.10: 10% of variance is at the group level; standard errors inflated by DEFF = 1 + (m-1) x 0.10
   - ICC > 0.20: strong clustering; hierarchical model is essential

4. Cross-level interactions:
   - Does the effect of x_ij (Level 1) vary with w_j (Level 2)?
   - Include: beta_j = gamma_10 + gamma_11 * w_j + u_1j
   - This is the hallmark of hierarchical modeling: testing whether context moderates individual-level effects

5. Convergence diagnostics (for MCMC estimation):
   - R-hat (Gelman-Rubin): should be < 1.01 for all parameters
   - Effective sample size (ESS): should be > 100 (preferably > 400) for each parameter
   - Trace plots: chains should mix well, with no trends or stuck periods
   - Posterior predictive check: does the model reproduce the observed data distribution?

Return: hierarchical model specification, ICC calculation, cross-level interaction interpretation, and MCMC convergence assessment.