--- name: hierarchical-models description: Patterns for hierarchical\/multilevel Bayesian models including random effects, partial pooling, and centered vs non-centered parameterizations. --- # Hierarchical Models ## When to Use - Nested\/grouped data (students in schools, patients in hospitals) - Repeated measurements on subjects - Meta-analysis with study-level variation - Partial pooling between complete pooling and no pooling ## Core Concept: Partial Pooling ``` Group means shrink toward overall mean based on: - Within-group sample size - Within-group variance - Between-group variance ``` ## Stan Implementation ### Centered Parameterization (Default) ```stan data { int N; \/\/ Total observations int J; \/\/ Number of groups array[N] int group; vector[N] y; } parameters { real mu; \/\/ Population mean real tau; \/\/ Between-group SD real sigma; \/\/ Within-group SD vector[J] theta; \/\/ Group means } model { \/\/ Hyperpriors mu ~ normal(0, 10); tau ~ cauchy(0, 2.5); sigma ~ exponential(1); \/\/ Group effects theta ~ normal(mu, tau); \/\/ Likelihood y ~ normal(theta[group], sigma); } ``` ### Non-Centered Parameterization (Better for weak data\/small tau) ```stan parameters { real mu; real tau; real sigma; vector[J] theta_raw; \/\/ Standard normal } transformed parameters { vector[J] theta = mu + tau * theta_raw; } model { theta_raw ~ std_normal(); \/\/ ... rest same } ``` **When to use non-centered**: Divergences, small tau, few observations per group. ## JAGS Implementation ``` model { for (i in 1:N) { y[i] ~ dnorm(theta[group[i]], tau.y) } for (j in 1:J) { theta[j] ~ dnorm(mu, tau.theta) } # Hyperpriors mu ~ dnorm(0, 0.0001) tau.theta <- pow(sigma.theta, -2) sigma.theta ~ dunif(0, 100) tau.y <- pow(sigma.y, -2) sigma.y ~ dunif(0, 100) } ``` ## Classic Example: Eight Schools ```stan data { int J; array[J] real y; \/\/ Observed effects array[J] real sigma; \/\/ Known SEs } parameters { real mu; real tau; vector[J] theta_raw; } transformed parameters { vector[J] theta = mu + tau * theta_raw; } model { mu ~ normal(0, 5); tau ~ cauchy(0, 5); theta_raw ~ std_normal(); y ~ normal(theta, sigma); } ``` ## Diagnostics - Check tau posterior (very small → use non-centered) - Divergences often indicate centered\/non-centered mismatch - Compare to no-pooling and complete-pooling models