hierarchical-models
Patterns for hierarchical/multilevel Bayesian models including random effects, partial pooling, and centered vs non-centered parameterizations.
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- hierarchical-models
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- 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<lower=0> N; // Total observations
int<lower=0> J; // Number of groups
array[N] int<lower=1,upper=J> group;
vector[N] y;
}
parameters {
real mu; // Population mean
real<lower=0> tau; // Between-group SD
real<lower=0> 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<lower=0> tau;
real<lower=0> 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<lower=0> J;
array[J] real y; // Observed effects
array[J] real<lower=0> sigma; // Known SEs
}
parameters {
real mu;
real<lower=0> 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
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