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Patterns for hierarchical/multilevel Bayesian models including random effects, partial pooling, and centered vs non-centered parameterizations.
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SOC 職業分類に基づく
Bayesian meta-analysis models including fixed effects, random effects, and network meta-analysis with Stan and JAGS implementations.
Foundational knowledge for writing current PyMC models including syntax, distributions, sampling, and ArviZ diagnostics. Use when creating or reviewing PyMC models.
Foundational knowledge for writing modern Stan models including program structure, type system, distributions, and best practices. Use when creating or reviewing Stan models.
Group sequential design methods for interim analyses, alpha spending, and futility stopping. Use when designing trials with interim looks or implementing spending functions.
Core Mediana package functions for Clinical Scenario Evaluation (CSE). Use when designing data models, analysis models, evaluation models, and running comprehensive trial simulations.
Core simtrial package functions for time-to-event clinical trial simulation. Use when generating survival data, performing weighted logrank tests, or running TTE simulations.
| name | hierarchical-models |
| description | Patterns for hierarchical/multilevel Bayesian models including random effects, partial pooling, and centered vs non-centered parameterizations. |
Group means shrink toward overall mean based on:
- Within-group sample size
- Within-group variance
- Between-group variance
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);
}
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.
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)
}
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);
}