Hierarchical Linear Models

Summary

Chapter 15 of BDA3 extends Hierarchical Models to the regression setting. Coefficients vary across groups, partially pooled toward a common distribution — the Bayesian approach to mixed effects / multilevel models.

Model Structure

For group $j=1,\dots ,J$:

$$y_{ij}\mid {\alpha }_j,{\beta }_j,{\sigma }^2\sim N({\alpha }_j+x_{ij}{\beta }_j,{\sigma }^2)$$ $$\left(\begin{array}{c}{\alpha }_j\\ {\beta }_j\end{array}\right)\sim N\left(\left(\begin{array}{c}{\mu }_{\alpha }\\ {\mu }_{\beta }\end{array}\right),{\Sigma }_{\alpha \beta }\right)$$

Key Concepts

• Varying intercepts: ${\alpha }_j$ shifts baseline for each group (random intercepts)
• Varying slopes: ${\beta }_j$ allows different effects per group (random slopes)
• Partial pooling: groups with less data borrow more from the population — same principle as the eight schools example
• Computation: reparameterization (centered vs. non-centered) critical for HMC efficiency

Applications

• Forecasting elections: varying intercepts/slopes across states (U.S. presidential elections)
• ANOVA connection: analysis of variance as a special case of hierarchical regression
• Batching of variance components: hierarchical structure for modeling heterogeneous variances

See Also

• Bayesian Linear Regression — the non-hierarchical foundation
• Hierarchical Models — the general theory (Ch 5)
• Generalized Linear Models — hierarchical GLMs
• Differences-in-Differences — frequentist fixed effects approach; HLM is the Bayesian alternative for panel data
• Standard Errors and Clustering — clustering as a frequentist approach to the same grouped-data structure
• Nonparametric Models Overview — GP and mixture model extensions when parametric hierarchical structure is insufficient
• Global-Local Shrinkage Priors — shrinkage-prior view of the partial pooling used here