12.5 Summary

Chapter 12 in review

Chapter 12 introduced multilevel models — models with parameters within parameters. Group-level parameters are drawn from a population distribution, creating partial pooling that improves estimates and handles clustered data naturally.

We covered:

1. Multilevel tadpoles (Section 12.1): Tank-level survival with a common distribution. Partial pooling shares information across tanks, especially helping small tanks.

2. Varying effects trade-off (Section 12.2): Complete pooling underfits, no pooling overfits, partial pooling balances. Shrinkage is adaptive — more for small groups, less for large ones.

3. Multiple clusters (Section 12.3): Crossed effects (actors × blocks) and nested effects (students within schools within districts). The model structure must reflect the data-generating process.

4. Posterior predictions (Section 12.4): Three types — existing groups, new groups, population average. New-group predictions have wider uncertainty because they include between-group variation.

Key formulas

• Multilevel model: $\text{logit}(p_j) = \alpha + \gamma_j$, $\gamma_j \sim \text{Normal}(0, \sigma_\gamma)$
• Partial pooling: $\hat{\gamma}_j = \frac{\sigma_\gamma^2}{\sigma_\gamma^2 + \sigma_y^2/n_j} \bar{y}_j + \frac{\sigma_y^2/n_j}{\sigma_\gamma^2 + \sigma_y^2/n_j} \bar{y}$
• Crossed effects: (1 | actor) + (1 | block)
• Nested effects: (1 | district) + (1 | district:school)

Key lessons

• Multilevel models add structure to the parameters themselves.
• Partial pooling = adaptive regularization — shrinkage learned from data.
• The model structure must reflect the data-generating process.
• New-group predictions have more uncertainty than existing-group predictions.
• Multilevel models are the default for clustered, nested, or crossed data.