13.5 Summary

Chapter 13 in review

Chapter 13 extended multilevel models to varying slopes and Gaussian processes. The common thread is covariance: the model learns how parameters relate to each other, not just their average values.

We covered:

1. Varying slopes (Section 13.1): Allow the effect of a predictor to differ across groups. The model estimates a covariance matrix with variances and a correlation $\rho$ between intercepts and slopes. The LKJ prior regularizes the correlation.

2. Admission and gender (Section 13.2): Simpson's paradox — the aggregate effect can reverse within-group effects. Varying slopes by department reveal the true structure and show that variation across departments is as important as the average.

3. Cross-classified chimpanzees (Section 13.3): Multiple crossed random effects (actor, block, treatment) with varying slopes by actor. More structure does not mean overfitting — partial pooling controls complexity.

4. Gaussian processes (Section 13.4): Varying intercepts for continuous categories (space, time). The covariance function determines how similarity decays with distance. The length scale $\ell$ controls smoothness.

Key formulas

• Varying slopes: $(\gamma_j, \beta_j) \sim \text{MVNormal}(0, \Sigma)$ with $\Sigma = \begin{pmatrix} \sigma_\gamma^2 & \rho\sigma_\gamma\sigma_\beta \\ \rho\sigma_\gamma\sigma_\beta & \sigma_\beta^2 \end{pmatrix}$
• LKJ prior: $p(\mathbf{R}) \propto |\mathbf{R}|^{\eta-1}$
• Squared exponential kernel: $\text{Cov}(f(x), f(x')) = \sigma^2 \exp(-(x-x')^2 / (2\ell^2))$

Key lessons

• Varying slopes are pooled interactions — interactions with shrinkage.
• The correlation between intercepts and slopes is part of the model.
• Simpson's paradox is a pooling problem — varying slopes solve it.
• Gaussian processes are varying intercepts for continuous dimensions.
• Modeling covariance is as important as modeling means.