11.4 Summary

Chapter 11 in review

Chapter 11 extended GLMs to ordered, zero-inflated, and overdispersed outcomes. These models are mixtures: they combine two processes in a single likelihood.

We covered:

Ordered categorical (Section 11.1): Cumulative link models for Likert-scale data. A latent continuous variable is cut into categories by estimated thresholds. Use MASS::polr().
Zero-inflated outcomes (Section 11.2): Mixture of structural zeros and a count process. ZIP allows zeros from two sources; hurdle models treat all zeros as one process.
Over-dispersed outcomes (Section 11.3): Beta-binomial and gamma-Poisson add dispersion parameters. Overdispersion = unobserved heterogeneity. These are continuous mixtures, connecting directly to multilevel models.

Key formulas

Cumulative link: $\log\frac{\Pr(y \leq k)}{1 - \Pr(y \leq k)} = \alpha_k - \phi_i$
ZIP: $P(y=0) = \pi + (1-\pi)e^{-\lambda}$
Beta-binomial variance: $\text{Var}(y) = n\bar{p}(1-\bar{p})\left(1 + \frac{n-1}{\theta+1}\right)$
Gamma-Poisson variance: $\text{Var}(y) = \mu + \mu^2/\phi$

Key lessons

Real data often violate simple distributional assumptions.
Mixture models provide a principled way to handle these violations.
The choice of likelihood is a modeling decision, not a default.
Overdispersion = unobserved heterogeneity — the dispersion parameter measures it.
Compare likelihoods with WAIC or posterior predictive checks.

Rethinking

Monsters are mixtures. The "monsters" in this chapter — ordered categories, zero-inflation, overdispersion — are all mixtures. They combine two processes in a single likelihood. This is a powerful idea: when a simple distribution does not fit, do not just add parameters to the linear predictor. Instead, ask what process could generate the data, and model that process. The cumulative link models a latent variable cut into categories. The ZIP models two kinds of zeros. The beta-binomial models varying probabilities. Each mixture tells a story about how the data came to be. This is the essence of statistical modeling: the likelihood is the story, and the parameters are the characters.

Over-dispersed outcomes

Practice