10.4 Summary

Chapter 10 in review

Chapter 10 applied GLMs to count data — binary outcomes, count outcomes, and the extensions needed when real data violate the standard assumptions.

We covered:

1. Binomial regression (Section 10.1): Logit link for binary and aggregated binomial outcomes. Coefficients are log-odds; interpret on the probability scale.

2. Poisson regression (Section 10.2): Log link for count outcomes. Use offsets for varying exposure. Always check for overdispersion (variance > mean).

3. Other count regressions (Section 10.3): Negative binomial for overdispersion, zero-inflated Poisson for excess zeros, hurdle models for two-stage processes. Choose based on the data-generating process.

Key formulas

• Logit link: $\log\frac{p_i}{1-p_i} = \alpha + \beta x_i$
• Log link: $\log(\lambda_i) = \alpha + \beta x_i$
• Offset: $\log(\lambda_i) = \log(\text{exposure}_i) + \alpha + \beta x_i$
• Negative binomial variance: $\text{Var}(y_i) = \lambda_i + \lambda_i^2 / \phi$
• ZIP: $P(y=0) = \pi + (1-\pi)e^{-\lambda}$

Key lessons

• The likelihood must match the data-generating process, or the model will mislead.
• Overdispersion is the rule, not the exception — always check.
• Interpret coefficients on the natural scale (probabilities, counts), not the link scale.
• Use posterior predictive checks to verify the model reproduces key features.
• Choose among Poisson, NB, ZIP, and hurdle based on theory, not just fit.