Generalized Linear Models and Beyond
Generalized linear models
Three-part structure
A GLM specifies a random component from the exponential family, a linear predictor , and a link function , where .
Canonical links
Common examples include logistic regression for binary data, Poisson regression for counts, and Gamma regression for positive skewed outcomes. The canonical link often simplifies estimation and interpretation.
Interpretation
In logistic regression, coefficients act on log-odds: .
Model fitting and deviance
Iteratively reweighted least squares
GLMs are commonly fit by iteratively reweighted least squares, which performs Newton-like optimization on the log-likelihood. Model comparison often uses deviance, likelihood ratio tests, AIC, and cross-validation.
Which model is most appropriate for binary outcomes with a logit link?
Logistic regression models binary probabilities through the logit link.
Correct answer: Logistic regression
What is the deviance in a GLM used for?
Deviance generalizes residual sum of squares to non-Gaussian settings.
Correct answer: It measures lack of fit relative to a saturated model and supports inference and model comparison.
Beyond standard GLMs
Extensions
Generalized additive models replace linear effects with smooth functions. Mixed effects GLMs handle clustered data. Quasi-likelihood and estimating equations relax full distributional assumptions.
Useful extensions
Which framework is especially common for correlated longitudinal data when robust marginal inference is desired?
Generalized estimating equations are designed for correlated clustered or longitudinal responses.
Correct answer: GEE
Why are mixed models and GEEs not interchangeable?
They answer different scientific questions and rely on different assumptions.
Correct answer: Mixed models target conditional subject-specific effects, while GEEs target population-averaged marginal effects.