Generalised linear models
CS2 tests the GLM as a unified framework and your ability to specify the random component, link and linear predictor correctly.
Structure
A GLM has independent responses from an exponential-family distribution with E(Y)=μ and Var(Y)=φV(μ), a linear predictor η=xβ, and a link g(μ)=η. Canonical links include identity for normal, log for Poisson, logit for binomial and inverse for gamma. Parameters are estimated by maximum likelihood using iteratively reweighted least squares. Deviance is 2(l_sat−l_model); Pearson X² uses (y−μ)²/V(μ). Nested models are compared by analysis of deviance.
Overdispersion
For Poisson or binomial data, extra variation inflates the variance. Estimate φ by X²/(n−p) or deviance/(n−p), use quasi-likelihood, or fit a random-effects model. An offset enters as a fixed coefficient of 1, usually log(exposure) for rates; a weight is not the same thing.
Worked example
Poisson claim counts with exposure E_i use log μ_i=log E_i+β₀+β₁x_i. If β₁=0.4 for a policy-type indicator, the rate ratio is e^{0.4}=1.49: that type has about 49% more claims per unit exposure. For logistic regression with logit p=−1.2+0.8x, the odds ratio is e^{0.8}=2.23.
Traps
Do not interpret a coefficient as a change in the mean when the link is nonlinear. Do not use ordinary R² for GLMs. Omitting an offset turns a rate model into a count model. Ignoring overdispersion understates standard errors. Watch for separation in logistic models and for non-integer counts in Poisson models.