What are generalized linear models?
Generalized linear models (GLMs) are a family of statistical models that keep the weighted-sum structure of linear regression but allow the outcome to follow distributions other than the bell curve. A link function connects the weighted sum to the outcome’s expected value, and a distribution describes how the outcome scatters around it. Linear regression is the GLM with an identity link and normal errors. Logistic regression is the GLM with a logit link and a binomial outcome. Poisson regression models counts, and gamma regression models positive, skewed amounts.
The appeal is that the coefficients stay interpretable while the model respects the nature of the data. A count cannot be negative and an amount is rarely symmetric, and a GLM built for them produces sensible predictions where plain linear regression does not.
Generalized linear models in practice
A commercial auto insurer prices policies with two GLMs. A Poisson model predicts the number of claims a vehicle will generate in a year from its class, radius of operation, driver history, and territory. A gamma model predicts the average cost of a claim given that one occurs. Multiplying the two gives the expected loss that the premium must cover. Regulators in many states expect to see exactly this structure in a rate filing, with each coefficient documented. Hospitals use Poisson and negative binomial GLMs in the same way to model readmission counts and length of stay.