GLM

Generalized linear model (GLM) is a framework for regression that extends ordinary linear regression to response variables whose distributions come from the exponential family and that may have non-constant variance. A GLM consists of three components: a random component specifying the distribution of the response variable, a systematic component given by a linear predictor, and a link function that connects the mean of the distribution to the linear predictor.

The random component assumes Y given predictors X follows a member of the exponential family, such as normal, binomial, or Poisson. The systematic component is η = Xβ, a linear combination of the predictors. The link function g relates the mean μ = E[Y|X] to the linear predictor via g(μ) = η. This structure allows the model to handle different types of data by selecting appropriate distributions and links (for example, identity link with normal errors reduces to linear regression; logit or probit links with binomial errors yield logistic regression; log link with Poisson errors yields Poisson regression).

Estimation is typically performed by maximum likelihood, often using iteratively reweighted least squares to solve the estimating equations. Model assessment includes checking goodness-of-fit, examining residuals, and evaluating the suitability of the chosen distribution and link function. GLMs are widely used for binary, count, and non-normally distributed continuous outcomes, with extensions such as quasi-likelihood methods for overdispersion and generalized additive models for nonlinearity. The framework was introduced by Nelder and Wedderburn in 1972 and has since become a foundational tool in statistics and data analysis.