Generalized Linear Models

A GLM has three connected pieces. The random component chooses a response distribution from an exponential-family model, such as Gaussian, binomial, Poisson or Gamma. The systematic component forms a linear predictor from the inputs, often written eta = X beta. The link function connects the expected response mu to that predictor through g(mu) = eta. The model remains linear in its coefficients on the link scale, even when the expected outcome itself is not a straight line in the predictors. With a Gaussian distribution and identity link, the conditional mean is the linear predictor, recovering the usual linear regression form. A binomial model with the logit link models log odds; applying the inverse logit gives a probability. A Poisson model with the log link models the logarithm of the expected count, so its predicted mean is positive. These links encode constraints and relationships suitable to different outcome types. For a hypothetical binary outcome, suppose a model's linear predictor is zero. The inverse logit maps zero to probability 0.5. If the linear predictor rises, the probability rises but remains below one. By contrast, an identity-link model could predict a binary response below zero or above one, which is why an ordinary Gaussian linear model may not suit that task. A GLM does not guarantee the selected family is correct; it makes the assumptions explicit so they can be assessed. The family determines a mean-variance relationship as well as support for the response. Poisson models commonly assume conditional variance equals the conditional mean, a condition that real count data may violate. A binomial model must reflect how trials and successes are represented. Link choices may be constrained by family and software implementation. Evaluate residuals or deviance diagnostics, uncertainty, calibration where relevant, and held-out performance. The GLM label does not mean that all outcome types share one error distribution or one interpretation of coefficients.