A generalization of linear regression allowing for nonlinear relationships via a "link function" and for the variance of the response to depend on the predicted value. (Not to be confused with "general linear model" which extends the ordinary linear model to general covariance structure and multivariate response.)
A generalized linear model extends regression models by allowing a more general (conditional) distribution for the observations, a variance function related to the mean, and by allowing non-linear relationship between the mean and the linear predictor, $X\beta$.
A generalized linear model consists of three components:
- Systematic part: $\eta_i = X_i'\beta$ . This is the linear predictor.
- Random part: $Y_1, Y_2, ..., Y_n$ that are independent random variables where $$ Y_i \sim D(\mu_i = EY_i)$$ where $D$ is an exponential family distribution. More generally we can have an additional parameter, the overdispersion parameter $\phi$ which controls the dispersion in $Y_i$
- Link function: an invertible function $g$, such that $\eta_i = g(\mu_i)$, or equivalently, $E(Y_i) = \mu_i = g^{-1}(\eta_i) = g^{-1}(X_i'\beta)$
The similar term "general linear model" is often confused with generalized linear models (both are typically abbreviated GLM). A general linear model is the standard multiple regression setting $Y = X\beta + \varepsilon$ (for a "design matrix" $X$, parameters $\beta$, and "error term" $\varepsilon$). Use the multiple-regression or linear-model tags for such cases (see discussion).
