Generalized Linear Model (GLM)
Generalized Linear Model · Also known as: GLM, generalized regression, exponential family regression, link-function model
The Generalized Linear Model is a unified regression framework that extends ordinary linear regression to outcomes from the exponential family — including binary, count, proportion, and continuous positive outcomes. A link function connects the linear predictor to the mean of the response, enabling principled modelling beyond the Gaussian case.
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When to use it
Use GLM whenever the outcome variable is not continuous and unbounded: binary outcomes (logistic GLM), event counts (Poisson or negative-binomial GLM), proportions constrained to [0,1] (binomial with logit/probit), or strictly positive continuous data with right-skew (Gamma or inverse-Gaussian GLM). GLM is appropriate when the relationship between predictors and the mean response is best captured on a transformed scale and when the variance-mean relationship follows the chosen family. Prefer OLS only when the Gaussian assumption is defensible. Do not use a single GLM when the data have clustered or hierarchical structure (use mixed-effects extensions) or when the number of events is very small relative to the number of predictors.
Strengths & limitations
- Unifies many regression models — logistic, Poisson, Gamma, and others — under one coherent framework with shared theory and software.
- The link function and distributional assumption are chosen to match the outcome type, giving better-calibrated predictions and more valid standard errors than OLS on non-Gaussian data.
- Maximum-likelihood estimation provides well-characterised asymptotic inference (Wald, likelihood-ratio, and score tests).
- Model fit is assessed through deviance and AIC, enabling principled model comparison and selection.
- Coefficients on the link scale have straightforward interpretations (e.g., log-odds ratios for logistic, rate ratios for Poisson).
- Computationally efficient via IRLS and available in every major statistical package.
- Requires the user to correctly specify both the distribution family and the link function; misspecification can bias inference.
- Standard GLM assumes independence of observations and does not accommodate clustered, repeated-measures, or hierarchical data without extensions (GEE or mixed-effects GLM).
- Maximum-likelihood standard errors rely on asymptotic theory and may be unreliable with small samples or rare events.
- Does not automatically handle overdispersion in count data (Poisson family); a negative-binomial or quasi-Poisson model is then preferable.
- Non-linear link functions make raw coefficients less immediately intuitive than OLS slopes; marginal effects or back-transformation is often required for communication.
Frequently asked
What is the difference between GLM and OLS?
OLS is the special case of GLM with a Gaussian (normal) distribution and an identity link. GLM generalises this by allowing other exponential-family distributions and link functions, making it suitable for binary, count, and other non-Gaussian outcomes where OLS assumptions are violated.
How do I choose the right family and link function?
Match the family to the data-generating process: Binomial for binary or proportion outcomes, Poisson or negative-binomial for counts, Gamma for strictly positive continuous data with right skew, and Gaussian for symmetric continuous outcomes. The canonical link (logit, log, inverse, identity) is the default and usually best, but alternatives such as log or identity links on the Binomial are valid when estimating risk or prevalence ratios.
What is deviance in a GLM?
Deviance measures how much the fitted model's log-likelihood falls short of the saturated (perfect-fit) model's log-likelihood, scaled by two: D = 2(ℓ_sat − ℓ_fitted). Smaller deviance means better fit. The difference in deviance between nested models follows a chi-squared distribution and is used for likelihood-ratio tests.
What is overdispersion and how do I handle it?
Overdispersion occurs when the observed variance in count or proportion data exceeds what the assumed distribution predicts — common with Poisson models for heterogeneous populations. Remedies include switching to a negative-binomial family, using quasi-likelihood with a dispersion parameter, or fitting a zero-inflated model if excess zeros are present.
Can GLM handle longitudinal or clustered data?
Standard GLM assumes independent observations and is not appropriate for clustered or repeated-measures data without modification. The two main extensions are Generalized Estimating Equations (GEE), which targets population-averaged effects, and mixed-effects GLM (GLMM), which models subject-specific random effects.
Sources
- Nelder, J. A., & Wedderburn, R. W. M. (1972). Generalized linear models. Journal of the Royal Statistical Society: Series A (General), 135(3), 370–384. DOI: 10.2307/2344614 ↗
- McCullagh, P., & Nelder, J. A. (1989). Generalized Linear Models (2nd ed.). Chapman and Hall/CRC. ISBN: 978-0412317606
How to cite this page
ScholarGate. (2026, June 3). Generalized Linear Model. ScholarGate. https://scholargate.app/en/statistics/generalized-linear-model
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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