Gamma Regression (GLM)
Gamma Regression (Generalized Linear Model) · Also known as: gamma GLM, gamma generalized linear model, Gamma Regresyonu (GLM)
Gamma regression is a generalized linear model that uses the gamma distribution to model a positive, right-skewed continuous outcome. Developed within the GLM framework of McCullagh and Nelder (1989), it is an alternative to ordinary linear regression for variables such as health-care costs, durations, and income.
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When to use it
Use gamma regression when the dependent variable is strictly positive (greater than zero) and right-skewed, and the coefficient of variation is roughly constant — that is, the variance grows in proportion to the square of the mean. It suits cross-sectional or panel data with at least about 30 observations and is well suited to skewed continuous outcomes such as costs, durations, and income distributions where normal regression is inappropriate. It is not suitable for outcomes that can be zero or negative.
Strengths & limitations
- Naturally models positive, right-skewed continuous outcomes without log-transforming the response.
- The log link gives multiplicative, interpretable effects and keeps all fitted means positive.
- Handles the constant coefficient-of-variation pattern, where variance scales with the square of the mean, better than ordinary least squares.
- Requires a strictly positive outcome; observations equal to zero cannot be modelled directly.
- Assumes the variance is proportional to the square of the mean (constant coefficient of variation); a different variance pattern misspecifies the model.
- Estimated by maximum likelihood with no closed form, so it relies on iterative fitting and needs an adequate sample (about 30 or more observations).
Frequently asked
When should I prefer gamma regression over ordinary linear regression?
When the outcome is strictly positive and right-skewed and its variability grows with its mean — typical of costs, durations, and income. Ordinary regression assumes constant variance and can predict negative values, which gamma regression avoids.
Which link function should I use?
The log link is usually preferred because it keeps fitted means positive and makes coefficients act multiplicatively on the outcome. The inverse link is the canonical link for the gamma family but is often harder to interpret.
Can gamma regression handle zeros in the outcome?
No. The gamma distribution is defined only for strictly positive values, so zeros or negative values cannot be modelled directly. A different model, such as a hurdle or two-part model, is needed when zeros occur.
How are the coefficients estimated?
By maximum likelihood. Because there is no closed-form solution, the estimates are obtained iteratively using iteratively reweighted least squares (IRLS), the standard fitting algorithm for generalized linear models.
Sources
- McCullagh, P. & Nelder, J. A. (1989). Generalized Linear Models (2nd ed.). Chapman and Hall. DOI: 10.1201/9780203753736 ↗
How to cite this page
ScholarGate. (2026, June 1). Gamma Regression (Generalized Linear Model). ScholarGate. https://scholargate.app/en/statistics/gamma-regression
Which method?
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