Generalised linear models
Objective. Link response distributions to linear predictors.
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Learn: the key idea
A generalised linear model links a response variable's mean to a linear predictor through a chosen link function, extending ordinary regression beyond normal errors. The response is assumed to come from an exponential-family distribution, such as Binomial or Poisson, with mean μ related to the linear predictor η = Xβ via g(μ) = η for link function g. For binary data, the logit link log(μ/(1−μ)) = Xβ gives logistic regression; for counts, the log link log(μ) = Xβ gives Poisson regression. Parameters are fitted by maximum likelihood using iteratively reweighted least squares, since no closed form exists in general. For example, modelling disease presence (0/1) against age uses logistic regression, where coefficients describe changes in log-odds per unit age increase.
Worked example
In a generalised linear model for count data with log link, log(μ) = 1.2 + 0.5x. Find μ when x = 2.
- 1Linear predictor: η = 1.2 + 0.5(2) = 2.2
- 2Log link means log(μ) = η
- 3So μ = e^2.2
- 4Evaluate: μ ≈ 9.03
Answer: μ ≈ 9.03
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Start practisingCommon mistake. Students fit a generalised linear model then interpret coefficients as if on the response scale, e.g. reading a logistic regression β as a probability change; it is a change in log-odds, so exponentiate for an odds ratio before interpreting.
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