11 Generalized linear models
Every model so far has had a response that is, at least approximately, normal with constant variance. Many responses are not. Whether an insect dies, whether a borrower defaults, whether a patient recovers: these are binary responses, \(0\) or \(1\). The number of faults on a machine, of accidents at a junction, of customers in an hour: these are counts. For such responses the mean is restricted (a probability lies between \(0\) and \(1\), a mean count is positive), and the variance depends on the mean. A generalized linear model (GLM) keeps the linear predictor \(\mathbf {x}^\top \boldsymbol \beta \) of regression but connects it to the mean through a link function, and replaces the normal distribution by one suited to the response. This section treats the two most used cases, logistic regression for binary data and Poisson regression for counts, and then the general theory that covers both, and the normal linear model as well.
11.2 Maximum likelihood estimation
11.3 Inference in logistic regression
11.4 The exponential family and generalized linear models
11.5 Poisson regression
11.6 Practice problems
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