Generalised linear models
Generalised linear models extend linear regression to responses whose distributions are not well represented by constant-variance normal errors.
They relate the mean response \(\mu=E[Y]\) to predictors through a link function:
\[g(\mu)=\beta_0+\beta_1x_1+\cdots+\beta_px_p.\]Examples
Logistic regression uses a Bernoulli or binomial response and the logit link:
\[\log\frac{p}{1-p}=\beta_0+\beta_1x_1+\cdots.\]Poisson regression is commonly used for count data and typically uses a logarithmic link.
Key idea. GLMs preserve a regression structure while adapting the probability distribution and link function to the type of biological response.