GLM

General linear models extend linear regression by allowing the dependent variable to follow different distributions and linking it to predictors through various functions.

GLMs consist of the following components:

• random component: distribution of the dependent variables
• systematic component: a linear combo of the predictor variables
• link function: connect random and systematic components by linking the mean to the linear predictors

Examples:

• linear regression
  • dependent variable ~ normal
  • identity link function: $g(\mu )=\mu$
• logistic regression
  • dependent variable ~ Bernoulli
  • logit link function: $\text{logit}(\pi )=\log (\frac{\pi }{1-\pi })$
• probit regression
  • dependent variable ~ Bernoulli
  • probit link function: ${\Phi }^{-1}(\pi )$
• count data models
  • dependent variable ~ poisson, negative binomial, etc.
  • log link function