200. Generalized Linear Models
Three components:
- Random Component
Specifies the distribution of the response variable .
can come from the exponential family of distributions (which includes Normal, Binomial, Poisson, Gamma, etc.)
- Systematic Component
Specifies the linear predictor, a linear combination of explanatory variables:
- Link Function
Connects the expected value of , denoted , to the linear predictor :
Where is called the link function
| Model | Response Type |
Distribution | Link Function |
| Linear Regression |
Continuous | Normal | (identity) |
| Logistic Regression |
Binary (0, 1) |
Binomial | (logit) |
| Poisson Regression |
Count | Poisson | |
| Gamma Regression |
Positive continuous |
Gamma | |
Example
Logistic Regression
| x | y |
| 1 | 0 |
| 2 | 0 |
| 3 | 0 |
| 4 | 1 |
| 5 | 1 |
| 6 | 1 |
Fit linear model
Logit link function:
Model
Equivalently