13.1 Logistic regression with a single predictor

Logistic function maps $(0,1)$ to $(-\infty, \infty)$:

$$\text{logit}(x) = log\left(\frac{x}{1-x}\right)$$

Also known as ‘log odds’, this can be used to map probabilities to the whole real line.

The inverse is $\text{logit}^{-1}$ or ‘sigmoid’ function:

$$\text{logit}^{-1} = \frac{e^x}{1 + e^x}$$ This maps the real line to probabilities.

In R we can use the logistic distribution:

logit <- qlogis
invlogit <- plogis

This mapping allows us to expand our linear regression into models with two outcomes $y_i \in \{0,1\}$

$$Pr(y_i = 1) = \text{logit}^{-1}(X_i\beta)$$

• Note that all the uncertainty comes for the probabilistic prediction of the binary outcome.