Logistic Regression

• Instead of modeling $Y$ directly, we may model $\mathrm{Pr}(Y=1\mid X)$
• Logit function

$$\mathrm{logit}(p_X)=\ln \frac{p_X}{1-p_X}={\beta }_0+{\beta }_{1}X\Rightarrow p_X=\frac{1}{1+e^{-({\beta }_0+{\beta }_{1}X)}}$$

• Terms
  • Probability: $\mathrm{Pr}(Y=1\mid X)$
  • Odds: $\mathrm{Pr}(Y=1\mid X)/\mathrm{Pr}(Y=0\mid X)$
  • Log-odds: ${\beta }_0+{\beta }_{1}X$
• e