6.16 The Logistic Model

• How should we model the relationship between $p(X)=Pr(Y=1|X)$ and $X$?
  • See Figure 4.2 in James et al. (2013, 131)
  • Use either linear probability model or logistic regression
• Linear probability model: $p(X)=\beta_{0}+\beta_{1}X$
  • Linear predictions of our outcome (probabilities), can be out of [0,1] range
• Logistic regression (uses logistic function): $p(X)=\frac{e^{\beta_{0}+\beta_{1}X}}{1+e^{\beta_{0}+\beta_{1}X}}$
  • odds: $\frac{p(X)}{1-p(X)}$ (range: $[0,\infty]$, the higher, the higher probability of recidivism)
  • log-odds: $log\left(\frac{p(X)}{1-p(X)}\right)$ (James et al. 2013, 132)
• Estimation of $\beta_{0}$ and $\beta_{1}$ usually relies on maximum likelihood
• See James et al. (2013 Chap. 4.3.4) for an overview
• Source: James et al. (2013 Chap. 4.3.1, 4.3.2, 4.3.4)

References

James, Gareth, Daniela Witten, Trevor Hastie, and Robert Tibshirani. 2013. An Introduction to Statistical Learning: With Applications in R. Springer Texts in Statistics. Springer.