8.1 The Logistic Model

• Predicting recidivsm (0/1): 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: \(\frac{p(X)}{1-p(X)}=e^{\beta_{0}+\beta_{1}X}\) (range: \([0,\infty]\), the higher, the higher probability of recidivism/default)
  • log-odds/logit: \(log\left(\frac{p(X)}{1-p(X)}\right) = \beta_{0}+\beta_{1}X\) (James et al. 2013, 132)
    • Increasing X by one unit, increases the log odds by \(\beta_{1}\) (usually output in R)
• 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, chaps. 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.