## 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: $$log\left(\frac{p(X)}{1-p(X)}\right) = \beta_{0}+\beta_{1}X$$
• 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.